Abstract

The behavior of the marginal power spectrum as a two-channel-multiplexed hologram is analyzed. Its “negative energies” make it quite different from the conventional holograms, i.e., it is not recordable in general and the objects to be reconstructed (the cross-spectral densities at both the aperture and the observation planes) are virtual. The holographic reconstruction results from the superposition of the spatial coherence wavelets that carry the marginal power spectrum. These features make the marginal power spectrum a powerful tool for analysis and synthesis of optical fields, for instance, in optical information processing (signal encryption) and beam shaping for microlithography.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
    [CrossRef]
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2008 (1)

2007 (4)

J. Herrera and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259-263 (2007).
[CrossRef]

R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[CrossRef]

R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castaneda, M. Usuga-Castaneda, and J. Herrera-Ramirez, “Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs,” Appl. Opt. 46, 5321-5328 (2007).
[CrossRef] [PubMed]

2006 (3)

R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 81-90 (2006).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62-67 (2006).
[CrossRef]

B. H. Liu, L. Y. Wu, and J. Zhang, “Phase-only SLM generating variable patterns applied in optical connections,” J. Phys.: Conf. Ser. 48, 902-906 (2006).
[CrossRef]

2005 (6)

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

P. E. Mix, Introduction to Nondestructive Testing (Wiley, 2005).

R. Castaneda and J. Garcia-Sucerquia, “Radiometry and spatial coherence wavelets,” Opt. Commun. 248, 147-165 (2005).
[CrossRef]

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Jena) 116, 44-48 (2005).
[CrossRef]

R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

2004 (2)

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353-1358 (2004).
[CrossRef]

L. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms (Kluwer Academic, 2004).

2003 (2)

R. Castaneda and J. Garcia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
[CrossRef]

R. Castañeda and J. Garcia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

2000 (2)

1997 (2)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E.Wolf, ed., (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

K. B. Wolf and A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36-42 (1997).
[CrossRef]

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

1993 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

1987 (1)

1986 (1)

1979 (1)

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica A 96, 315-320 (1979).
[CrossRef]

1974 (1)

1962 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Bastiaans, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

Carrasquilla, J.

R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

Castaneda, R.

R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[CrossRef]

R. Castaneda, M. Usuga-Castaneda, and J. Herrera-Ramirez, “Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs,” Appl. Opt. 46, 5321-5328 (2007).
[CrossRef] [PubMed]

R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 81-90 (2006).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62-67 (2006).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, “Radiometry and spatial coherence wavelets,” Opt. Commun. 248, 147-165 (2005).
[CrossRef]

R. Castaneda and J. Garcia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
[CrossRef]

Castañeda, R.

R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

R. Castañeda and J. Garcia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

Coppola, G.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

De Nicola, S.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Dickey, F. M.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (CRC, 2000).
[CrossRef]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E.Wolf, ed., (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

Ferraro, P.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Finizio, A.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Garcia, J.

R. Castañeda and J. Garcia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

R. Castaneda and J. Garcia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
[CrossRef]

García, J.

R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

Garcia-Sucerquia, J.

J. Herrera and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259-263 (2007).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62-67 (2006).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 81-90 (2006).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Jena) 116, 44-48 (2005).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, “Radiometry and spatial coherence wavelets,” Opt. Commun. 248, 147-165 (2005).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353-1358 (2004).
[CrossRef]

Herrera, J.

J. Herrera and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259-263 (2007).
[CrossRef]

R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62-67 (2006).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Jena) 116, 44-48 (2005).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353-1358 (2004).
[CrossRef]

Herrera-Ramirez, J.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (CRC, 2000).
[CrossRef]

Iodice, M.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Jüptner, W.

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

Leith, E.

Liu, B. H.

B. H. Liu, L. Y. Wu, and J. Zhang, “Phase-only SLM generating variable patterns applied in optical connections,” J. Phys.: Conf. Ser. 48, 902-906 (2006).
[CrossRef]

Lohmann, A. W.

Magro, C.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marchand, E. W.

Mix, P. E.

P. E. Mix, Introduction to Nondestructive Testing (Wiley, 2005).

Mukunda, N.

Pierattini, G.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Rivera, A. L.

K. B. Wolf and A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36-42 (1997).
[CrossRef]

Schnars, U.

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

Simon, R.

Spoto, G. E.

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica A 96, 315-320 (1979).
[CrossRef]

Testorf, M.

Tricoles, G.

Upatnieks, J.

Usuga-Castaneda, M.

Velasquez, D.

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Jena) 116, 44-48 (2005).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353-1358 (2004).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

E. W. Marchand and E. Wolf, “Walthers definition of generalized radiance,” J. Opt. Soc. Am. 64, 1219-1226 (1974).
[CrossRef]

Wolf, K. B.

K. B. Wolf and A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36-42 (1997).
[CrossRef]

Wu, L. Y.

B. H. Liu, L. Y. Wu, and J. Zhang, “Phase-only SLM generating variable patterns applied in optical connections,” J. Phys.: Conf. Ser. 48, 902-906 (2006).
[CrossRef]

Yaroslavsky, L.

L. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms (Kluwer Academic, 2004).

Zhang, J.

B. H. Liu, L. Y. Wu, and J. Zhang, “Phase-only SLM generating variable patterns applied in optical connections,” J. Phys.: Conf. Ser. 48, 902-906 (2006).
[CrossRef]

Appl. Opt. (3)

J. Microlithogr., Microfabr., Microsyst. (1)

G. Coppola, M. Iodice, A. Finizio, S. De Nicola, G. Pierattini, P. Ferraro, C. Magro, and G. E. Spoto, “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Microlithogr., Microfabr., Microsyst. 4, 013012 (2005).
[CrossRef]

J. Mod. Opt. (2)

R. Castaneda and J. Garcia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
[CrossRef]

R. Castañeda and J. Garcia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys.: Conf. Ser. (1)

B. H. Liu, L. Y. Wu, and J. Zhang, “Phase-only SLM generating variable patterns applied in optical connections,” J. Phys.: Conf. Ser. 48, 902-906 (2006).
[CrossRef]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Commun. (7)

J. Herrera and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259-263 (2007).
[CrossRef]

J. Garcia-Sucerquia, J. Herrera, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62-67 (2006).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, “Radiometry and spatial coherence wavelets,” Opt. Commun. 248, 147-165 (2005).
[CrossRef]

R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[CrossRef]

R. Castañeda, J. García and J. Carrasquilla, “Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

K. B. Wolf and A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36-42 (1997).
[CrossRef]

Optik (Jena) (1)

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Jena) 116, 44-48 (2005).
[CrossRef]

Physica A (1)

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica A 96, 315-320 (1979).
[CrossRef]

Proc. SPIE (1)

J. Garcia-Sucerquia, J. Herrera, and D. Velasquez, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353-1358 (2004).
[CrossRef]

Other (7)

P. E. Mix, Introduction to Nondestructive Testing (Wiley, 2005).

L. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms (Kluwer Academic, 2004).

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, 2005).

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (CRC, 2000).
[CrossRef]

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E.Wolf, ed., (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (4)

Fig. 1
Fig. 1

Illustrating the center and difference coordinates for denoting pairs of points at both the AP and the OP planes.

Fig. 2
Fig. 2

Phase-space diagrams (graphs of the marginal power spectra) for diffraction through a slit of width 2 a . The slit is uniformly illuminated by a plane wave with a Gaussian degree of spatial coherence. Upper row: diagrams for Fraunhofer diffraction. a, the standard deviation of the degree of coherence is significantly greater than the aperture width (coherent illumination); b, the standard deviation is 0.5 a (partially coherent illumination); and c, the standard deviation is about 0.2 a (nearly incoherent illumination). Lower row: diagrams for Fresnel diffraction (the slit subtends two Fresnel zones) under the same coherence conditions as in upper row. d, coherent illumination; e, partially coherent illumination; and f, nearly incoherent illumination.

Fig. 3
Fig. 3

Morphological features of the phase-space diagrams for a spatially partially coherent optical field by a, the Fraunhofer diffraction and b, the Fresnel diffraction.

Fig. 4
Fig. 4

Power spectrum at OP for the phase-space diagrams: a, in the upper row and b, in the lower row of Fig. 2. The vertical axis is in arbitrary units. The energy redistribution is apparent, taking into account that the diffracted wave is uniform. c, Comparison between the shapes of the power spectra by both Fraunhofer and Fresnel diffraction for great enough slit width in comparison to the size of the support of the complex degree of spatial coherence.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

W ( r A + r D 2 , r A r D 2 , ξ A ; ω ) = S ( r A , ξ A ; ω ) exp ( i k z r D ξ A ) ,
S ( r A , ξ A ; ω ) = W ( r A , r A , ξ A ; ω ) = AP μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 , ω ) t * ( ξ A ξ D 2 ) exp ( i k z ξ A ξ D ) exp ( i k z ξ D r A ) d 2 ξ D ,
W ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) = μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 ; ω ) t * ( ξ A ξ D 2 ) ,
W ( r A + r D 2 , r A r D 2 ; ω ) = ( 1 λ z ) 2 exp ( i k z r A r D ) AP AP W ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) exp ( i k z ξ A ξ D ) exp [ i k z ( r A ξ D + r D ξ A ) ] d 2 ξ A d 2 ξ D
W ( r A + r D 2 , r A r D 2 ; ω ) = ( 1 λ z ) 2 exp ( i k z r A r D ) AP W ( r A + r D 2 , r A r D 2 , ξ A ; ω ) d 2 ξ A = ( 1 λ z ) 2 exp ( i k z r A r D ) AP S ( r A , ξ A ; ω ) exp ( i k z r D ξ A ) d 2 ξ A .
S ( r A ; ω ) = ( 1 λ z ) 2 AP S ( r A , ξ A ; ω ) d 2 ξ A .
OP S ( r A ; ω ) d 2 r A = AP S ( ξ A ; ω ) t ( ξ A ) 2 d 2 ξ A = ( 1 λ z ) 2 OP AP S ( r A , ξ A ; ω ) d 2 ξ A d 2 r A .
Ψ ( ξ A ± ξ D 2 ; ω ) = S ( ξ A ± ξ D 2 ; ω ) t ( ξ A ± ξ D 2 ) exp ( i k 2 z ξ A ± ξ D 2 2 )
S ( r A , ξ A ; ω ) = AP μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) Ψ ( ξ A + ξ D 2 ; ω ) Ψ * ( ξ A ξ D 2 ; ω ) exp ( i k z ξ D r A ) d 2 ξ D ,
S ( r A , ξ A ; ω ) = C S ( ξ A ; ω ) t ( ξ A ) 2 + 2 AP ξ D 0 μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 , ω ) t ( ξ A ξ D 2 ) cos [ k z ( r A ξ A ) ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) Δ ϕ ] d 2 ξ D ,
OP cos [ k z ( r A ξ A ) ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) Δ ϕ ] d 2 r A = 0 ,
S ( r A ; ω ) = C ( 1 λ z ) 2 AP S ( ξ A ; ω ) t ( ξ A ) 2 d 2 ξ A + 2 ( 1 λ z ) 2 AP AP ξ D 0 μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 , ω ) t ( ξ A ξ D 2 ) cos [ k z ( r A ξ A ) ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) Δ ϕ ] d 2 ξ D d 2 ξ A .
S ( ξ A ; ω ) t ( ξ A ) 2 = ( 1 λ z ) 2 OP S ( r A , ξ A ; ω ) d 2 r A .
S ( ξ A ; ω ) = AP μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 , ω ) t * ( ξ A ξ D 2 ) d 2 ξ D ,
S ( ξ A ; ω ) = AP S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 , ω ) t * ( ξ A ξ D 2 ) d 2 ξ D ,
S ( ξ A ; ω ) = C S ( ξ A ; ω ) t ( ξ A ) 2 ,
W ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) = ( 1 λ z ) 2 exp ( i k z ξ A ξ D ) OP S ( r A , ξ A ; ω ) exp ( i k z ξ D r A ) d 2 r A ,
W ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) = μ ( ξ A + ξ D 2 , ξ A ξ D 2 ; ω ) S ( ξ A + ξ D 2 ; ω ) t ( ξ A + ξ D 2 ) S ( ξ A ξ D 2 ; ω ) t * ( ξ A ξ D 2 ) ,
W ( ξ A , ξ A ; ω ) = S ( ξ A ; ω ) t ( ξ A ) 2 = ( 1 λ z ) 2 OP S ( r A , ξ A ; ω ) d 2 r A ,

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