Abstract

The phase space representation of imaging with optical fields in any state of spatial coherence is developed by using spatial coherence wavelets. It leads to new functions for describing the optical transfer and response of imaging systems when the field is represented by Wigner distribution functions. Specific imaging cases are analyzed in this context, and special attention is devoted to the imaging of two point sources.

© 2008 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  2. J. W. Goodman, Statistical Optics (Wiley, 1985).
  3. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  4. J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).
  5. K. Izuka, Engineering Optics (Springer-Verlag, 1985).
  6. L.Mandel and E.Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (1850-1966), SPIE Milestone Series, Vol. MS 19 (SPIE, 1990).
  7. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  8. 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]
  9. J. García and R. Castaneda, “Spatial partially coherent imaging,” J. Mod. Opt. 49, 2093-2104 (2002).
    [CrossRef]
  10. R. Castaneda, “Partially coherent imaging and spatial coherence wavelets,” Opt. Commun. 230, 7-18 (2004).
    [CrossRef]
  11. R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and phase space representation of diffraction,” Appl. Opt. 41, xxxx-xxxx (2008).
  12. R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
  13. R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
    [CrossRef]
  14. R. Simon and N. Mukunda, “Optical phase space, Wigner representation and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440-2463 (2000).
    [CrossRef]
  15. E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica A (Amsterdam) 96, 315-320 (1979).
    [CrossRef]
  16. 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]
  17. J. P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136-138 (1978).
    [CrossRef]
  18. R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
    [CrossRef]
  19. R. Castaneda, J. Carrasquilla, and J. Garcia-Sucerquia, “Young's experiment with electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 2519-2529 (2006).
    [CrossRef]
  20. R. Castaneda, “Electromagnetic spatial coherence wavelets and the classical laws on polarization,” Opt. Commun. 267, 4-13 (2006).
    [CrossRef]

2008 (1)

R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and phase space representation of diffraction,” Appl. Opt. 41, xxxx-xxxx (2008).

2007 (1)

2006 (2)

R. Castaneda, J. Carrasquilla, and J. Garcia-Sucerquia, “Young's experiment with electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 2519-2529 (2006).
[CrossRef]

R. Castaneda, “Electromagnetic spatial coherence wavelets and the classical laws on polarization,” Opt. Commun. 267, 4-13 (2006).
[CrossRef]

2004 (1)

R. Castaneda, “Partially coherent imaging and spatial coherence wavelets,” Opt. Commun. 230, 7-18 (2004).
[CrossRef]

2003 (3)

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).

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

2002 (1)

J. García and R. Castaneda, “Spatial partially coherent imaging,” J. Mod. Opt. 49, 2093-2104 (2002).
[CrossRef]

2000 (1)

1995 (1)

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

1990 (1)

L.Mandel and E.Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (1850-1966), SPIE Milestone Series, Vol. MS 19 (SPIE, 1990).

1985 (2)

K. Izuka, Engineering Optics (Springer-Verlag, 1985).

J. W. Goodman, Statistical Optics (Wiley, 1985).

1979 (1)

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

1978 (2)

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

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

1975 (1)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

1968 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Carrasquilla, J.

R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and phase space representation of diffraction,” Appl. Opt. 41, xxxx-xxxx (2008).

R. Castaneda, J. Carrasquilla, and J. Garcia-Sucerquia, “Young's experiment with electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 2519-2529 (2006).
[CrossRef]

Castaneda, R.

R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and phase space representation of diffraction,” Appl. Opt. 41, xxxx-xxxx (2008).

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, “Electromagnetic spatial coherence wavelets and the classical laws on polarization,” Opt. Commun. 267, 4-13 (2006).
[CrossRef]

R. Castaneda, J. Carrasquilla, and J. Garcia-Sucerquia, “Young's experiment with electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 2519-2529 (2006).
[CrossRef]

R. Castaneda, “Partially coherent imaging and spatial coherence wavelets,” Opt. Commun. 230, 7-18 (2004).
[CrossRef]

R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).

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

J. García and R. Castaneda, “Spatial partially coherent imaging,” J. Mod. Opt. 49, 2093-2104 (2002).
[CrossRef]

Castañeda, R.

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
[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]

García, J.

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
[CrossRef]

J. García and R. Castaneda, “Spatial partially coherent imaging,” J. Mod. Opt. 49, 2093-2104 (2002).
[CrossRef]

Garcia-Sucerquia, J.

García-Sucerquia, J.

R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).

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

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Guigay, J. P.

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

Herrera-Ramirez, J.

Izuka, K.

K. Izuka, Engineering Optics (Springer-Verlag, 1985).

Mandel, L.

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

Mukunda, N.

Simon, R.

Sudarshan, E. C. G.

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

Usuga-Castaneda, M.

Wolf, E.

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

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Appl. Opt. (2)

R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and phase space representation of diffraction,” Appl. Opt. 41, xxxx-xxxx (2008).

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]

J. Mod. Opt. (3)

J. García and R. Castaneda, “Spatial partially coherent imaging,” J. Mod. Opt. 49, 2093-2104 (2002).
[CrossRef]

R. Castaneda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).

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

J. Opt. Soc. Am. A (2)

Opt. Commun. (4)

R. Castaneda, “Electromagnetic spatial coherence wavelets and the classical laws on polarization,” Opt. Commun. 267, 4-13 (2006).
[CrossRef]

R. Castaneda, “Partially coherent imaging and spatial coherence wavelets,” Opt. Commun. 230, 7-18 (2004).
[CrossRef]

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

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
[CrossRef]

Physica A (Amsterdam) (1)

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

Other (8)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. W. Goodman, Statistical Optics (Wiley, 1985).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

K. Izuka, Engineering Optics (Springer-Verlag, 1985).

L.Mandel and E.Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (1850-1966), SPIE Milestone Series, Vol. MS 19 (SPIE, 1990).

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

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]

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Figures (7)

Fig. 1
Fig. 1

Center and difference coordinates on the A P and O P planes.

Fig. 2
Fig. 2

Degrees of freedom of the separation vector ξ D for a given radiator pair within the support of the complex degree of spatial coherence centered at ξ A .

Fig. 3
Fig. 3

(a)–(c) Phase-space diagrams for a Young mask placed at A P (pinhole distance b and interference in Fraunhofer domain) under uniform illumination with (a)  | μ ( + , ) | = 1 and α ( + , ) = 0 , (b)  | μ ( + , ) | = 0.2 and α ( + , ) = π , (c)  | μ ( + , ) | = 0 . The white vertical bars on the left and the right correspond to carrier rays, and the vertical bar in the middle corresponds to modulating rays (white and black fringes represent the “positive and negative energies” provided by these rays). (d)–(f) Phase-space diagrams for a slit of width b placed at A P under uniform illumination; the fringe patterns correspond to modulating rays, which are superimposed on the carrier rays with (d) | μ ( + , ) | = 1 , α ( + , ) = 0 and Fraunhofer diffraction, (b) | μ ( + , ) | = 0.2 , α ( + , ) = 0 and Fresnel diffraction (the quadratic Fresnel phase causes the shear of the pattern), (c) | μ ( + , ) | = 0 and Fraunhofer diffraction.

Fig. 4
Fig. 4

Conceptual scheme for the phase space representation of imaging. The (defocused) elementary transfer function P ( ξ A , ξ D ; 1 F ) transfers each surrounding of the spatial coherence moiré from e p to E P , and the (defocused) elementary response function p ( r A , ξ A ; 1 / F ) weights the object marginal power spectrum along each ray r A ξ A for producing the image marginal power spectrum along the ray ξ A r A .

Fig. 5
Fig. 5

(a) Support spread function h ( x A , x D ; 0 ) for the best focus plane of a one-dimensional aberration-free imaging system with uniform pupil. (b) Profile of h ( x A 0 ; 0 ) in arbitrary units (a.u), corresponding to the PSF of the system.

Fig. 6
Fig. 6

Phase-space diagrams of the image marginal power spectra provided by a one-dimensional aberration-free imaging system with a uniform pupil function, under (a) fully spatially coherent and (b) incoherent illuminations, for a given pinhole distance and for the best focus plane. The horizontal fringe pattern in the middle of Fig. 6a denotes the cosinelike modulation due to the spatial coherence of the illumination. It does not appear in Fig. 6b.

Fig. 7
Fig. 7

Image power spectra of a double pinhole mask under (a)–(c) spatially incoherent illumination and (d)–(f) spatially coherent illumination. The pinhole separation is smaller than the Rayleigh resolution limit in (a) and (d), equal to this limit in (b) and (e), and greater than it in (c) and (f).

Equations (37)

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W ( r A + r D / 2 , r A r D / 2 , ξ A ) = S ( r A , ξ A ) exp ( i k z r D · ξ A ) ,
S ( r A , ξ A ) = A P μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) S ( ξ A + ξ D / 2 ) t ( ξ A + ξ D / 2 ) × S ( ξ A ξ D / 2 ) t * ( ξ A ξ D / 2 ) exp [ i k z ( ξ A r A ) · ξ D ] d 2 ξ D
S ( r A , ξ A ) = A P μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) Ψ ( ξ A + ξ D / 2 ) Ψ * ( ξ A ξ D / 2 ) exp [ i k z r A · ξ D ] d 2 ξ D
W ( r A + r D / 2 , r A r D / 2 ) = ( 1 λ z ) 2 exp ( i k z r A · r D ) A P W ( r A + r D / 2 , r A r D / 2 , ξ A ) d 2 ξ A = ( 1 λ z ) 2 exp ( i k z r A · r D ) A P S ( r A , ξ A ) exp ( i k z r D · ξ A ) d 2 ξ A
S ( r A ) = ( 1 λ z ) 2 A P S ( r A , ξ A ) d 2 ξ A .
S ( r A , ξ A ) = C S ( ξ A ) | t ( ξ A ) | 2 + 2 A P ξ D 0 | μ ( + , ) | S ( + ) | t ( + ) | S ( ) | t ( ) | cos [ k z ( ξ A r A ) · ξ D + α ( + , - ) + Δ ϕ ] d 2 ξ D .
O P S ( r A ) d 2 r A = C ( 1 λ z ) 2 A P S ( ξ A ) | t ( ξ A ) | 2 d 2 ξ A O P d 2 r A + 2 ( 1 λ z ) 2 A P A P ξ D 0 | μ ( + , ) | S ( + ) | t ( + ) | S ( ) | t ( ) | × O P cos [ i k z ( ξ A r A ) · ξ D + α ( + , ) + Δ ϕ ] d 2 r A d 2 ξ A d 2 ξ D .
O P cos [ k z ( ξ A r A ) · ξ D + α ( + , ) + Δ ϕ ] d 2 r A = 0.
W e p e w ( ξ A + ξ D / 2 , ξ A ξ D / 2 , r A ) = exp ( i k z ξ D · r A ) e w μ e w ( r A + r D / 2 , r A r D / 2 ) × S e w ( r A + r D / 2 ) t ( r A + r D / 2 ) S e w ( r A r D / 2 ) t * ( r A r D / 2 ) exp [ i k z ( r A ξ A ) · r D ] d 2 r D
W E W E P ( r A + r D / 2 , r A r D / 2 , ξ A ) = exp ( i k z r D · ξ A ) E P μ E P ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) S E P ( ξ A + ξ D / 2 ) × P ( ξ A + ξ D / 2 ) S E P ( ξ A ξ D / 2 ) P * ( ξ A ξ D / 2 ) exp [ i k ( 1 z 1 f ) ξ A · ξ D ] exp ( i k z r A · ξ D ) d 2 ξ D .
W e p ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) = S E P ( ξ A + ξ D / 2 ) S E P ( ξ A ξ D / 2 ) μ E P ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) = ( 1 λ z ) 2 exp ( i k z ξ A · ξ D ) e w W e p e w ( ξ A + ξ D / 2 , ξ A ξ D / 2 , r A ) d 2 r A .
W E W E P ( r A + r D / 2 , r A r D / 2 , ξ A ) = ( 1 λ z ) 2 exp ( i k z r D · ξ A ) × E P e w W e p e w ( ξ A + ξ D / 2 , ξ A ξ D / 2 , r A ) d 2 r A P ( ξ A , ξ D ; 1 / F ) exp ( i k z r A · ξ D ) d 2 ξ D ,
P ( ξ A , ξ D ; 1 / F ) = P ( ξ A + ξ D / 2 ) P * ( ξ A ξ D / 2 ) exp ( i k F ξ A · ξ D )
S E W E P ( r A , ξ A ) = 1 λ z e w S e p e w ( ξ A , r A ) p ( z z r A + r A , ξ A ; 1 / F ) d 2 r A .
p ( r A , ξ A ; 1 / F ) = 1 λ z E P P ( ξ A , ξ D ; 1 / F ) exp ( i k z r A ξ D ) d 2 ξ D ,
p ( r A , ξ A ; 1 / F ) = 1 λ z E P Q ( ξ A + ξ D / 2 ) Q * ( ξ A ξ D / 2 ) exp ( i k z r A ξ D ) d 2 ξ D ,
Q ( ξ A ± ξ D / 2 ) = P ( ξ A ± ξ D / 2 ) exp ( i k 2 F | ξ A ± ξ D / 2 | 2 ) ,
W E W ( r A + r D 2 , r r D 2 ) = ( z z ) 2 W e w [ z z ( r A + r D 2 ) , z z ( r A r D 2 ) ] ,
S E W ( r A ) = ( z z ) 2 S e w ( z z r A )
μ E W ( r A + r D 2 , r A r D 2 ) = μ e w [ z z ( r A + r D 2 ) z z ( r A r D 2 ) ]
W e p e w ( ξ A + ξ D / 2 , ξ A ξ D / 2 , r A ) = λ z S e w ( r A ) | t ( r A ) | 2 exp ( i k z ξ D r A ) ,
S e p e w ( ξ A , r A ) = λ z S e w ( r A ) | t ( r A ) | 2 ,
W E W E P ( r A + r D / 2 , r A r D / 2 , ξ A ) = S E W E P ( r A , ξ A ) exp ( i k z r D · ξ A ) ,
S E W E P ( r A , ξ A ) = e w S e w ( r A ) | t ( r A ) | 2 p ( z z r A + r A , ξ A ; 1 / F ) d 2 r A
W E W ( r A + r D / 2 , r A r D ) = 1 λ z exp ( i k z r A r D ) e w S e w ( r A ) | t ( r A ) | 2 h ( z z r A + r A , r D ; 1 / F ) d 2 r A ,
S E W ( r A ) = 1 λ z e w S e w ( r A ) | t ( r A ) | 2 h ( z z r A + r A , 0 ; 1 / F ) d 2 r A ,
h ( r A , r D ; 1 / F ) = 1 λ z E P p ( r A , ξ A ; 1 / F ) exp ( i k z r D ξ A ) d 2 ξ A .
μ E W ( r A + r D 2 , r A r D 2 ) = exp ( i k z r A r D ) e w S e w ( r A ) | t ( r A ) | 2 h ( z z r A + r A , r D ; 1 / F ) d 2 r A e w S e w ( r A ) | t ( r A ) | 2 h ( z z r A + r A , 0 ; 1 / F ) d 2 r A .
H ( r D , ξ D ; 1 / F ) = 1 λ z E P P ( ξ A + ξ D / 2 ) P * ( ξ A ξ D / 2 ) exp ( i k F ξ A · ξ D ) exp ( i k z r D · ξ A ) d 2 ξ A ,
h ( r A , r D ; 1 / F ) = 1 λ z E P H ( r D , ξ D ; 1 / F ) exp ( i k z r A · ξ D ) d 2 ξ D .
h ( r A , 0 ; 1 / F ) = 1 λ z E P H ( 0 , ξ D ; 1 / F ) exp ( i k z r A · ξ D ) d 2 ξ D
H ( 0 , ξ D ; 1 / F ) = 1 λ z E P P ( ξ A + ξ D / 2 ) P * ( ξ A ξ D / 2 ) exp ( i k F ξ A · ξ D ) d 2 ξ A .
h ( r A , 0 ; 1 / F ) = 1 λ z E P p ( r A , ξ A ; 1 / F ) d 2 ξ A ,
S e p e w ( ξ A , r A ) = ( λ z ) 2 S 0 { δ ( r A + b / 2 ) + δ ( r A b / 2 ) + 2 δ ( r A ) | μ e w ( b ) | cos [ k z ( r A ξ A ) · b + α e w ( b ) ] } .
S e w E P ( r A , ξ A ) = λ z S 0 { p ( z z r A b / 2 , ξ A ; 1 / F ) + p ( z z r A + b / 2 , ξ A ; 1 / F ) + 2 | μ e w ( b ) | p ( z z r A , ξ A ; 1 / F ) cos [ k z b ξ A α e w ( b ) ] }
W E W ( r A , + r D / 2 , r A , r D / 2 ) = z z S 0 exp ( i k z r A · r D ) { h ( z z r A b / 2 , r D ; 1 / F ) + h ( z z r A + b / 2 , r D ; 1 / F ) + 2 | μ e w ( b ) | h ( z z r A , z z r D + b ; 1 / F ) cos [ υ ( z z r A , z z r D + b ; 1 / F ) + α e w ( b ) ] } ,
S E W ( r A ) | = z z S 0 { h ( z z r A b / 2 , 0 ; 1 / F ) + h ( z z r A + b / 2 , 0 ; 1 / F ) + 2 | μ e w ( b ) | h ( z z r A , b ; 1 / F ) cos [ υ ( k z r A , b ; 1 / F ) + α e w ( b ) ] }

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