Abstract

The diffraction theory of Kirchhoff is applied to the semi-infinite aperture of a black half-screen. The derivative of the spherical Green’s function is taken into account without any approximation. The uniformly evaluated scattering integral is compared with the physical optics solution. It is shown that the non-omitted term causes the existence of fictitious diffracted waves.

© 2009 Optical Society of America

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  1. T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
    [CrossRef]
  2. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  3. A. J. Fresnel, “Mémoire sur la diffraction de la lumière,” Ann. Chim. Phys. 1, 239-281 (1816).
  4. G. A. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
    [CrossRef]
  5. S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
    [CrossRef]
  6. W. C. Elmore and M. A. Heald, Physics of Waves (McGraw-Hill, 1969).
  7. C. Tai, “Kirchhoff theory: Scalar, vector or dyadic?” IEEE Trans. Antennas Propag. 20, 114-115 (1972).
    [CrossRef]
  8. E. Hecht, Optics (Addison-Wesley, 2002).
  9. R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B: Photophys. Laser Chem. 90, 379-382 (2008).
    [CrossRef]
  10. H. M. MacDonald, “The effect produced by an obstacle on a train of electric waves,” Philos. Trans. R. Soc. London, Ser. A 212, 299-337 (1913).
    [CrossRef]
  11. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. Pura Appl. 16, 21-48 (1888).
  12. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
    [CrossRef]
  13. H. Poincaré, Théorie Mathématique de la Lumière (De la Faculté des Sciences de Paris, 1892).
  14. N. Mukunda, “Consistency of Rayleigh's diffraction formulas with Kirchhoff's boundary conditions,” J. Opt. Soc. Am. 52, 336-337 (1962).
    [CrossRef]
  15. F. Kottler, “Zur Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 375, 405-456 (1923).
    [CrossRef]
  16. F. Kottler, “Elektromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 376, 457-508 (1923).
    [CrossRef]
  17. J. S. Asvestas, “Diffraction by a black screen,” J. Opt. Soc. Am. 65, 155-158 (1975).
    [CrossRef]
  18. E. W. Marchand and E. Wolf, “Consistent formulation of Kirchhoff's diffraction theory,” J. Opt. Soc. Am. 56, 1712-1721 (1966).
    [CrossRef]
  19. S. Ganci, “Equivalence between two consistent formulations of Kirchhoff's diffraction theory,” J. Opt. Soc. Am. A 5, 1626-1628 (1988).
    [CrossRef]
  20. S. Ganci, “A note on the Kirchhoff formulation of diffraction by a plane screen,” J. Mod. Opt. 45, 873-876 (1998).
    [CrossRef]
  21. S. Ganci, “A general scalar solution for the half-plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
    [CrossRef]
  22. S. Ganci, “Half-plane diffraction in a case of oblique incidence,” J. Mod. Opt. 43, 2543-2551 (1996).
    [CrossRef]
  23. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  24. Y. Z. Umul, “Modified diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 25, 1850-1860 (2008).
    [CrossRef]
  25. A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  26. T. Gravelsaeter and J. J. Stamnes, “Diffraction by circular apertures. 1. Method of linear phase and amplitude approximation,” Appl. Opt. 21, 3644-3651 (1982).
    [CrossRef] [PubMed]
  27. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. 378, 339-364 (1924).
    [CrossRef]
  28. A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931-936 (1938).
    [CrossRef]
  29. W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448-454 (1975).
    [CrossRef]
  30. G. A. Forbes and A. A. Asatryan, “Reducing canonical diffraction problems into singularity-free one-dimensional integrals,” J. Opt. Soc. Am. A 15, 1320-1328 (1998).
    [CrossRef]
  31. A. S. Marathay and J. F. McCalmont, “On the usual approximation used in the Rayleigh-Sommerfeld diffraction theory,” J. Opt. Soc. Am. A 21, 510-516 (2004).
    [CrossRef]
  32. Y. Z. Umul, “Rubinowicz transform of the MTPO surface integrals,” Opt. Commun. 281, 5641-5646 (2008).
    [CrossRef]
  33. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2003).
  34. Y. Z. Umul, “The relation between the boundary diffraction wave theory and physical optics,” Opt. Commun. 281, 4844-4848 (2008).
    [CrossRef]
  35. Y. Z. Umul, “Simplified uniform theory of diffraction,” Opt. Lett. 30, 1614-1616 (2005).
    [CrossRef] [PubMed]
  36. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, 1983).
  37. S. W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

2008 (4)

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B: Photophys. Laser Chem. 90, 379-382 (2008).
[CrossRef]

Y. Z. Umul, “Modified diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 25, 1850-1860 (2008).
[CrossRef]

Y. Z. Umul, “Rubinowicz transform of the MTPO surface integrals,” Opt. Commun. 281, 5641-5646 (2008).
[CrossRef]

Y. Z. Umul, “The relation between the boundary diffraction wave theory and physical optics,” Opt. Commun. 281, 4844-4848 (2008).
[CrossRef]

2006 (1)

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

2005 (1)

2004 (2)

1998 (2)

1996 (1)

S. Ganci, “Half-plane diffraction in a case of oblique incidence,” J. Mod. Opt. 43, 2543-2551 (1996).
[CrossRef]

1995 (1)

S. Ganci, “A general scalar solution for the half-plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
[CrossRef]

1988 (1)

1982 (1)

1976 (1)

S. W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

1975 (2)

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448-454 (1975).
[CrossRef]

J. S. Asvestas, “Diffraction by a black screen,” J. Opt. Soc. Am. 65, 155-158 (1975).
[CrossRef]

1972 (1)

C. Tai, “Kirchhoff theory: Scalar, vector or dyadic?” IEEE Trans. Antennas Propag. 20, 114-115 (1972).
[CrossRef]

1966 (1)

1962 (2)

1938 (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931-936 (1938).
[CrossRef]

1924 (1)

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. 378, 339-364 (1924).
[CrossRef]

1923 (2)

F. Kottler, “Zur Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 375, 405-456 (1923).
[CrossRef]

F. Kottler, “Elektromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 376, 457-508 (1923).
[CrossRef]

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

1913 (1)

H. M. MacDonald, “The effect produced by an obstacle on a train of electric waves,” Philos. Trans. R. Soc. London, Ser. A 212, 299-337 (1913).
[CrossRef]

1896 (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

1888 (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. Pura Appl. 16, 21-48 (1888).

1883 (1)

G. A. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

1816 (1)

A. J. Fresnel, “Mémoire sur la diffraction de la lumière,” Ann. Chim. Phys. 1, 239-281 (1816).

1802 (1)

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

Asatryan, A. A.

Asvestas, J. S.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2003).

Deschamps, G.

S. W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

Elmore, W. C.

W. C. Elmore and M. A. Heald, Physics of Waves (McGraw-Hill, 1969).

Forbes, G. A.

Fresnel, A. J.

A. J. Fresnel, “Mémoire sur la diffraction de la lumière,” Ann. Chim. Phys. 1, 239-281 (1816).

Ganci, S.

S. Ganci, “A note on the Kirchhoff formulation of diffraction by a plane screen,” J. Mod. Opt. 45, 873-876 (1998).
[CrossRef]

S. Ganci, “Half-plane diffraction in a case of oblique incidence,” J. Mod. Opt. 43, 2543-2551 (1996).
[CrossRef]

S. Ganci, “A general scalar solution for the half-plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
[CrossRef]

S. Ganci, “Equivalence between two consistent formulations of Kirchhoff's diffraction theory,” J. Opt. Soc. Am. A 5, 1626-1628 (1988).
[CrossRef]

Gordon, W. B.

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448-454 (1975).
[CrossRef]

Gravelsaeter, T.

Heald, M. A.

W. C. Elmore and M. A. Heald, Physics of Waves (McGraw-Hill, 1969).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, 1983).

Janssens, K.

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

Keller, J. B.

Kirchhoff, G. A.

G. A. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Kottler, F.

F. Kottler, “Zur Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 375, 405-456 (1923).
[CrossRef]

F. Kottler, “Elektromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 376, 457-508 (1923).
[CrossRef]

Kukhlevsky, S. V.

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

Kumar, R.

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B: Photophys. Laser Chem. 90, 379-382 (2008).
[CrossRef]

Lee, S. W.

S. W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

MacDonald, H. M.

H. M. MacDonald, “The effect produced by an obstacle on a train of electric waves,” Philos. Trans. R. Soc. London, Ser. A 212, 299-337 (1913).
[CrossRef]

Maggi, G. A.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. Pura Appl. 16, 21-48 (1888).

Marathay, A. S.

Marchand, E. W.

McCalmont, J. F.

Mechler, M.

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

Mukunda, N.

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (De la Faculté des Sciences de Paris, 1892).

Rubinowicz, A.

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931-936 (1938).
[CrossRef]

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. 378, 339-364 (1924).
[CrossRef]

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

Samek, O.

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Stamnes, J. J.

Tai, C.

C. Tai, “Kirchhoff theory: Scalar, vector or dyadic?” IEEE Trans. Antennas Propag. 20, 114-115 (1972).
[CrossRef]

Umul, Y. Z.

Y. Z. Umul, “Modified diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 25, 1850-1860 (2008).
[CrossRef]

Y. Z. Umul, “Rubinowicz transform of the MTPO surface integrals,” Opt. Commun. 281, 5641-5646 (2008).
[CrossRef]

Y. Z. Umul, “The relation between the boundary diffraction wave theory and physical optics,” Opt. Commun. 281, 4844-4848 (2008).
[CrossRef]

Y. Z. Umul, “Simplified uniform theory of diffraction,” Opt. Lett. 30, 1614-1616 (2005).
[CrossRef] [PubMed]

Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
[CrossRef] [PubMed]

Wolf, E.

Young, T.

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

Ann. Chim. Phys. (1)

A. J. Fresnel, “Mémoire sur la diffraction de la lumière,” Ann. Chim. Phys. 1, 239-281 (1816).

Ann. Mat. Pura Appl. (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. Pura Appl. 16, 21-48 (1888).

Ann. Phys. (5)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

F. Kottler, “Zur Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 375, 405-456 (1923).
[CrossRef]

F. Kottler, “Elektromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. 376, 457-508 (1923).
[CrossRef]

G. A. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. 378, 339-364 (1924).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B: Photophys. Laser Chem. (2)

S. V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, “Analytical model of the enhanced light transmission through subwavelength metal slits: Green's function formalism versus Rayleigh's expansion,” Appl. Phys. B: Photophys. Laser Chem. 84, 19-24 (2006).
[CrossRef]

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B: Photophys. Laser Chem. 90, 379-382 (2008).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

C. Tai, “Kirchhoff theory: Scalar, vector or dyadic?” IEEE Trans. Antennas Propag. 20, 114-115 (1972).
[CrossRef]

S. W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

J. Math. Phys. (1)

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448-454 (1975).
[CrossRef]

J. Mod. Opt. (3)

S. Ganci, “A note on the Kirchhoff formulation of diffraction by a plane screen,” J. Mod. Opt. 45, 873-876 (1998).
[CrossRef]

S. Ganci, “A general scalar solution for the half-plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
[CrossRef]

S. Ganci, “Half-plane diffraction in a case of oblique incidence,” J. Mod. Opt. 43, 2543-2551 (1996).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Math. Ann. (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Opt. Commun. (2)

Y. Z. Umul, “Rubinowicz transform of the MTPO surface integrals,” Opt. Commun. 281, 5641-5646 (2008).
[CrossRef]

Y. Z. Umul, “The relation between the boundary diffraction wave theory and physical optics,” Opt. Commun. 281, 4844-4848 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Philos. Trans. R. Soc. London (1)

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

H. M. MacDonald, “The effect produced by an obstacle on a train of electric waves,” Philos. Trans. R. Soc. London, Ser. A 212, 299-337 (1913).
[CrossRef]

Phys. Rev. (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931-936 (1938).
[CrossRef]

Other (5)

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, 1983).

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2003).

H. Poincaré, Théorie Mathématique de la Lumière (De la Faculté des Sciences de Paris, 1892).

E. Hecht, Optics (Addison-Wesley, 2002).

W. C. Elmore and M. A. Heald, Physics of Waves (McGraw-Hill, 1969).

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Figures (5)

Fig. 1
Fig. 1

Geometry of propagation.

Fig. 2
Fig. 2

Scattering of waves by a half-plane.

Fig. 3
Fig. 3

Comparison of the diffracted waves.

Fig. 4
Fig. 4

Complex contour of C 1 .

Fig. 5
Fig. 5

Complex contour of C 2 .

Equations (53)

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

u = u 0 exp ( j k z )
1 4 π S ( u G n G u n ) d S = { u ( P ) , P V 0 , P V }
u ( P ) = I 1 + I f ,
I 1 = j k u 0 4 π ρ = 0 ϕ = 0 2 π ( 1 + sin β ) exp ( j k R ) R ρ d ρ d ϕ
I f = u 0 4 π ρ = 0 ϕ = 0 2 π sin β exp ( j k R ) R 2 ρ d ρ d ϕ
G n = ( j k + 1 R ) exp ( j k R ) R R n ,
I 1 = u 0 exp ( j k z )
I f = u 0 2 j k exp ( j k z ) z ,
A = { ( x , y , z ) ; x ( , 0 ) , y = 0 , z ( , ) }
B = { ( τ , n , z ) ; τ ( 0 , ) , n = 0 , z ( , ) } .
1 4 π A + B ( u G n G u n ) d S = { u i ( P ) , P V 0 , P V } ,
u i = u 0 exp [ j k ρ cos ( ϕ ϕ 0 ) ] .
1 4 π A ( u G n G u n ) d S = { u i ( P ) + u d ( P ) , P V u d ( P ) , P V } ,
u d ( P ) = 1 4 π B ( u G n G u n ) d S
u d ( P ) = I d 1 + I d f ,
I d 1 = j k u 0 4 π τ = 0 z = sin β exp ( j k τ ) exp ( j k R ) R d z d τ
I d f = u 0 4 π τ = 0 z = sin β exp ( j k τ ) exp ( j k R ) R 2 d z d τ ,
R = ρ 2 + ( τ ) 2 2 ρ τ cos α + ( z z ) 2
I d 1 = exp ( j π 4 ) k u 0 2 2 π τ = 0 sin β exp ( j k τ ) exp ( j k R 1 ) R 1 d τ
I d f = exp ( j π 4 ) u 0 2 2 π k τ = 0 sin β exp ( j k τ ) exp ( j k R 1 ) R 1 3 2 d τ ,
g ( τ ) = τ + R 1 .
| d g d τ | τ s = 1 + τ s ρ cos α R 1 = 0 ,
u d 1 = exp ( j π 4 ) 2 2 π sin α 1 cos α exp ( j k ρ ) k ρ
u d f = exp ( j π 4 ) 2 2 π sin α 1 cos α exp ( j k ρ ) ( k ρ ) 3 2 .
u d 1 = exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 1 + cos ( ϕ ϕ 0 ) exp ( j k ρ ) k ρ
u d f = exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 1 + cos ( ϕ ϕ 0 ) exp ( j k ρ ) ( k ρ ) 3 2 ,
u d f = j u d 1 k ρ .
u d t = u d 1 ( 1 j k ρ ) ,
u d 1 = f exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 1 + cos ( ϕ ϕ 0 ) exp ( j k ρ ) k ρ
u d f = f exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 1 + cos ( ϕ ϕ 0 ) exp ( j k ρ ) ( k ρ ) 3 2 ,
f = p [ 1 exp ( 2 π k ρ | cos ϕ ϕ 0 2 | ) ] ,
u d 1 = f exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 2 cos ( ϕ ϕ 0 ) 2 exp ( j k ρ ) k ρ
u d f = f exp ( j π 4 ) 2 2 π sin ( ϕ ϕ 0 ) 2 cos ( ϕ ϕ 0 ) 2 exp ( j k ρ ) ( k ρ ) 3 2
u d 1 = exp ( j π 4 ) 2 2 π f cos ( ϕ ϕ 0 ) 2 exp ( j k ρ ) k ρ
u d f = exp ( j π 4 ) 2 2 π f cos ( ϕ ϕ 0 ) 2 exp ( j k ρ ) ( k ρ ) 3 2 ,
u d i = exp [ j k ρ cos ( ϕ ϕ 0 ) ] sign ( ξ i ) F [ | ξ i | ]
F [ x ] = exp ( j π 4 ) π x exp ( j t 2 ) d t .
u ( P ) = u i ( P ) ( 1 j 2 k z ) ,
u ( P ) = u i ( P ) ,
u d t = u d 1 ( 1 j k ρ ) ,
G ( R ) = j k 2 π C 1 C 2 exp ( j k R sin β cos α ) d β d α
G ( R ) = exp ( j k R ) R
u = exp ( j k x )
x = ρ cos ϕ
x = r sin θ cos ϕ
u = 1 π C 1 exp ( j k ρ cos α ) d α ,
sin α 1 s h α 2 > 0 ,
1 π C 1 exp ( j k ρ cos α ) d α 2 π exp ( j π 4 ) exp ( j k ρ ) k ρ ,
u = j k 2 π C 1 C 2 exp ( j k r sin β cos α ) d β d α
j k 2 π C 1 C 2 exp ( j k r sin β cos α ) d β d α exp ( j k r ) r
G ( R ) = j k 2 π C 1 C 2 exp ( j k R sin β cos α ) d β d α
d G ( R ) d R = k 2 2 π C 1 C 2 sin β cos α exp ( j k R sin β cos α ) d β d α ,
d G ( R ) d R j k exp ( j k R ) R ,

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