Abstract

The diffraction theory of Kirchhoff is reinterpreted and a new form of a surface diffraction integral is developed by using the axioms of the modified theory of physical optics, which leads to the exact scattered fields by conducting bodies. The new integral is arranged according to the interpretation of Young, and the diffracted waves are expressed in terms of a line integral. The method is applied to the diffraction problem by a semi-infinite edge contour.

© 2008 Optical Society of America

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References

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  1. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford U. Press, 1950).
  2. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).
  3. C. R. Schultheisz, “Numerical solution of the Huygens-Fresnel-Kirchhoff diffraction of spherical waves by a circular aperture,” J. Opt. Soc. Am. A 11, 774-778 (1994).
    [CrossRef]
  4. S. Ganci, “Equivalence between two consistent formulations of Kirchhoff's diffraction theory,” J. Opt. Soc. Am. A 5, 1626-1628 (1988).
    [CrossRef]
  5. P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley-IEEE, 2007).
    [CrossRef]
  6. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).
  7. P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
    [CrossRef]
  8. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  9. Y. Z. Umul, “Modified theory of physical optics approach to wedge diffraction problems,” Opt. Express 13, 216-224 (2005).
    [CrossRef] [PubMed]
  10. Y. Z. Umul, “MTPO based potential function of the boundary diffraction wave theory,” Opt. Laser Technol. 40, 769-774 (2008).
    [CrossRef]
  11. Y. Z. Umul, “The theory of the boundary diffraction wave for wedge diffraction,” J. Mod. Opt. 55, 1417-1426 (2008).
    [CrossRef]
  12. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
    [CrossRef]
  13. A. Rubinowicz, “Die beugungswelle in der Kirchoffschen theorie der beugungsercheinungen,” Ann. Phys. 4, 257-278 (1917).
    [CrossRef]
  14. S. Ganci, “A general solution for the half plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
    [CrossRef]
  15. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).
  16. Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
    [CrossRef]
  17. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  18. Y. Z. Umul, “Edge-dislocation waves in the edge diffraction process by an impedance half-plane,” J. Opt. Soc. Am. A 24, 507-511 (2007).
    [CrossRef]
  19. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave--Part I,” J. Opt. Soc. Am. 52, 615-625 (1962).
    [CrossRef]
  20. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave--Part II,” J. Opt. Soc. Am. 52, 626-637 (1962).
    [CrossRef]
  21. A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 201-240 (1965).
  22. A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
    [CrossRef]
  23. F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, 1962).
  24. A. Sommerfeld, “Matematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  25. S. Ganci, “Half-plane diffraction in a case of oblique incidence,” J. Mod. Opt. 43, 2543-2551 (1996).
    [CrossRef]
  26. Y. Rahmat-Samii, “Keller's cone encountered at a hotel,” IEEE Antennas Propag. Mag. 49, 88-89 (2007).
    [CrossRef]
  27. T. B. A. Senior and P. L. E. Ushlengi, “Experimental detection of the edge-diffraction cone,” J. Electron Microsc. 60, 1448-1449 (1972).
  28. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]
  29. Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
    [CrossRef]

2008 (2)

Y. Z. Umul, “MTPO based potential function of the boundary diffraction wave theory,” Opt. Laser Technol. 40, 769-774 (2008).
[CrossRef]

Y. Z. Umul, “The theory of the boundary diffraction wave for wedge diffraction,” J. Mod. Opt. 55, 1417-1426 (2008).
[CrossRef]

2007 (4)

2006 (1)

Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

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

1999 (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

1998 (1)

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

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 solution for the half plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
[CrossRef]

1994 (1)

1991 (1)

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

1988 (1)

1974 (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

1972 (1)

T. B. A. Senior and P. L. E. Ushlengi, “Experimental detection of the edge-diffraction cone,” J. Electron Microsc. 60, 1448-1449 (1972).

1965 (1)

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 201-240 (1965).

1962 (4)

1957 (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

1950 (1)

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford U. Press, 1950).

1949 (1)

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).

1917 (1)

A. Rubinowicz, “Die beugungswelle in der Kirchoffschen theorie der beugungsercheinungen,” Ann. Phys. 4, 257-278 (1917).
[CrossRef]

1896 (1)

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

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford U. Press, 1950).

Born, M.

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

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford U. Press, 1950).

Dubra, A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Ferrari, J. A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Ganci, S.

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

S. Ganci, “A general 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]

Hildebrand, F. B.

F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, 1962).

Keller, J. B.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Miyamoto, K.

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Rahmat-Samii, Y.

Y. Rahmat-Samii, “Keller's cone encountered at a hotel,” IEEE Antennas Propag. Mag. 49, 88-89 (2007).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 201-240 (1965).

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

A. Rubinowicz, “Die beugungswelle in der Kirchoffschen theorie der beugungsercheinungen,” Ann. Phys. 4, 257-278 (1917).
[CrossRef]

Schultheisz, C. R.

Senior, T. B. A.

T. B. A. Senior and P. L. E. Ushlengi, “Experimental detection of the edge-diffraction cone,” J. Electron Microsc. 60, 1448-1449 (1972).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).

Sommerfeld, A.

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

Stutzman, W. L.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

Thiele, G. A.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley-IEEE, 2007).
[CrossRef]

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

Umul, Y. Z.

Y. Z. Umul, “MTPO based potential function of the boundary diffraction wave theory,” Opt. Laser Technol. 40, 769-774 (2008).
[CrossRef]

Y. Z. Umul, “The theory of the boundary diffraction wave for wedge diffraction,” J. Mod. Opt. 55, 1417-1426 (2008).
[CrossRef]

Y. Z. Umul, “Edge-dislocation waves in the edge diffraction process by an impedance half-plane,” J. Opt. Soc. Am. A 24, 507-511 (2007).
[CrossRef]

Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
[CrossRef]

Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

Y. Z. Umul, “Modified theory of physical optics approach to wedge diffraction problems,” Opt. Express 13, 216-224 (2005).
[CrossRef] [PubMed]

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

Ushlengi, P. L. E.

T. B. A. Senior and P. L. E. Ushlengi, “Experimental detection of the edge-diffraction cone,” J. Electron Microsc. 60, 1448-1449 (1972).

Wolf, E.

Am. J. Phys. (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Ann. Phys. (1)

A. Rubinowicz, “Die beugungswelle in der Kirchoffschen theorie der beugungsercheinungen,” Ann. Phys. 4, 257-278 (1917).
[CrossRef]

Electromagnetics (1)

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

Y. Rahmat-Samii, “Keller's cone encountered at a hotel,” IEEE Antennas Propag. Mag. 49, 88-89 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

J. Electron Microsc. (1)

T. B. A. Senior and P. L. E. Ushlengi, “Experimental detection of the edge-diffraction cone,” J. Electron Microsc. 60, 1448-1449 (1972).

J. Mod. Opt. (3)

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

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

Y. Z. Umul, “The theory of the boundary diffraction wave for wedge diffraction,” J. Mod. Opt. 55, 1417-1426 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Math. Ann. (1)

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

Nature (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

Y. Z. Umul, “MTPO based potential function of the boundary diffraction wave theory,” Opt. Laser Technol. 40, 769-774 (2008).
[CrossRef]

Proc. IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Prog. Opt. (1)

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 201-240 (1965).

Other (6)

F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, 1962).

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley-IEEE, 2007).
[CrossRef]

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford U. Press, 1950).

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

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).

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Figures (13)

Fig. 1
Fig. 1

Edge diffraction process.

Fig. 2
Fig. 2

Surface of integration for the Kirchhoff theory.

Fig. 3
Fig. 3

Geometry of the scattering.

Fig. 4
Fig. 4

Geometry for the vector potential.

Fig. 5
Fig. 5

Surfaces of scattering.

Fig. 6
Fig. 6

Reflection geometry by a plane.

Fig. 7
Fig. 7

Scattering surface and its boundary.

Fig. 8
Fig. 8

Geometry for the evaluation of the line integral of GO waves.

Fig. 9
Fig. 9

Geometry for the GO field evaluation.

Fig. 10
Fig. 10

Geometry of the semi-infinite edge contour.

Fig. 11
Fig. 11

Geometry of the corner diffracted ray.

Fig. 12
Fig. 12

Edge diffracted field at C.

Fig. 13
Fig. 13

Compensation of the edge diffracted wave at the corner.

Equations (96)

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2 u + k 2 u = 0 ,
u ( P ) = S [ u ( Q ) Q G ( P , Q ) G ( P , Q ) Q u ( Q ) ] n d S ,
cos ν = cos ϕ α 2 ,
u ( P ) = 1 4 π S [ u ( Q ) Q G ( P , Q ) G ( P , Q ) Q u ( Q ) ] n 1 d S ,
u ( P ) = 1 4 π S [ u ( Q ) G ( P , Q ) n 1 G ( P , Q ) u ( Q ) n 1 ] d S .
G ( P , Q ) n 1 = G ( P , Q ) R cos ( n 1 , R ) ,
u ( Q ) n 1 = u ( Q ) R i cos ( n 1 , R i ) ,
s r = R R ,
s i = R i R i ,
G ( P , Q ) n 1 = G ( P , Q ) R cos ν = G ( P , Q ) R sin β α 2 ,
u ( Q ) n 1 = u ( Q ) R i cos ( ν + β α ) = u ( Q ) R i sin β α 2 ,
u ( P ) = 1 4 π S [ u ( Q ) G ( P , Q ) R + G ( P , Q ) u ( Q ) R i ] sin β α 2 d S ,
u ( P ) = 1 4 π S [ u ( Q ) G ( P , Q ) R s r G ( P , Q ) u ( Q ) R i s i ] n 1 d S ,
cos ( n 1 , R ) = s r n 1 ,
cos ( n 1 , R i ) = s i n 1 ,
u s ( P ) = 1 4 π S G ( P , Q ) u ( Q ) R i ( s i n 1 ) d S ,
u h ( P ) = 1 4 π S u ( Q ) G ( P , Q ) R ( s r n 1 ) d S .
V = 1 4 π ( u G G u )
× V = 1 2 π u × G .
2 π × × V = ( G ) u ( u ) G + 4 π k 2 V ,
e 1 = R i R i , e 2 = R R .
u = u R i e 1 ,
G = G R e 2 .
2 π × × V = G R 2 u R i 2 R i R e 1 u R i 2 G R 2 R R i e 2 + 4 π k 2 V ,
2 π × × V = k 2 cos α ( u G R e 1 G u R i e 2 ) + 4 π k 2 V ,
u j k u e 1 ,
G j k G e 2 ,
2 π × × V = j k u G k 2 cos α ( e 1 e 2 ) + 4 π k 2 V ,
V j k 4 π u G ( e 2 e 1 ) .
× × V = 2 k 2 V [ 1 + cos ( R , R i ) ] ,
V = × W .
× { ( × V ) 2 k 2 [ 1 + cos ( R , R i ) ] W } = 0 ,
( × V ) 2 k 2 [ 1 + cos ( R , R i ) ] W = φ .
W = u × G 4 π k 2 [ 1 + cos ( R , R i ) ] φ 2 k 2 [ 1 + cos ( R , R i ) ] ,
W = u G ( e 1 × e 2 ) 4 π [ 1 + cos ( R , R i ) ] φ 2 k 2 [ 1 + cos ( R , R i ) ] ,
W = u G sin ( R , R i ) e d 4 π [ 1 + cos ( R , R i ) ] ,
u ( P ) = S V n 1 d S ,
u ( P ) = S ( × W ) n 1 d S ,
S ( × W ) n 1 d S = C W t d l n 1 n ,
φ = β α 2 .
S ( × W ) n 1 d S = S 1 ( × W ) n 1 d S 1 n 1 n ,
S 1 ( × W ) n 1 d S 1 n 1 n = C 1 W t 1 d l 1 n 1 n ,
u d ( P ) = C W t d l n 1 n ,
u d ( P ) = 1 4 π C u ( Q ) G ( P , Q ) sin ( R , R i ) 1 + cos ( R , R i ) e d t n 1 n d l ,
s j = cos α e x + sin α e y ,
s r = cos β e x + sin β e y ,
u i = u 0 e j k ( x cos α + y sin α ) .
n 1 = sin ( β α 2 ) e x + cos ( β α 2 ) e y .
W = u 0 4 π e j x y cos α e j k R R sin ( β α ) 1 cos ( β α ) e z
n 1 n = cos β α 2 ,
u d = u 0 4 π e j k R e R e sin ( β e α ) 1 cos ( β e α ) 1 cos β e α 2 d z ,
u d = 1 4 π u 0 sin ϕ α 2 e j k R e R e d z ,
u d = u 0 e j ( π 4 ) 2 2 π 1 sin ϕ α 2 e j k ρ k ρ ,
S ( × W ) n 1 d S = C W t d l n 1 n + u GO ,
W = u G ( s j × s r ) 4 π [ 1 s j s r ] ,
S ( × W ) n 1 d S = C 1 W t d l n 1 n C 2 W t d l n 1 n .
u GO = C 2 W t d l n 1 n .
W t n 1 n = f ( P , Q ) g ( P , Q ) ,
u GO = lim ε 0 0 2 π f ( P , Q ) g ( P , Q ) ε d φ ,
u GO = 0 2 π lim ε 0 f ( P , Q ) g ( P , Q ) ε d φ .
u GO = f ( P , Q s ) 0 2 π lim ε 0 ε g ( P , Q ) d φ ,
u GO = f ( P , Q s ) g ( P , Q s ) 0 2 π d φ ,
u GO = 2 π f ( P , Q s ) g ( P , Q s ) .
W = u 0 4 π e j k ( x s ε ) cos α e j k R R sin ( β α ) 1 cos ( β α ) e z ,
W n 1 n = u 0 4 π e j k ( x s ε ) cos α e j k R R 1 sin β α 2 e z ,
u GO = u 0 4 π lim ε 0 0 2 π e j k ( x s ε ) cos α e j k R R ε sin β α 2 d φ ,
u GO = u 0 4 π e j k x s cos α e j k R s R s 0 2 π lim ε 0 ε sin β α 2 d φ ,
sin β α 2 ε 2 R ,
u GO = u 0 4 π e j k ( x cos α + y sin α ) R s 0 2 π lim ε 0 2 R d φ ,
u GO = u 0 e j k ( x cos α + y sin α ) .
u i = u 0 e j k ( x cos α + y sin α ) .
s i = e x cos α e y sin α ,
s j = e x cos α + e y sin α ,
s r = e x cos β + e y sin β sin η e z sin β cos η ,
s j × s r = e x sin α sin β cos η e y cos α sin β cos η + e z ( sin α cos β cos α sin β sin η ) ,
s j s r = cos α cos β + sin α sin β sin η ,
n 1 n = sin α + sin β sin η 2 ( 1 s i s r ) ,
n 1 × s i = n 1 × s r .
u d = 0 W z n 1 n x = 0 d z .
W = u 0 4 π e j k x cos α e j k R R s j × s r 1 s j s r ,
f 1 = ( cos α sin β sin η sin α cos β ) 1 cos α cos β + sin α sin β sin η ( 1 cos α cos β sin α sin β sin η ) ( sin α + sin β sin η ) ,
u d = u 0 2 2 π 0 f 1 e j k R 1 R 1 d z ,
u d = u 0 e j ( π 4 ) 2 2 π 1 cos ϕ + α 2 e j k ρ k ρ ,
a f ( x ) e j k g ( x ) d x 1 j k f ( a ) g ( a ) e j k g ( a ) ,
u c = u 0 j k 2 2 π f 1 z = 0 cos θ e j k r r ,
f 1 z = 0 = ( cos α sin β c sin η c sin α cos β c ) 1 cos α cos β c + sin α sin β c sin η c ( 1 cos α cos β c sin α sin β c sin η c ) ( sin α + sin β c sin η c ) ,
f 1 z = 0 = sin θ sin ( ϕ + α ) 1 + sin θ cos ( ϕ α ) [ 1 + sin θ cos ( ϕ + α ) ] ( sin α + sin θ sin ϕ ) ,
cos β c = sin θ cos ϕ ,
sin β c sin η c = sin θ sin ϕ ,
sin β c cos η c = cos θ ,
u d = u 0 e j k ( x cos α y sin α ) sgn ( ξ d ) F [ ξ d ] ,
ξ d = 2 k ρ cos ϕ + α 2 .
F [ x ] = e j ( π 4 ) π x e j t 2 d t .
u c = u 0 2 2 e j k ρ cos ( ϕ + α ) f 1 z = 0 sin θ cos ϕ + α 2 cos θ + ( π 2 ) 2 cos θ sgn ( ξ d ) F [ ξ d ] sgn ( ξ c ) F [ ξ c ]
ξ c = 2 k r cos θ + ( π 2 ) 2 .
u t = u d + u c .

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