Abstract

The expression of the edge diffracted fields, in terms of the Fresnel integral, is transformed into a path integral. The obtained integral considers the integration of the incident field along the ray path of the transition region. The similarities of the path integral with Kirchhoff’s theory of diffraction and the modified theory of physical optics are examined.

© 2008 Optical Society of America

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  1. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  2. Y. Z. Umul, “Uniform line integral representation of edge-diffracted fields,” J. Opt. Soc. Am. A 25, 133-137 (2008).
    [CrossRef]
  3. R. Borghi, “Summing Pauli asymptotic series to solve the wedge problem,” J. Opt. Soc. Am. A 25, 211-218 (2008).
    [CrossRef]
  4. R. Kumar, “Structure of the boundary diffraction wave revisited,” Appl. Phys. B: Lasers Opt. 90, 379-382 (2008).
    [CrossRef]
  5. Y. Z. Umul, “Alternative interpretation of the edge diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-587 (2008).
    [CrossRef]
  6. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
    [CrossRef]
  7. O. M. Bucci and G. Pelosi, “From wave theory to ray optics,” IEEE Antennas Propag. Mag. 36, 35-42 (1994).
    [CrossRef]
  8. G. Kirchhoff, “Zur theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883) (in German).
    [CrossRef]
  9. A. Sommerfeld, “Matematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896) (in German).
    [CrossRef]
  10. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen's Principle (Oxford U. Press, 1953).
  11. A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917) (in German).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).
  13. W. C. Elmore and M. A. Heald, Physics of Waves (Mc-Graw Hill, 1969).
  14. 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]
  15. 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]
  16. A. Rubinowicz, “Simple derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717-722 (1962).
    [CrossRef]
  17. A. Rubinowicz, “Geometric derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717-718 (1962).
    [CrossRef]
  18. A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 199-240 (1965).
    [CrossRef]
  19. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426-444 (1957).
    [CrossRef]
  20. J. B. Keller, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570-579 (1957).
    [CrossRef]
  21. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  22. J. B. Keller, “One hundred years of diffraction theory,” IEEE Trans. Antennas Propag. AP-33, 123-126 (1985).
    [CrossRef]
  23. D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
    [CrossRef]
  24. R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” J. Math. Phys. 10, 2291-2305 (1969).
    [CrossRef]
  25. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]
  26. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Press, 1976).
  27. P. Ya. Ufimtsev, “Theory of acoustical edge waves,” J. Acoust. Soc. Am. 86, 463-474 (1989).
    [CrossRef]
  28. P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
    [CrossRef]
  29. P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).
    [CrossRef]
  30. Y. Z. Umul, “Modified theory of physical optics approach to wedge diffraction problems,” Opt. Express 13, 216-224 (2005).
    [CrossRef] [PubMed]
  31. Y. Z. Umul, “Edge-dislocation waves in the diffraction process by an impedance half-plane,” J. Opt. Soc. Am. A 24, 507-511 (2007).
    [CrossRef]
  32. Y. Z. Umul, “Diffraction by a black half-plane: modified theory of physical optics approach,” Opt. Express 13, 7276-7287 (2005).
    [CrossRef] [PubMed]
  33. F. Gori, “Diffraction from a half-plane. A new derivation of the Sommerfeld solution,” Opt. Commun. 48, 67-70 (1983).
    [CrossRef]
  34. S. Ganci, “A general scalar solution for the half-plane problem,” J. Mod. Opt. 42, 1707-1711 (1995).
    [CrossRef]
  35. Y. Z. Umul, “MTPO based potential function of the boundary diffraction wave theory,” Opt. Laser Technol. 40, 769-774 (2008).
    [CrossRef]
  36. A. H. Serbest, “An extension to GTD for an edge on a curved perfectly conducting surface,” IEEE Trans. Antennas Propag. AP-34, 837-841 (1986).
    [CrossRef]
  37. J. B. Keller, “Rays, waves and asymptotics,” Bull. Am. Math. Soc. 84, 727-750 (1978).
    [CrossRef]

2008

2007

2005

2004

1995

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

1994

O. M. Bucci and G. Pelosi, “From wave theory to ray optics,” IEEE Antennas Propag. Mag. 36, 35-42 (1994).
[CrossRef]

1991

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

1989

P. Ya. Ufimtsev, “Theory of acoustical edge waves,” J. Acoust. Soc. Am. 86, 463-474 (1989).
[CrossRef]

1986

A. H. Serbest, “An extension to GTD for an edge on a curved perfectly conducting surface,” IEEE Trans. Antennas Propag. AP-34, 837-841 (1986).
[CrossRef]

1985

J. B. Keller, “One hundred years of diffraction theory,” IEEE Trans. Antennas Propag. AP-33, 123-126 (1985).
[CrossRef]

1983

F. Gori, “Diffraction from a half-plane. A new derivation of the Sommerfeld solution,” Opt. Commun. 48, 67-70 (1983).
[CrossRef]

1978

J. B. Keller, “Rays, waves and asymptotics,” Bull. Am. Math. Soc. 84, 727-750 (1978).
[CrossRef]

1974

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

1969

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” J. Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

1968

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
[CrossRef]

1965

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 199-240 (1965).
[CrossRef]

1962

1957

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426-444 (1957).
[CrossRef]

J. B. Keller, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570-579 (1957).
[CrossRef]

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

1917

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917) (in German).
[CrossRef]

1896

A. Sommerfeld, “Matematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896) (in German).
[CrossRef]

1883

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

Ahluwalia, D. S.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
[CrossRef]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen's Principle (Oxford U. Press, 1953).

Boersma, J.

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” J. Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
[CrossRef]

Borghi, R.

Born, M.

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

Bucci, O. M.

O. M. Bucci and G. Pelosi, “From wave theory to ray optics,” IEEE Antennas Propag. Mag. 36, 35-42 (1994).
[CrossRef]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen's Principle (Oxford U. Press, 1953).

Elmore, W. C.

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

Ganci, S.

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

Gori, F.

F. Gori, “Diffraction from a half-plane. A new derivation of the Sommerfeld solution,” Opt. Commun. 48, 67-70 (1983).
[CrossRef]

Heald, M. A.

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

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Press, 1976).

Keller, J. B.

J. B. Keller, “One hundred years of diffraction theory,” IEEE Trans. Antennas Propag. AP-33, 123-126 (1985).
[CrossRef]

J. B. Keller, “Rays, waves and asymptotics,” Bull. Am. Math. Soc. 84, 727-750 (1978).
[CrossRef]

J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
[CrossRef] [PubMed]

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426-444 (1957).
[CrossRef]

J. B. Keller, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570-579 (1957).
[CrossRef]

Kirchhoff, G.

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

Kouyoumjian, R. G.

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

Kumar, R.

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

Lewis, R. M.

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” J. Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
[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 screen,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Pelosi, G.

O. M. Bucci and G. Pelosi, “From wave theory to ray optics,” IEEE Antennas Propag. Mag. 36, 35-42 (1994).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 199-240 (1965).
[CrossRef]

A. Rubinowicz, “Simple derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717-722 (1962).
[CrossRef]

A. Rubinowicz, “Geometric derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717-718 (1962).
[CrossRef]

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

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917) (in German).
[CrossRef]

Serbest, A. H.

A. H. Serbest, “An extension to GTD for an edge on a curved perfectly conducting surface,” IEEE Trans. Antennas Propag. AP-34, 837-841 (1986).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Matematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896) (in German).
[CrossRef]

Ufimtsev, P. Ya.

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

P. Ya. Ufimtsev, “Theory of acoustical edge waves,” J. Acoust. Soc. Am. 86, 463-474 (1989).
[CrossRef]

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

Umul, Y. Z.

Wolf, E.

Ann. Phys.

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

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917) (in German).
[CrossRef]

Appl. Phys. B: Lasers Opt.

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

Bull. Am. Math. Soc.

J. B. Keller, “Rays, waves and asymptotics,” Bull. Am. Math. Soc. 84, 727-750 (1978).
[CrossRef]

Electromagnetics

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

IEEE Antennas Propag. Mag.

O. M. Bucci and G. Pelosi, “From wave theory to ray optics,” IEEE Antennas Propag. Mag. 36, 35-42 (1994).
[CrossRef]

IEEE Trans. Antennas Propag.

J. B. Keller, “One hundred years of diffraction theory,” IEEE Trans. Antennas Propag. AP-33, 123-126 (1985).
[CrossRef]

A. H. Serbest, “An extension to GTD for an edge on a curved perfectly conducting surface,” IEEE Trans. Antennas Propag. AP-34, 837-841 (1986).
[CrossRef]

J. Acoust. Soc. Am.

P. Ya. Ufimtsev, “Theory of acoustical edge waves,” J. Acoust. Soc. Am. 86, 463-474 (1989).
[CrossRef]

J. Appl. Phys.

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426-444 (1957).
[CrossRef]

J. B. Keller, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570-579 (1957).
[CrossRef]

J. Math. Phys.

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” J. Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Ann.

A. Sommerfeld, “Matematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896) (in German).
[CrossRef]

Nature

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

Opt. Commun.

F. Gori, “Diffraction from a half-plane. A new derivation of the Sommerfeld solution,” Opt. Commun. 48, 67-70 (1983).
[CrossRef]

Opt. Express

Opt. Laser Technol.

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

Proc. IEEE

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

Prog. Opt.

A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Prog. Opt. 4, 199-240 (1965).
[CrossRef]

SIAM J. Appl. Math.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783-807 (1968).
[CrossRef]

Other

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Press, 1976).

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

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

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

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen's Principle (Oxford U. Press, 1953).

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Figures (7)

Fig. 1
Fig. 1

Diffraction of plane waves by a half-plane.

Fig. 2
Fig. 2

Integration path of the diffracted field.

Fig. 3
Fig. 3

Half-plane and integration surfaces.

Fig. 4
Fig. 4

Integration surface of B.

Fig. 5
Fig. 5

Diffraction geometry for a cylinder.

Fig. 6
Fig. 6

Edge diffracted wave for Eq. (56).

Fig. 7
Fig. 7

Edge diffracted wave for Eq. (60).

Equations (61)

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

u ( P ) = u 0 { exp [ j k ρ cos ( ϕ α ) ] F [ ξ i ] exp [ j k ρ cos ( ϕ + α ) ] F [ ξ r ] }
F [ x ] = exp ( j π 4 ) π x exp ( j t 2 ) d t ,
ξ i = 2 k ρ cos ϕ α 2
ξ r = 2 k ρ cos ϕ + α 2 ,
u is = u i GO + u id ,
u i GO = u 0 exp [ j k ρ cos ( ϕ α ) ] U ( ξ i )
u id = u 0 exp [ j k ρ cos ( ϕ α ) ] sign ( ξ i ) F [ ξ i ] ,
u id = u 0 exp ( j π 4 ) π exp [ j k ρ cos ( ϕ α ) ] sign ( ξ i ) ξ i exp ( j t 2 ) d t .
u id = u 0 exp ( j π 4 ) π sign ( ξ i ) ξ i exp { j [ t 2 k ρ cos ( ϕ α ) ] } d t .
t 2 k ρ cos ( ϕ α ) = k ( r + R )
d t = k 2 1 cos β r + R + ρ cos ( ϕ α ) d r
d R d r = cos β
R = ρ 2 + r 2 2 r ρ cos γ
R sin γ = r sin ( β γ ) = ρ sin β
r + R + ρ cos ( ϕ α ) = R ( 1 cos β )
d t = k 2 R sin β 2 d r .
u id = u 0 k exp ( j π 4 ) 2 π sign ( ξ i ) 0 exp [ j k ( r + R ) ] k R sin β 2 d r .
exp ( j k R ) k R exp ( j π 4 ) 2 π exp ( j k R 1 ) R 1 d z ,
u id = u 0 j k 2 π sign ( ξ i ) z = r = 0 exp [ j k ( r + R 1 ) ] R 1 sin β 2 d r d z
u ( P ) = S V n 1 d S ,
V = 1 4 π ( u G G u ) ,
A = { ( x , y , z ) ; x ( , 0 ) , y = 0 , z ( , ) } ,
B = { ( r , z ) ; r ( 0 , ) , z ( , ) } ,
C = { ( x , y , z ) ; x ( 0 , ) , y = 0 , z ( , ) } ,
A + B V n 1 d S = u i GO ( P ) ,
u i GO ( P ) = { u i ( P ) , P V 1 0 , P V 2 ,
A V n 1 d S = u i GO ( P ) + u id ( P ) ,
u id ( P ) = B V n 1 d S .
B + C V n 1 d S = u id ( P )
C V n 1 d S = 0
u id ( P ) = B V n 1 d S
u id ( P ) = ε B V n 1 d S
ε = { 1 , P V 1 1 , P V 2 ,
u id ( P ) = ε 4 π B ( u G n 1 G u n 1 ) d S
G n 1 = G R 1 R n 1 = G R 1 sin β 2
u n 1 = u r r n 1 = u r sin β 2
u id ( P ) = ε 4 π B ( u G R 1 + G u r ) sin β 2 d S
u i ( Q ) = u 0 exp ( j k r )
u id ( P ) = u 0 ε 4 π z = r = 0 exp ( j k r ) ( G R 1 j k G ) sin β 2 d r d z .
G R 1 j k G j 2 k G
u id ( P ) = u 0 j k 2 π ε z = r = 0 exp [ j k ( r + R 1 ) ] R 1 sin β 2 d r d z ,
u id ( P ) = u 0 j k 2 π ε Γ r = 0 exp [ j k ( r + R 1 ) ] R 1 R 1 n 1 d r d l
u i = u 0 exp [ j k ρ cos ( ϕ ϕ 0 ) ] .
u i ( Q e ) = u 0 exp ( j k a cos ϕ 0 ) .
u d = ε k u 0 exp [ j ( π 4 + k a cos ϕ 0 ) ] 2 π 0 sin β 2 exp [ j k ( r + R ) ] k R d r
ψ = r + R ,
ψ = 1 cos β
ψ s = r + R s ,
ψ s = r + R cos β
ξ 2 = k ( ψ ψ s )
ξ = 2 k R sin β 2 ,
d ξ = k 2 R sin β 2 d r
d β d r = sin β R
ξ e = 2 k R e sin α e 2 .
u d = ε u 0 exp [ j k ( a cos ϕ 0 R e cos α e ) ] F [ ξ e ]
u d = ε u 0 exp [ j k ρ cos ( ϕ ϕ 0 ) ] F [ ξ e ]
ρ sin ϕ 0 = R e cos α e sin ϕ = a sin ( ϕ ϕ 0 )
sign ( x ) F [ x ] exp [ j ( π 4 + x 2 ) ] 2 π x .
u d exp ( j π 4 ) 2 2 π 1 sin ( α e 2 ) exp ( j k R e ) k R e .
ξ e = 2 k R e cos α ϕ 0 2 .
u id = exp ( j π 4 ) 2 2 π 1 cos ϕ α 2 e j k ρ k ρ .

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