Abstract

The diffraction of a plane electromagnetic wave by a rectangular groove corrugated in a perfectly conducting ground plane is rigorously formulated in terms of a Fourier-type integral for two orthogonal polarizations. I successfully expand the angular spectrum of the diffracted field in a series of Bessel functions. Furthermore, the Watson transform is adopted to compute the far-zone diffracted field. Results indicate that the groove width is inversely related to the beamwidth, while the groove depth is responsible for the power distribution among the mainlobe and the sidelobes. The solution remains computationally stable for a wide range of the parameters.

© 1992 Optical Society of America

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References

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  1. G. W. Lehman, “Diffraction of electromagnetic waves by planar dielectric structures. I. Transverse electric excitation,”J. Math. Phys. 11, 1522–1535 (1970).
    [CrossRef]
  2. S. C. Kashyap, M. A. K. Hamid, “Diffraction characteristics of a slit in a thick conducting screen,”IEEE Trans. Antennas Propag. AP-19, 499–507 (1971).
    [CrossRef]
  3. R. A. Hurd, Y. Hayashi, “Low-frequency scattering by a slit in a conducting plane,” Radio Sci. 15, 1171–1178 (1980).
    [CrossRef]
  4. D. T. Auckland, R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. MTT-26, 499–505 (1978).
    [CrossRef]
  5. D. T. Auckland, R. F. Harrington, “A nonmodal formulation for electromagnetic transmission through a filled slot of arbitrary cross section in a thick conducting screen,” IEEE Trans. Microwave Theory Tech. MTT-28, 548–555 (1980).
    [CrossRef]
  6. J. -M. Jin, J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280–1286 (1990).
    [CrossRef]
  7. J.-M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-baked apertures,”IEEE Trans. Antennas Propag. 39, 97–104 (1991).
    [CrossRef]
  8. T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
    [CrossRef]
  9. S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane-TE case,”IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).
  10. R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,”IEEE Trans. Antennas Propag. AP-24, 870–873 (1976).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 578–580 and 590.
  12. W. Magnus, F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1949).
  13. fortran Subroutines for Mathematical Applications, Version 1.1 (IMSL, Inc., 2500 City West Boulevard, Houston, Texas, 1989).
  14. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).
  15. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).
  16. Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. A 5, 65–73 (1988).
    [CrossRef] [PubMed]

1991 (1)

J.-M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-baked apertures,”IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

1990 (3)

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane-TE case,”IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

J. -M. Jin, J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280–1286 (1990).
[CrossRef]

1988 (1)

1980 (2)

D. T. Auckland, R. F. Harrington, “A nonmodal formulation for electromagnetic transmission through a filled slot of arbitrary cross section in a thick conducting screen,” IEEE Trans. Microwave Theory Tech. MTT-28, 548–555 (1980).
[CrossRef]

R. A. Hurd, Y. Hayashi, “Low-frequency scattering by a slit in a conducting plane,” Radio Sci. 15, 1171–1178 (1980).
[CrossRef]

1978 (1)

D. T. Auckland, R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. MTT-26, 499–505 (1978).
[CrossRef]

1976 (1)

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,”IEEE Trans. Antennas Propag. AP-24, 870–873 (1976).
[CrossRef]

1971 (1)

S. C. Kashyap, M. A. K. Hamid, “Diffraction characteristics of a slit in a thick conducting screen,”IEEE Trans. Antennas Propag. AP-19, 499–507 (1971).
[CrossRef]

1970 (1)

G. W. Lehman, “Diffraction of electromagnetic waves by planar dielectric structures. I. Transverse electric excitation,”J. Math. Phys. 11, 1522–1535 (1970).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

Auckland, D. T.

D. T. Auckland, R. F. Harrington, “A nonmodal formulation for electromagnetic transmission through a filled slot of arbitrary cross section in a thick conducting screen,” IEEE Trans. Microwave Theory Tech. MTT-28, 548–555 (1980).
[CrossRef]

D. T. Auckland, R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. MTT-26, 499–505 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 578–580 and 590.

Gallagher, N. C.

Hamid, M. A. K.

S. C. Kashyap, M. A. K. Hamid, “Diffraction characteristics of a slit in a thick conducting screen,”IEEE Trans. Antennas Propag. AP-19, 499–507 (1971).
[CrossRef]

Harrington, R. F.

D. T. Auckland, R. F. Harrington, “A nonmodal formulation for electromagnetic transmission through a filled slot of arbitrary cross section in a thick conducting screen,” IEEE Trans. Microwave Theory Tech. MTT-28, 548–555 (1980).
[CrossRef]

D. T. Auckland, R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. MTT-26, 499–505 (1978).
[CrossRef]

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,”IEEE Trans. Antennas Propag. AP-24, 870–873 (1976).
[CrossRef]

Hayashi, Y.

R. A. Hurd, Y. Hayashi, “Low-frequency scattering by a slit in a conducting plane,” Radio Sci. 15, 1171–1178 (1980).
[CrossRef]

Hurd, R. A.

R. A. Hurd, Y. Hayashi, “Low-frequency scattering by a slit in a conducting plane,” Radio Sci. 15, 1171–1178 (1980).
[CrossRef]

Jeng, S.-K.

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane-TE case,”IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

Jin, J. -M.

J. -M. Jin, J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280–1286 (1990).
[CrossRef]

Jin, J.-M.

J.-M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-baked apertures,”IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

Kashyap, S. C.

S. C. Kashyap, M. A. K. Hamid, “Diffraction characteristics of a slit in a thick conducting screen,”IEEE Trans. Antennas Propag. AP-19, 499–507 (1971).
[CrossRef]

Kok, Y. L.

Lehman, G. W.

G. W. Lehman, “Diffraction of electromagnetic waves by planar dielectric structures. I. Transverse electric excitation,”J. Math. Phys. 11, 1522–1535 (1970).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1949).

Mautz, J. R.

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,”IEEE Trans. Antennas Propag. AP-24, 870–873 (1976).
[CrossRef]

Natzke, J. R.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Oberhettinger, F.

W. Magnus, F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1949).

Sarabandi, K.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Volakis, J. L.

J.-M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-baked apertures,”IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

J. -M. Jin, J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280–1286 (1990).
[CrossRef]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 578–580 and 590.

IEEE Trans. Antennas Propag. (6)

J. -M. Jin, J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280–1286 (1990).
[CrossRef]

J.-M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-baked apertures,”IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,”IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane-TE case,”IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,”IEEE Trans. Antennas Propag. AP-24, 870–873 (1976).
[CrossRef]

S. C. Kashyap, M. A. K. Hamid, “Diffraction characteristics of a slit in a thick conducting screen,”IEEE Trans. Antennas Propag. AP-19, 499–507 (1971).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

D. T. Auckland, R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. MTT-26, 499–505 (1978).
[CrossRef]

D. T. Auckland, R. F. Harrington, “A nonmodal formulation for electromagnetic transmission through a filled slot of arbitrary cross section in a thick conducting screen,” IEEE Trans. Microwave Theory Tech. MTT-28, 548–555 (1980).
[CrossRef]

J. Math. Phys. (1)

G. W. Lehman, “Diffraction of electromagnetic waves by planar dielectric structures. I. Transverse electric excitation,”J. Math. Phys. 11, 1522–1535 (1970).
[CrossRef]

J. Opt. Soc. Am. A (1)

Radio Sci. (1)

R. A. Hurd, Y. Hayashi, “Low-frequency scattering by a slit in a conducting plane,” Radio Sci. 15, 1171–1178 (1980).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 578–580 and 590.

W. Magnus, F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1949).

fortran Subroutines for Mathematical Applications, Version 1.1 (IMSL, Inc., 2500 City West Boulevard, Houston, Texas, 1989).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).

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Figures (10)

Fig. 1
Fig. 1

Perfectly conducting ground plane corrugated by a rectangular groove with groove width c and depth h. With the z axis passing through the center line of the groove, point P is positioned in polar coordinates (r, θ′).

Fig. 2
Fig. 2

Unit-amplitude plane wave with wave number k′ is incident upon the groove-corrugated ground plane (Fig. 1). Two incident angles are defined as ϕ and θ. The effective wave number is k = k′ cos ϕ and α0 = k sin θ, β0 = k cos θ.

Fig. 3
Fig. 3

Fast polarization case in which the field magnitudes are plotted along the interface (y = 0) between regions 1 and 2, where the parameters are set to be θ = 30°, ϕ = 0°, c = 0.3, h = 0.2, and λ = 1: (a) z component of the electric field, (b) x component of the magnetic field.

Fig. 4
Fig. 4

Slow polarization case in which the field magnitudes are plotted along the interface (y = 0) between regions 1 and 2, where the parameters are set to be θ = 30°, ϕ = 0°, c = 0.3, h = 0.2, and λ = 1: (a) x component of the electric field, (b) z component of the magnetic field.

Fig. 5
Fig. 5

Fast polarization case in which the equimagnitude contours of the fields are generated, where the parameters are θ = 30°, ϕ = 0°, c = 0.3, h = 0.2, and λ = 1: (a) z component of the electric-field, (b) x component of the magnetic field.

Fig. 6
Fig. 6

Slow polarization case in which the equimagnitude contours of the fields are generated, where the parameters are θ = 30°, ϕ = 0°, c = 0.3, h = 0.2, and λ = 1: (a) x component of the electric field, (b) z component of the magnetic field.

Fig. 7
Fig. 7

Slow polarization case in which the x component of the electric field (or the equivalent magnetic current) is computed, where the parameters are θ = 0°, ϕ = 0°, c = 1.2, h = 0.8, and λ = 1: (a) boundary-value method, (b) finite element method (solid curve) and the mode-matching method (filled circles).

Fig. 8
Fig. 8

Fast polarization case in which the far-field amplitudes are plotted against the polar angle θ′ for fixed values of θ = 30°, ϕ = 0°, λ = 1, and the following variables for 0.6 ≤ c ≤ 2.6: (a) h = 0.05, (b) h = 0.5, (c) h = 1, (d) h = 1.5.

Fig. 9
Fig. 9

Slow polarization case in which the far-field amplitudes are plotted against the polar angle θ′ for fixed values of θ = 30°, ϕ = 0°, λ = 1, and the following variables for 0.6 ≤ c ≤ 2.6: (a) h = 0.05, (b) h = 0.5, (c) h = 1, (d) h = 1.5.

Fig. 10
Fig. 10

Contour C, as shown in the complex α plane, is deformed into that of steepest descents S(θ′) to evaluate the integrals given in Eqs. (D1) and (D2). The singular point θ0 = cos−1 [cos(ϕ)sin(θ)] is implied11 along the contour C.

Equations (65)

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E = ( i sin ϕ k u x , i sin ϕ k u y , u cos 2 ϕ ) , H = ( - i ω μ u y , i ω μ u x , 0 )
E = ( i ω u y , - i ω u x , 0 ) , H = ( i sin ϕ k u x , i sin ϕ k u y , u cos 2 ϕ )
2 u + k 2 u = 0.
u ( x , y ) = exp [ i ( α 0 x - β 0 y ) ] + - r ( u ) × exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y } exp ( 2 i π u x ) d u ,
E z ( x , y ) = cos 2 ϕ ( exp [ i ( α 0 x - β 0 y ) ] + - r f ( u ) × exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y + 2 i π u x } d u ) ,
H x ( x , y ) = ( - i ) ω μ ( - i β 0 exp [ i ( α 0 x - β 0 y ) ] + - i [ k 2 - ( 2 π u ) 2 ] 1 / 2 r f ( u ) × exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y + 2 i π u x } d u ) ,
H z ( x , y ) = cos 2 ϕ ( exp [ i ( α 0 x - β 0 y ) ] + - r s ( u ) × exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y + ( 2 i π u x ) } d u ) ,
E x ( x , y ) = ( i ) ω ( - i β 0 exp [ i ( α 0 x - β 0 y ) + - i [ k 2 - ( 2 π u ) 2 ] 1 / 2 r s ( u ) × exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y + ( 2 i π u x ) } d u ) ,
E z ( x , y ) = m = 1 a m sin [ A m ( y + h ) ] sin [ m π c ( x + c 2 ) ] ,             0 x c 2
E x ( x , y ) = i ω μ k 2 m = 0 a m ( - A m ) sin [ A m ( y + h ) ] × cos [ m π c ( x + c 2 ) ] ,             0 x c 2
H x ( x , y ) = ( - i ω ) k 2 m = 1 a m A m cos [ A m ( y + h ) ] × sin [ m π c ( x + c 2 ) ] ,             0 x c / 2
H z ( x , y ) = m = 0 a m cos [ A m ( y + h ) ] cos { A m [ m π c ( x + c 2 ) ] } ,             0 x c / 2
- r ¯ f ( u ) exp ( i 2 π u x ) d u = { m = 1 ( a m sin ( A m h ) sin [ ( m π / c ) ( x + c / 2 ) ] 0 x c / 2 0 elsewhere ,
- r ¯ f ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 exp ( i 2 π u x ) d u = 2 i β 0 exp ( i α 0 x ) + m = 1 a m A m cos ( A m h ) sin [ m π c ( x + c 2 ) ] ,             0 x c 2 .
- r ¯ s ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 exp ( i 2 π u x ) d u = { m = 0 ( a m ( - A m ) sin ( A m h ) cos [ ( m π / c ) ( x + c / 2 ) ] 0 x c / 2 0 elsewhere ,
- r ¯ s ( u ) exp ( i 2 π u x ) d u = - 2 exp ( i α 0 x ) + m = 0 a m cos ( A m h ) cos [ m π c ( x + c 2 ) ] ,             0 x c 2 ,
r ¯ f ( u ) = n = 1 c n J n ( π c u ) u ,
r ¯ s ( u ) = n = 0 c n J n ( π c u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 .
n = 1 c n ϕ n ( x ) = m = 1 a m sin ( A m h ) sin [ m π c ( x + c 2 ) ] ,             0 x c 2 ,
n = 1 c n Φ n ( x ) = β 0 exp ( i α 0 x ) + m = 1 a m A m cos ( A m h ) 2 i × sin [ m π c ( x + c 2 ) ] ,             0 x c 2 ,
ϕ n ( x ) = { ( 2 / n ) cos [ n sin - 1 ( 2 x / c ) ] n = 1 , 3 , 5 , ( 2 i / n ) sin [ n sin - 1 ( 2 x / c ) ] n = 2 , 4 , 6 , ,
Φ n ( x ) = { 0 [ k 2 - ( 2 π u ) 2 ] 1 / 2 [ J n ( π c u ) / u ] cos ( 2 π u x ) d u n = 1 , 3 , 5 , i 0 [ k 2 - ( 2 π u ) 2 ] 1 / 2 [ J n ( π c u ) / u ] sin ( 2 π u x ) d u n = 2 , 4 , 6 , .
n = 0 c n ϕ n ( x ) = m = 0 a m ( - A m ) sin ( A m h ) cos [ m π c ( x + c 2 ) ] ,             0 x c / 2 ,
n = 0 c n Φ n ( x ) = exp ( i α 0 x ) + m = 0 a m cos ( A m h ) ( - 2 ) × cos [ m π c ( x + c 2 ) ] ,             0 x c / 2 ,
ϕ n ( x ) = { i sin [ n sin - 1 ( 2 x / c ) ] π [ ( c / 2 ) 2 - x 2 ] 1 / 2 n = 1 , 3 , 5 , cos [ n sin - 1 ( 2 x / c ) ] π [ ( c / 2 ) 2 - x 2 ] 1 / 2 n = 0 , 2 , 4 , ,
Φ n ( x ) = { - 0 J n ( π c u ) [ k 2 - ( 2 π u ) 2 ] 1 / 2 sin ( 2 π u x ) d u n = 1 , 3 , 5 , i 0 J n ( π c u ) [ k 2 - ( 2 π u ) 2 ] 1 / 2 cos ( 2 π u x ) d u n = 0 , 2 , 4 , .
n = k p m n c n = n = k s m n a n ,
n = k q m n c n = b m + n = k d m n a n ,
P · c ¯ = S · a ¯ ,
Q · c ¯ = b ¯ + D · a ¯ .
( Q - D · S - 1 · P ) · c ¯ = b ¯ .
E z s ( x , y ) = E z d ( x , y ) + E z r ( x , y ) = - r ¯ f ( u ) exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y } exp ( 2 i π u x ) d u - exp [ i ( α 0 x + β 0 y ) ] .
E x s ( x , y ) = E x d ( x , y ) + E x r ( x , y ) = - r ¯ s ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 exp { i [ k 2 - ( 2 π u ) 2 ] 1 / 2 y } × exp ( 2 i π u x ) d u + i β 0 exp [ i ( α 0 x + β 0 y ) ] .
E z , x d ( r , θ ) = - 2 π exp [ i ( k r - π / 4 ) ] k r P F , S ( θ ) .
P F ( θ ) = n = 1 c n J n ( k c cos θ 2 ) tan θ
P S ( θ ) = n = 0 c n J n ( k c cos θ 2 ) ( sin θ λ )
r ¯ f e ( u ) = n = 1 , 3 , 5 , c n J n ( π c u ) u ,             u 0 ,
r ¯ f o ( u ) = n = 2 , 4 , 6 , c n J n ( π c u ) u ,             u 0.
- r ¯ f ( u ) exp ( i 2 π u x ) d x = 2 0 r ¯ f e ( u ) cos ( 2 π u x ) d u + 2 i 0 r ¯ f o ( u ) sin ( 2 π u x ) d u .
0 J μ ( a u ) cos ( b u ) d u u = { 1 μ cos [ μ sin - 1 ( b a ) ] a > b a μ cos ( μ π / 2 ) μ [ b + ( b 2 - a 2 ) 1 / 2 ] μ a < b ,
0 J μ ( a u ) sin ( b u ) d u u = { 1 μ sin [ μ sin - 1 ( b a ) ] a > b a μ sin ( μ π / 2 ) μ [ b + ( b 2 - a 2 ) 1 / 2 ] μ a < b ,
- r ¯ f ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 exp ( i 2 π u x ) d u = 2 i 0 [ k 2 - ( 2 π u ) 2 ] 1 / 2 r ¯ f e ( u ) cos ( 2 π u x ) d u - 2 0 [ k 2 - ( 2 π u ) 2 ] 1 / 2 r ¯ f o ( u ) sin ( 2 π u x ) d u .
r ¯ s e ( u ) = n = 0 , 2 , 4 , c n J n ( π c u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 ,             u 0 ,
r ¯ s o ( u ) = n = 1 , 3 , 5 , c n J n ( π c u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 ,             u 0.
- r ¯ s ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 exp ( i 2 π u x ) d u = 2 0 r ¯ s e ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 cos ( 2 π u x ) d u + 2 i 0 r ¯ s o ( u ) i [ k 2 - ( 2 π u ) 2 ] 1 / 2 sin ( 2 π u x ) d u .
0 J μ ( a x ) cos ( b x ) d x = { cos [ μ sin - 1 ( b / a ) ] ( a 2 - b 2 ) 1 / 2 a > b - a μ sin ( μ π / 2 ) ( b 2 - a 2 ) 1 / 2 [ b + ( b 2 - a 2 ) 1 / 2 ] μ a < b ,
0 J μ ( a x ) sin ( b x ) d x = { sin [ μ sin - 1 ( b / a ) ] ( a 2 - b 2 ) 1 / 2 a > b a μ cos ( μ π / 2 ) ( b 2 - a 2 ) 1 / 2 [ b + ( b 2 - a 2 ) 1 / 2 ] μ a < b ,
- r ¯ s ( u ) exp ( i 2 π u x ) d x = 2 0 r ¯ s e ( u ) cos ( 2 π u x ) d u + 2 i 0 r ¯ s o ( u ) sin ( 2 π u x ) d u .
P m n = { 4 c 0 c / 2 ϕ n ( x ) sin [ m π c ( x + c 2 ) ] d x ( m + n ) even 0 ( m + n ) odd ,
q m n = { 4 c 0 c / 2 Φ n ( x ) sin [ m π c ( x + c 2 ) ] d x ( m + n ) even 0 ( m + n ) odd ,
s m n = δ m n sin ( A n h ) ,
d m n = δ m n A n cos ( A n h ) / ( 2 i ) ,
b m = i β 0 [ exp ( - i m π 2 ) sinc ( α 0 c - m π 2 ) - exp ( i m π 2 ) sinc ( α 0 c + m π 2 ) ] ,
P m n = { 4 c 0 c / 2 ϕ n ( x ) cos [ m π c ( x + c 2 ) ] d x ( m + n ) even , m 0 2 c 0 c / 2 ϕ n ( x ) d x n even , m = 0 0 ( m + n ) odd ,
q m n = { 4 c 0 c / 2 Φ n ( x ) cos [ m π c ( x + c 2 ) ] d x ( m + n ) even , m 0 2 c 0 c / 2 Φ n ( x ) d x n even , m = 0 0 ( m + n ) odd ,
s m n = - δ m n A n sin ( A n h ) ,
d m n = - δ m n cos ( A n h ) / 2 ,
b m = { exp ( - i m π 2 ) sinc ( α 0 c - m π 2 ) + exp ( i m π 2 ) sinc ( α 0 c + m π 2 ) m 0 sinc ( α 0 c 2 ) m = 0 ,
E z d ( r , θ ) = C n = 1 c n J n ( k c cos α / 2 ) cos α / λ × exp [ i k r cos ( θ - α ) ] ( - sin α λ ) d α ,
E x d ( r , θ ) = C n = 0 c n J n ( k c cos α 2 ) × exp [ i k r cos ( θ - α ) ] ( - sin α λ ) d α ,
τ = 2 exp ( i π 4 ) sin ( α - θ 2 )
E z d ( r , θ ) = - 2 exp [ i ( k r - π 4 ) ] S ( θ ) n = 1 c n J n ( k c cos α 2 ) × tan ( α ) exp ( - k r τ 2 ) [ 1 + i ( τ 2 / 2 ) ] 1 / 2 d τ ,
E x d ( r , θ ) = - 2 exp [ i ( k r - π 4 ) ] S ( θ ) n = 0 c n J n ( k c cos α 2 ) × sin ( α ) λ exp ( - k r τ 2 ) [ 1 + i ( τ 2 / 2 ) ] 1 / 2 d τ .
E z d ( r , θ ) = - ( 2 π k r ) 1 / 2 exp [ i ( k r - π 4 ) ] × n = 1 c n J n ( k c cos θ 2 ) tan ( θ )
E x d ( r , θ ) = - ( 2 π k r ) 1 / 2 exp [ i ( k r - π 4 ) ] × n = 0 c n J n ( k c cos θ 2 ) sin ( θ ) λ

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