Abstract

Scattering of an obliquely incident plane wave by a general-shaped groove engraved on a perfectly conducting plane is rigorously solved. The scattered field is represented by a Fourier-integral representation. To analytically represent the fields in a general-shaped groove, the groove is divided into L number of layers. Fields are then expressed in each layer as summations of 2D spatial harmonic fields with unknown coefficients. Matching the boundary conditions between layers provides a linear set of equations connecting all the unknown harmonic coefficients. Judicious use of Fourier transform on the equations resulting from matching boundary conditions at the groove aperture provides a series representation of the scattered field in the spectral domain with unknown harmonic coefficients of the first layer in the groove. A stable solution is obtained by solving the complete system of equations with an adaptive choice for the number of modes in each layer.

© 2007 Optical Society of America

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  1. G. S. White and J. F. Marchiando, "Scattering from a v-shaped groove in the resonance domain," Appl. Opt. 22, 2308-2312 (1983).
    [CrossRef] [PubMed]
  2. K. Toshitomi, "Scattering of an electromagnetic beam wave of rectangular grooves on a perfect conductor," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 67, 447-448 (1984).
  3. Y.-L. Kok, "Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane," J. Opt. Soc. Am. A 9, 302-311 (1992).
    [CrossRef]
  4. H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
    [CrossRef]
  5. K. Barkeshli and J. L. Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions," IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
    [CrossRef]
  6. M. K. Hinders and A. D. Yaghjian, "Dual-series solution to scattering from a semicircular channel in a ground plane," IEEE Microw. Guid. Wave Lett. 1, 239-242 (1991).
    [CrossRef]
  7. W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).
  8. H. A. Ragheb, "Electromagnetic scattering from a coaxial dielectric circular cylinder loading a semicircular gap in a ground plane," IEEE Trans. Microwave Theory Tech. 43, 1303-1309 (1995).
    [CrossRef]
  9. T. Shen, W. Dou, and Z. Sun, "Gaussian beam scattering from a semicircular channel in a ground plane," in Proceedings of the 1997 Progress in Electromagnetics Research Symposium (PIERS) (Massachusetts Institute of Technology, 1997), Vol. 16, pp. 67-85.
  10. J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
    [CrossRef]
  11. J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
    [CrossRef]
  12. H. J. Eom, T. J. Park, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A 10, 905-911 (1993).
    [CrossRef]
  13. M. S. Mirotznik, D. W. Parther, and J. N. Mait, "Boundary integral methods applied to the analysis of diffractive optical elements," J. Opt. Soc. Am. A 14, 34-43 (1997).
    [CrossRef]
  14. E. N. Glytsis, K. Hirayama, and T. K. Gaylord, "Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings," J. Opt. Soc. Am. A 14, 907-917 (1997).
    [CrossRef]
  15. D. A. Pommet, M. G. Moharam, E. Bran, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  16. P. M. Morse and H. Feshbach, Method of Theoretical Physics (McGraw-Hill, 1953), Vol. I.
  17. H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems (Springer-Verlag, 2004).
  18. K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

2002 (1)

J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
[CrossRef]

1998 (1)

J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
[CrossRef]

1997 (2)

1995 (2)

D. A. Pommet, M. G. Moharam, E. Bran, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

H. A. Ragheb, "Electromagnetic scattering from a coaxial dielectric circular cylinder loading a semicircular gap in a ground plane," IEEE Trans. Microwave Theory Tech. 43, 1303-1309 (1995).
[CrossRef]

1993 (2)

H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
[CrossRef]

H. J. Eom, T. J. Park, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

1992 (2)

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

Y.-L. Kok, "Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane," J. Opt. Soc. Am. A 9, 302-311 (1992).
[CrossRef]

1991 (1)

M. K. Hinders and A. D. Yaghjian, "Dual-series solution to scattering from a semicircular channel in a ground plane," IEEE Microw. Guid. Wave Lett. 1, 239-242 (1991).
[CrossRef]

1990 (1)

K. Barkeshli and J. L. Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions," IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

1984 (1)

K. Toshitomi, "Scattering of an electromagnetic beam wave of rectangular grooves on a perfect conductor," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 67, 447-448 (1984).

1983 (1)

Barkeshli, K.

K. Barkeshli and J. L. Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions," IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

Boerner, W. M.

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

Bran, E.

Byun, W. J.

J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
[CrossRef]

J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
[CrossRef]

Choi, T. I.

K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

Dou, W.

T. Shen, W. Dou, and Z. Sun, "Gaussian beam scattering from a semicircular channel in a ground plane," in Proceedings of the 1997 Progress in Electromagnetics Research Symposium (PIERS) (Massachusetts Institute of Technology, 1997), Vol. 16, pp. 67-85.

Eom, H. J.

H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
[CrossRef]

H. J. Eom, T. J. Park, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems (Springer-Verlag, 2004).

K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

Feshbach, H.

P. M. Morse and H. Feshbach, Method of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

Gaylord, T. K.

Glytsis, E. N.

Hinders, M. K.

M. K. Hinders and A. D. Yaghjian, "Dual-series solution to scattering from a semicircular channel in a ground plane," IEEE Microw. Guid. Wave Lett. 1, 239-242 (1991).
[CrossRef]

Hirayama, K.

Kok, Y.-L.

Mait, J. N.

Marchiando, J. F.

Mirotznik, M. S.

Moharam, M. G.

Morse, P. M.

P. M. Morse and H. Feshbach, Method of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

Myung, N. H.

J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
[CrossRef]

J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
[CrossRef]

Park, T. J.

H. J. Eom, T. J. Park, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
[CrossRef]

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

Parther, D. W.

Pommet, D. A.

Ragheb, H. A.

H. A. Ragheb, "Electromagnetic scattering from a coaxial dielectric circular cylinder loading a semicircular gap in a ground plane," IEEE Trans. Microwave Theory Tech. 43, 1303-1309 (1995).
[CrossRef]

Shen, T.

T. Shen, W. Dou, and Z. Sun, "Gaussian beam scattering from a semicircular channel in a ground plane," in Proceedings of the 1997 Progress in Electromagnetics Research Symposium (PIERS) (Massachusetts Institute of Technology, 1997), Vol. 16, pp. 67-85.

Sun, Z.

T. Shen, W. Dou, and Z. Sun, "Gaussian beam scattering from a semicircular channel in a ground plane," in Proceedings of the 1997 Progress in Electromagnetics Research Symposium (PIERS) (Massachusetts Institute of Technology, 1997), Vol. 16, pp. 67-85.

Toshitomi, K.

K. Toshitomi, "Scattering of an electromagnetic beam wave of rectangular grooves on a perfect conductor," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 67, 447-448 (1984).

Volakis, J. L.

K. Barkeshli and J. L. Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions," IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

White, G. S.

Yaghjian, A. D.

M. K. Hinders and A. D. Yaghjian, "Dual-series solution to scattering from a semicircular channel in a ground plane," IEEE Microw. Guid. Wave Lett. 1, 239-242 (1991).
[CrossRef]

Yamaguchi, Y.

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

Yoshitomi, K.

H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
[CrossRef]

H. J. Eom, T. J. Park, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

Yu, J. W.

J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
[CrossRef]

J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE Microw. Guid. Wave Lett. (1)

M. K. Hinders and A. D. Yaghjian, "Dual-series solution to scattering from a semicircular channel in a ground plane," IEEE Microw. Guid. Wave Lett. 1, 239-242 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

K. Barkeshli and J. L. Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions," IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

J. W. Yu, W. J. Byun, and N. H. Myung, "Multiple scattering from two dielectric-filled semi-circular channels in a conducting plane: TM case," IEEE Trans. Antennas Propag. 50, 1250-1253 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

H. A. Ragheb, "Electromagnetic scattering from a coaxial dielectric circular cylinder loading a semicircular gap in a ground plane," IEEE Trans. Microwave Theory Tech. 43, 1303-1309 (1995).
[CrossRef]

J. W. Yu, W. J. Byun, and N. H. Myung, "TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane," IEEE Trans. Microwave Theory Tech. 46, 1336-1339 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Radio Sci. (1)

H. J. EomT. J. Park, and K. Yoshitomi, "An analysis of transverse electric scattering from rectangular channel in a conducting plane," Radio Sci. 28, 663-673 (1993).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E (2)

W. M. Boerner, T. J. Park, H. J. Eom, and Y. Yamaguchi, "TM scattering from a dielectric-loaded semi-circular trough in a conducting plane," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 57B, 87-91 (1992).

K. Toshitomi, "Scattering of an electromagnetic beam wave of rectangular grooves on a perfect conductor," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 67, 447-448 (1984).

Other (4)

P. M. Morse and H. Feshbach, Method of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems (Springer-Verlag, 2004).

K. Yoshitomi, T. I. Choi, H. J. Eom, and T. J. Park, "TM-scattering from a rectangular channel in a conducting plane," in Proceedings of IEEE Antennas and Propagation Society International Symposium (1992), AP-S, 1992 Digest, Vol. 1, pp. 400-403.

T. Shen, W. Dou, and Z. Sun, "Gaussian beam scattering from a semicircular channel in a ground plane," in Proceedings of the 1997 Progress in Electromagnetics Research Symposium (PIERS) (Massachusetts Institute of Technology, 1997), Vol. 16, pp. 67-85.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the general-shaped groove of the scattering problem.

Fig. 2
Fig. 2

Schematic of two interlayers of the general-shaped groove.

Fig. 3
Fig. 3

Scattered field at the interface of rectangular groove with width ( 2 a ) and depth ( D ) equal to λ and λ 4 , respectively, and normal incident field, (a) Results for L = 40 , 60 , 100 have complete overlap, and (b) comparison between results from Ref. [18], MWS, and L = 40 .

Fig. 4
Fig. 4

Scattered field at the interface of rectangular groove with width ( 2 a ) and depth ( D ) equal to λ and λ 4 , respectively, and normal incident field for a different number of harmonics in the rectangular groove interlayers and number of layers L = 80 .

Fig. 5
Fig. 5

Scattered field at the interface of rectangular groove with aperture width ( 2 a ) and groove depth ( D ) equal to λ and 1.5 λ , respectively, and several values of w l for normal incident field.

Fig. 6
Fig. 6

Plot of the scattered field at the groove interface for isosceles right triangular (IRT) groove with its aperture width ( 2 a ) equal to 1.2 λ and normal incidence field for different number of groove layers.

Fig. 7
Fig. 7

Plot of the scattered field at the IRT groove with its aperture width ( 2 a ) equal to 1.2 λ , and various θ i n c .

Fig. 8
Fig. 8

Plot of the scattered field at the IRT groove with its aperture width ( 2 a ) equal to 1.2 λ , and various depths and θ i n c = 0 .

Fig. 9
Fig. 9

Plot of the scattered field at the IRT groove, its with depth, D = 2 a ( a = 0.6 λ ) , and various aperture widths for θ i n c = 0 .

Tables (2)

Tables Icon

Table 1 Modal Field Coefficients of the Scattered Fields from a Rectangular Groove with Normal Incident Plane Wave for Different Numbers of Layers

Tables Icon

Table 2 Field Coefficients of the Scattered Field from a Rectangular Groove with Normal Incident Plane Wave and Number of Layers L = 80

Equations (47)

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E y i n c ( x , z ) = exp ( j k x x + j k z z ) ,
E y r e f ( x , z ) = exp ( j k x x j k z z ) ,
E y s ( x , z ) = 1 2 π E ̃ y s ( z , ζ ) exp ( j ζ x ) d ζ .
E y s ( x , z ) = 1 2 π E ̃ y s ( ζ ) exp ( j ζ x j k 0 z ) d ζ ,
κ 0 = { k 0 2 ζ 2 , k 0 > ζ k 0 2 ζ 2 , k 0 < ζ .
H x i n c ( x , z ) = k z ω μ 0 exp ( j k x x + j k z z ) ,
H x r e f ( x , z ) = k z ω μ 0 exp ( j k x x j k z z ) ,
H x s ( x , z ) = 1 2 π ω μ 0 E ̃ y s ( ζ ) exp ( j ζ x j κ 0 z ) κ 0 d ζ .
E y l ( x , z ) = n = 1 sin β n l ( x + a l ) ( A n l e j ζ n l ( z + D l 1 ) + B n l e j ζ n l ( z + D l ) ) ,
H x l ( x , z ) = 1 ω μ r μ 0 n = 1 ζ n l sin β n l ( x + a l ) ( A n l e j ζ n l ( z + D l 1 ) B n l e j ζ n l ( z + D l ) ) .
E y L ( x , z ) = n = 1 A n L sin β n L ( x + a L ) sin ζ n L ( z + D ) ,
H x L ( x , z ) = 1 ω μ r μ 0 n = 1 ζ n L A n L sin β n L ( x + a L ) cos ζ n L ( z + D ) .
n = 1 A n L sin β n L ( x + a L ) sin ( ζ n L d ) = n = 1 sin β n L 1 ( x + a L 1 ) ( A n L 1 e j ζ n L 1 d + B n L 1 ) .
n = 1 ζ n L A n L sin β n L ( x + a L ) cos ( ζ n L d ) = n = 1 ζ n L 1 sin β n L 1 ( x + a L 1 ) ( A n L 1 e j ζ n L 1 d B n L 1 ) .
n = 1 sin β n l + 1 ( x + a l + 1 ) ( A n l + 1 + B n l + 1 e j ζ n l + 1 d ) = n = 1 sin β n l ( x + a l ) ( A n l e j ζ n l d + B n l ) .
n = 1 ζ n l + 1 sin β n l + 1 ( x + a l + 1 ) ( A n l + 1 B n l + 1 e j ζ n l + 1 d ) = n = 1 ζ n l sin β n l ( x + a l ) ( A n l e j ζ n l d B n l ) .
A p L 1 e j ζ p L 1 d + B p L 1 = 1 w L 1 n = 1 A n L sin ( ζ n L d ) I E p , n L 1 , L ,
I E p , n L 1 , L = a L b L sin β p L 1 ( x + a L 1 ) sin β n L ( x + a L ) d x .
A p L = n = 1 ζ n L 1 ζ p L I H p , n L , L 1 w L e j ζ n L 1 d cos ( ζ p L d ) A n L 1 n = 1 ζ n L 1 ζ p L I H p , n L , L 1 w L cos ( ζ p L d ) B n L 1 ,
I H p , n L , L 1 = a L b L sin β n L 1 ( x + a L 1 ) sin β p L ( x + a L ) d x .
[ exp L 1 I BA L 1 BB L 1 ] ( A L 1 B L 1 ) ( AA L I ) [ A L ] = 0 .
a a L ( p , n ) = I E p , n L 1 , L w L 1 sin ( ζ n L d ) ,
b a L 1 ( p , n ) = ζ n L 1 ζ p L I H p , n L , L 1 w L e j ζ n L 1 d cos ( ζ p L d ) ,
b b L 1 ( p , n ) = ζ n L 1 ζ p L I H p , n L , L 1 w L cos ( ζ p L d ) .
A p l e j ζ p l d + B p l = 1 w l n = 1 I E p , n l , l + 1 ( A n l + 1 + B n l + 1 e j ζ n l + 1 d ) ,
I E p , n l , l + 1 = a l + 1 b l + 1 sin β n l + 1 ( x + a l + 1 ) sin β p l ( x + a l ) d x .
A p l + 1 B p l + 1 e j ζ p l + 1 d = n = 1 ζ n l I H p , n l + 1 , l ζ p l + 1 w l + 1 ( A n l e j ζ n l d B n l ) ,
I H p , n l + 1 , l = a l + 1 b l + 1 sin β n l ( x + a l ) sin β p l + 1 ( x + a l + 1 ) d x .
[ exp l I BA l BB l ] ( A l B l ) + [ AA l + 1 AB l + 1 I exp l + 1 ] ( A l + 1 B l + 1 ) = 0 .
a a l + 1 ( p , n ) = I E p , n l , l + 1 w l ,
a b l + 1 ( p , n ) = I E p , n l , l + 1 w l e j ζ n l + 1 d ,
b a l ( p , n ) = ζ n l I H p , n l + 1 , l ζ p l + 1 w l + 1 e j ζ n l d ,
b b l ( p , n ) = ζ n l I H p , n l + 1 , l ζ p l + 1 w l + 1 ,
E y s ( x , 0 ) = { E y 1 ( x , 0 ) , x < a 0 , x > a .
E ̃ y s ( ζ ) = n = 1 ( A n 1 + B n 1 e j ζ n d ) β n 1 a 2 G n ( a ζ ) ,
G n ( ζ ) = e j ζ ( 1 ) n e j ζ ζ 2 ( β n 1 a ) 2 .
2 k z ω μ 0 exp ( j k x x ) 1 2 π ω μ 0 E ̃ y s ( ζ ) e j ζ x κ 0 d ζ = 1 ω μ r μ 0 n = 1 ζ n 1 sin β n 1 ( x + a l ) ( A n 1 B n 1 e j ζ n 1 d ) .
2 k z exp ( j k x x ) 1 2 π [ n = 1 ( A n 1 + B n 1 e j ζ n 1 d ) β n 1 a 2 G n ( a ζ ) e j ζ x κ 0 ] d ζ = 1 μ r n = 1 ζ n 1 sin β n 1 ( x + a l ) ( A n 1 B n 1 e j ζ n 1 d ) .
2 k z β m 1 a 2 G m ( k x a ) 1 2 π n = 1 a 2 β n 1 β m 1 ( A n 1 + B n 1 e j ζ n 1 d ) R m , n ( k 0 ) = 1 μ r a ζ m 1 ( A m 1 B m 1 e j ζ m 1 d ) ,
R m , n ( k 0 ) = a 2 G n ( a ζ ) G m ( a ζ ) κ 0 d ζ .
2 k z a 2 G m ( k x a ) = 1 2 π n = 1 ( a 2 β n 1 R m , n ( k 0 ) ) A n + a μ r ζ m 1 β m 1 A m 1 + 1 2 π n = 1 ( a 2 β n 1 R m , n ( k 0 ) e j ζ n 1 d ) B n 1 a μ r ζ m 1 β m 1 e j ζ m 1 d B m 1 .
[ C ] = [ CA CB ] ( A 1 B 1 ) ,
C ( m ) = 2 k z a 2 G m ( k x a ) ,
C A ( m , n ) = 1 2 π a 2 β n 1 R m , n ( k 0 ) + a μ r ζ m 1 β m 1 δ m n ,
C B ( m , n ) = 1 2 π a 2 β n 1 R m , n ( k 0 ) e j ζ n 1 d a μ r ζ m 1 β m 1 e j ζ m 1 d δ m n .
M = [ CA CB 0 0 0 0 0 0 exp 1 I AA 2 AB 2 0 0 0 0 BA 1 BB 1 I exp 2 0 0 0 0 0 0 exp 2 I AA 3 AB 3 0 0 0 0 BA 2 BB 2 I exp 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 exp L 1 I AA L 0 0 0 0 0 0 0 BA L 1 BB L 1 I ] .
N M = 2 l = 1 L 1 N l + N L ,

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