Abstract

A numerical approach is presented to analyze the two-dimensional scattering properties from a multilayered periodic dielectric structure of an arbitrary number of arbitrarily shaped unit cells. The approach is enhanced by the periodic moment method, the lattice sums technique, and the Poisson summation formula. The matrix element’s evaluation accounts for the overall coupling between layers. The choosing of lattice parameters allows designs for a wide range of applications, including the electromagnetic bandgap filtering of an E-polarized wave, which is simulated and reported here.

© 2010 Optical Society of America

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References

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  1. K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystal (Taylor & Francis, 2006).
  2. Y. W. Kong and S. T. Chew, “EBG-based dual mode resonator filter,” IEEE Microw. Wireless Compon. Lett. 14, 124–126(2004).
    [CrossRef]
  3. R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
    [CrossRef]
  4. S. Y. Lin and J. G. Fleming, “A three-dimensional optical photonic crystal,” J. Lightwave Technol. 17, 1944–1947 (1999).
    [CrossRef]
  5. R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
    [CrossRef]
  6. T. S. Chu and T. Itoh, “Generalized scattering matrix method for analysis of cascaded and offset microstrip step discontinuities,” IEEE Trans. Microw. Theory Tech. 34, 280–284 (1986).
    [CrossRef]
  7. J. Lech and R. Mazur, “Electromagnetic curtain effect and tunneling properties of multilayered periodic structures,” Antennas Wirel. Propag. Lett. 7, 201–205 (2008).
    [CrossRef]
  8. K. Yasumoto and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
    [CrossRef]
  9. G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
    [CrossRef]
  10. E. Popov and B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926–4933 (2000).
    [CrossRef]
  11. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102–110 (2000).
    [CrossRef]
  12. H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401–444.
  13. A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
    [CrossRef]
  14. R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
    [CrossRef]
  15. M. Yokota and M. Sesay, “Two-dimensional scattering of a plane wave from a periodic array of dielectric cylinders with arbitrary shape,” J. Opt. Soc. Am. A 25, 1691–1696 (2008).
    [CrossRef]
  16. K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
    [CrossRef]
  17. R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
    [CrossRef]
  18. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1991).
  19. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall1983).
  20. Y. Saad and M. H. Schultz, “GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
    [CrossRef]
  21. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics Media (IEEE, 1997).
    [CrossRef]
  22. W. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

2008 (2)

J. Lech and R. Mazur, “Electromagnetic curtain effect and tunneling properties of multilayered periodic structures,” Antennas Wirel. Propag. Lett. 7, 201–205 (2008).
[CrossRef]

M. Yokota and M. Sesay, “Two-dimensional scattering of a plane wave from a periodic array of dielectric cylinders with arbitrary shape,” J. Opt. Soc. Am. A 25, 1691–1696 (2008).
[CrossRef]

2004 (3)

K. Yasumoto and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Y. W. Kong and S. T. Chew, “EBG-based dual mode resonator filter,” IEEE Microw. Wireless Compon. Lett. 14, 124–126(2004).
[CrossRef]

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

2000 (3)

1999 (3)

S. Y. Lin and J. G. Fleming, “A three-dimensional optical photonic crystal,” J. Lightwave Technol. 17, 1944–1947 (1999).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

1988 (1)

R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

1986 (2)

T. S. Chu and T. Itoh, “Generalized scattering matrix method for analysis of cascaded and offset microstrip step discontinuities,” IEEE Trans. Microw. Theory Tech. 34, 280–284 (1986).
[CrossRef]

Y. Saad and M. H. Schultz, “GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

1985 (1)

R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
[CrossRef]

Andres, M. V.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Boria, V. E.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Bozhkov, B.

Chew, S. T.

Y. W. Kong and S. T. Chew, “EBG-based dual mode resonator filter,” IEEE Microw. Wireless Compon. Lett. 14, 124–126(2004).
[CrossRef]

Chew, W.

W. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Chu, T. S.

T. S. Chu and T. Itoh, “Generalized scattering matrix method for analysis of cascaded and offset microstrip step discontinuities,” IEEE Trans. Microw. Theory Tech. 34, 280–284 (1986).
[CrossRef]

Cocchi, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Coccioli, R.

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

Coves, A.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Fleming, J. G.

Gil, J.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Gimeno, B.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Hall, R. C.

R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Harrington, R.

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

Hikari, M.

Ikuno, H.

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401–444.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1991).

Itoh, T.

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

T. S. Chu and T. Itoh, “Generalized scattering matrix method for analysis of cascaded and offset microstrip step discontinuities,” IEEE Trans. Microw. Theory Tech. 34, 280–284 (1986).
[CrossRef]

Klock, P.

R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
[CrossRef]

Kong, Y. W.

Y. W. Kong and S. T. Chew, “EBG-based dual mode resonator filter,” IEEE Microw. Wireless Compon. Lett. 14, 124–126(2004).
[CrossRef]

Koshiba, M.

Kushta, T.

K. Yasumoto and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Lampe, R.

R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
[CrossRef]

Lech, J.

J. Lech and R. Mazur, “Electromagnetic curtain effect and tunneling properties of multilayered periodic structures,” Antennas Wirel. Propag. Lett. 7, 201–205 (2008).
[CrossRef]

Lin, S. Y.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall1983).

Ma, K.-P.

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

Mayes, P.

R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
[CrossRef]

Mazur, R.

J. Lech and R. Mazur, “Electromagnetic curtain effect and tunneling properties of multilayered periodic structures,” Antennas Wirel. Propag. Lett. 7, 201–205 (2008).
[CrossRef]

Mittra, R.

R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics Media (IEEE, 1997).
[CrossRef]

Mitzner, K. M.

R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Monorchio, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Naka, Y.

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401–444.

Pelosi, G.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics Media (IEEE, 1997).
[CrossRef]

Popov, E.

Ray, S. L.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics Media (IEEE, 1997).
[CrossRef]

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

San Blas, A. A.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

Sesay, M.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall1983).

Tsuji, Y.

Yang, F.-R.

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

Yasumoto, K.

K. Yasumoto and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystal (Taylor & Francis, 2006).

Yokota, M.

Yoshitomi, K.

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

Antennas Wirel. Propag. Lett. (1)

J. Lech and R. Mazur, “Electromagnetic curtain effect and tunneling properties of multilayered periodic structures,” Antennas Wirel. Propag. Lett. 7, 201–205 (2008).
[CrossRef]

Appl. Opt. (1)

IEEE Microw. Wireless Compon. Lett. (1)

Y. W. Kong and S. T. Chew, “EBG-based dual mode resonator filter,” IEEE Microw. Wireless Compon. Lett. 14, 124–126(2004).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

K. Yasumoto and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag. 52, 2091–2099 (2004).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (3)

R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech. 33, 734–736(1985).
[CrossRef]

T. S. Chu and T. Itoh, “Generalized scattering matrix method for analysis of cascaded and offset microstrip step discontinuities,” IEEE Trans. Microw. Theory Tech. 34, 280–284 (1986).
[CrossRef]

R. Coccioli, F.-R. Yang, K.-P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech. 47, 2123–2130 (1999).
[CrossRef]

J. Lightwave Technol. (2)

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

SIAM J. Sci. Stat. Comput. (1)

Y. Saad and M. H. Schultz, “GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

Other (7)

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics Media (IEEE, 1997).
[CrossRef]

W. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1991).

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall1983).

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401–444.

K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystal (Taylor & Francis, 2006).

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Figures (13)

Fig. 1
Fig. 1

(a) Geometry for multilayered periodic structure of arbitrary shape and (b) reference region of unit cells.

Fig. 2
Fig. 2

Model of a rectangular grating denoting grating period by d, width by l 1 and l 2 , layer thickness by h, and relative dielectric constants by ε r 1 and ε r 2 .

Fig. 3
Fig. 3

Power reflection coefficient | R 0 | 2 for a grating consisting of two dielectric slabs.

Fig. 4
Fig. 4

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics for a two-layered structure showing symmetrical and unsymmetrical material properties.

Fig. 5
Fig. 5

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics for a six-layered structure consisting of a circular cylinder with radius a = 0.3 d and layer separation h = 0.7 d .

Fig. 6
Fig. 6

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics showing the effect of increasing the number of layers for rounded square cylinders.

Fig. 7
Fig. 7

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics for a five-layered structure consisting of a rounded square cylinder with a = 0.3 d and h = 0.7 d .

Fig. 8
Fig. 8

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics for a three-layered structure consisting of a rounded square cylinder and circular cylinder arrays with radius a = 0.3 d and h = 0.7 d .

Fig. 9
Fig. 9

Power reflection coefficient | R 0 | 2 of the fundamental space harmonics for a layered structure consisting of an elliptical cylinder array with major axis a = 0.3 d , eccentricity e = 0.8 , and h = 0.7 d .

Fig. 10
Fig. 10

Filter characteristics for a six-layered structure consisting of an elliptical cylinder array with major axis a = 0.3 d , minor axis b = 0.7 a , and layer separation h = 0.7 d .

Fig. 11
Fig. 11

Filter characteristics for a six-layered structure consisting of an elliptical cylinder array with major axis a = 0.15 d , eccentricity e = 0.9 , h = d , and d = 0.79 λ 0 .

Fig. 12
Fig. 12

High-pass filter characteristics for a six-layered structure consisting of a circular cylinder array with parameters: a = 0.15 d , h = 0.5 d , and d = 0.63 λ 0 .

Fig. 13
Fig. 13

Switching characteristic for a six-layered structure consisting of a circular cylinder array with parameters a = 0.15 d and h = 0.5 d .

Equations (19)

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

J eq ( x + l d , y ) = J eq ( x , y ) exp ( j k x l d ) ,
E ( r ) = E inc ( r ) j k 0 2 4 v = 1 V l = S 0 v H 0 ( 2 ) ( k 0 ρ l v ) E ( r v ) exp ( j k x l d ) ( ε r ( r v ) 1 ) d S 0 v ,
v = 1 V n = 1 N C m n u v E n v = E m i u , u = 1 , , V ; m = 1 , , N ,
( K 11 K 12 K 1 V K 21 K 22 K 2 V K V 1 K V 2 K V V ) ( L 1 L 2 L V ) = ( M 1 M 2 M V ) ,
K u v = ( C 11 u v C 12 u v C 1 N u v C 21 u v C 22 u v C 2 N u v C N 1 u v C N 2 u v C N N u v ) ,
C m n u v = j π 2 [ ε r ( n ) 1 ] k 0 a n J 1 ( k 0 a n ) l = exp ( j k x l d ) H 0 ( 2 ) ( k 0 ρ m n u v l ) , m n ,
C n n u u = 1 + j π 2 [ ε r ( n ) 1 ] [ { k 0 a n H 1 ( 2 ) ( k 0 a n ) 2 j π } + k 0 a n J 1 ( k 0 a n ) l = exp ( j k x l d ) H 0 ( 2 ) ( k 0 ρ n n u u l ) ] , m = n , u = v ,
ρ m n u v l = ( x m u x n v l d ) 2 + ( y m u y n u ) 2 ,
l = exp ( j k x l d ) H 0 ( 2 ) ( k ρ m n u u l ) = H 0 ( 2 ) ( k ρ m n u u 0 ) + l = 1 { H 0 ( 2 ) ( k ρ m n u u l ) exp ( j k x l d ) + H 0 ( 2 ) ( k ρ m n u u l ) exp ( j k x l d ) } .
l = exp ( j k x l d ) H 0 ( 2 ) ( k 0 ρ m n u u l ) = H 0 ( 2 ) ( k 0 ρ m n u u 0 ) + S 0 ( k 0 d , θ i ) J 0 ( k 0 ρ m n u u 0 ) + 2 p = 1 S p ( k 0 d , θ i ) J p ( k 0 ρ m n u u 0 ) cos ( p ϕ ) ,
l = exp ( j k x l d ) H 0 ( 2 ) ( k 0 ρ m n u u l ) = 2 j l = exp [ j k x , l ( x m u x n v ) j κ ( k x , l ) y m u y n v ] κ ( k x , l ) d ,
k x , l = k x + l 2 π d ,
κ ( k x , l ) = k 0 2 k x , l 2 , Im ( κ ( k x , l ) ) 0 ,
E s r = l = b l + exp [ j k x , l x j κ ( k x , l ) y ] , y > 0 ,
E s t = l = b l - exp [ j k x , l x + j κ ( k x , l ) y ] , y < V h .
b l + = j π d v = 1 V n = 1 N E n v [ ε r ( n ) 1 ] k 0 a n J 1 ( k 0 a n ) 1 κ ( k x , l ) exp [ + j k x , l x n v + j κ ( k x , l ) y n v ] ,
b l = j π d v = 1 2 n = 1 N E n v [ ε r ( n ) 1 ] k 0 a n J 1 ( k 0 a n ) 1 κ ( k x , l ) exp [ + j k x , l x n v j κ ( k x , l ) y n v ] .
| R l | 2 = | b l + | 2 Re [ κ ( k x , l ) ] k 0 cos θ i ,
| T l | 2 = | δ l 0 + b l | 2 Re [ κ ( k x , l ) ] k 0 cos θ i ,

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