Abstract

We have developed an efficient iterative algorithm for electromagnetic scattering of arbitrary but relatively smooth dielectric objects. The algorithm iteratively adapts the equivalent surface currents until the electromagnetic fields inside and outside the dielectric objects match the boundary conditions. Theoretical convergence is analyzed for two examples that solve scattering of plane waves incident upon air/dielectric slabs of semi-infinite and finite thicknesses. We applied the iterative algorithm for simulation of sinusoidally-perturbed dielectric slab on one side and the method converged for such unsmooth surfaces. We next simulated the shift in radiation pattern of a 6-inch dielectric lens for different offsets of the feed antenna on the focal plane. The result is compared to that of the Geometrical Optics (GO).

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References

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd. ed. (Artech House, 2005).
  2. J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite Element Method Electromagnetics (IEEE Computer Society Press, 1998).
    [Crossref]
  3. R. F. Harrington, Field Computation by Moment Methods (Wiley IEEE Press, 1993).
    [Crossref]
  4. W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).
  5. C. M. Kelso, P. D. Flammer, J. A. DeSanto, and R. T. Collins, “Integral equations applied to wave propagation in two dimensions: modeling the tip of a near-field scanning optical microscope,” J. Opt. Soc. Am. A 18(8), 1993–2001 (2001).
    [Crossref]
  6. Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
    [Crossref]
  7. M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
    [Crossref]
  8. X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. 51(9), 2065–2071 (2009).
    [Crossref]
  9. N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
    [Crossref]
  10. C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).
  11. S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. 57(7), 2068–2075 (2009).
    [Crossref]
  12. S. B. Sorensen and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007.
  13. J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
    [Crossref]
  14. Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).
  15. A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
    [Crossref]
  16. S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006).
    [Crossref]

2010 (3)

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
[Crossref]

Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).

2009 (3)

S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. 57(7), 2068–2075 (2009).
[Crossref]

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. 51(9), 2065–2071 (2009).
[Crossref]

2006 (1)

S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006).
[Crossref]

2003 (1)

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

2001 (1)

An, X.

X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. 51(9), 2065–2071 (2009).
[Crossref]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).

Boriskin, A. V.

A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
[Crossref]

Chan, C. H.

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

Chatterjee, A.

J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite Element Method Electromagnetics (IEEE Computer Society Press, 1998).
[Crossref]

Chew, W. C.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).

Chien, H.-T.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Collins, R. T.

DeSanto, J. A.

Dou, W. B.

Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).

Elmer, T. W.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Flammer, P. D.

Godi, G.

A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
[Crossref]

Gopalsami, N.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd. ed. (Artech House, 2005).

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Wiley IEEE Press, 1993).
[Crossref]

Heifetz, A.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Jin, J. M.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).

Kelso, C. M.

Kempel, L. C.

J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite Element Method Electromagnetics (IEEE Computer Society Press, 1998).
[Crossref]

Kim, W.-G.

J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
[Crossref]

Kim, Y.-H.

J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
[Crossref]

Koehl, E. R.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Lee, J. H.

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

Li, S. Q.

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

Liao, S.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. 57(7), 2068–2075 (2009).
[Crossref]

S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006).
[Crossref]

Lin, Y.

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

Liu, J.

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

Liu, Q. H.

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

Lu, Z. Q.

X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. 51(9), 2065–2071 (2009).
[Crossref]

Michielssen, E.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).

Nosich, A. I.

A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
[Crossref]

Pontoppidan, K.

S. B. Sorensen and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007.

Raptis, A. C.

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Sauleau, R.

A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
[Crossref]

Simsek, E.

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

Song, J.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).

Sorensen, S. B.

S. B. Sorensen and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd. ed. (Artech House, 2005).

Thakur, J. P.

J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
[Crossref]

Tsang, L.

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

Vernon, R. J.

S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. 57(7), 2068–2075 (2009).
[Crossref]

S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006).
[Crossref]

Volakis, J. L.

J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite Element Method Electromagnetics (IEEE Computer Society Press, 1998).
[Crossref]

Wang, Z. X.

Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).

Xia, M. Y.

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

Zhang, B.

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009).
[Crossref]

IEEE Trans. Antennas Propag. (2)

M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. 51(6), 1142–1149 (2003).
[Crossref]

S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. 57(7), 2068–2075 (2009).
[Crossref]

Int. J. Infrared Millim. Waves (1)

Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).

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

Microwave Opt. Technol. Lett. (1)

X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. 51(9), 2065–2071 (2009).
[Crossref]

PIER (1)

J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010).
[Crossref]

Proc. SPIE (1)

N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010).
[Crossref]

Prog. Electromagn. Res. (1)

S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006).
[Crossref]

Other (7)

A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag.56, 485–492 (2008).
[Crossref]

S. B. Sorensen and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007.

C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd. ed. (Artech House, 2005).

J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite Element Method Electromagnetics (IEEE Computer Society Press, 1998).
[Crossref]

R. F. Harrington, Field Computation by Moment Methods (Wiley IEEE Press, 1993).
[Crossref]

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).

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Figures (8)

Fig. 1
Fig. 1

The equivalent current approach to the scattering problem.

Fig. 2
Fig. 2

The iterative algorithm implementation procedures (also see text for details).

Fig. 3
Fig. 3

The equivalent currents approach to the scattering problem.

Fig. 4
Fig. 4

The application of iterative algorithm on dielectric slab with sinusoidal shape on one side.

Fig. 5
Fig. 5

E// and H// of outside (air) and inside (dielectric) on the sinusoidal surface of the dielectric slab along x-direction after 7 iterations. Both amplitude and real part are shown.

Fig. 6
Fig. 6

The application of iterative algorithm on horn-lens radiation pattern simulation.

Fig. 7
Fig. 7

E// and H// of outside (air) and inside (dielectric) on the convex surface of the dielectric lens along x-direction after 7 iterations. Only amplitude is shown.

Fig. 8
Fig. 8

Radiation patterns for different antenna offsets along x-direction: from 0λ to 4λ, 1λ increment: lines are results from iterative algorithm and dots are GO results.

Tables (1)

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Table 1 Memory and Computation Time (C. T.) for Different Methods

Equations (31)

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J s = 2 n ^ × H ; M s = 2 n ^ × E
H = H i + H sca ; E = E i + E sca
A = S d s J s exp j k | r r s | 4 π | r r s |
F = ɛ S d s M s exp j k | r r s | 4 π | r r s |
E sca = E A + E F H sca = H A + H F
E F ( r ) = × F ( r ) ɛ H F ( r ) = 1 j ω μ × E F ( r ) = × × F ( r ) j ω μ ɛ
H A ( r ) = × A ( r ) μ E A ( r ) = 1 j ω ɛ × H A ( r ) = × × A ( r ) j ω μ ɛ
J s = n ^ × H = n ^ × [ H i + H sca ] = n ^ × H o M s = n ^ × E = n ^ × [ E i + E sca ] = n ^ × E o
E sca , k = E sca , k 1 + n ^ × 1 Y [ H i , k + H sca , k 1 H o , k 1 ] H sca , k = H sca , k 1 n ^ × 1 Z [ E i , k + E sca , k 1 E o , k 1 ]
J s , k = J s , k 1 + 1 Z [ E i + E sca , k 1 E o , k 1 ] M s , k = M s , k 1 + 1 Y [ H i + H sca , k 1 H o , k 1 ]
[ M s , k J s , k ] = t [ M i η 0 η r J i ] + C = [ M s , k 1 J s , k 1 ]
C = = [ α = β = θ = γ = ] ; [ M i J i ] = [ E i × n ^ n ^ × H i ]
[ M s J s ] = t k = 0 [ α = β = θ = γ = ] k [ M i η 0 η r J i ] + [ α = β = θ = γ = ] [ M s , 1 J s , 1 ]
[ α = β = θ = γ = ] = Q = [ Λ = 1 0 0 Λ = 2 ] Q = 1 [ α = β = θ = γ = ] k = Q = [ Λ = 1 k 0 0 Λ = 2 k ] Q = 1 [ α = β = θ = γ = ] = 0 =
[ M s J s ] = t Q = [ 1 1 Λ = 1 0 0 1 1 Λ = 2 ] Q = 1 [ M i η 0 η r J i ]
M s , k 1 = ( z ^ ) × E k 1 = y ^ E k 1 ; J s , k 1 = ( z ^ ) × H k 1 = x ^ H k 1
E sca , k = x ^ [ 1 2 E k 1 η 0 2 H k 1 ] ; H sca , k = y ^ [ 1 2 η 0 E k 1 + 1 2 H k 1 ]
E o , k = x ^ [ 1 2 E k 1 + η r 2 H k 1 ] ; H o , k = y ^ [ 1 2 η r E k 1 + 1 2 H k 1 ]
[ M s J s ] = t [ M i η 0 η r J i ]
[ E H ] = t [ E i η 0 η r H i ]
M s , k 1 ± = y ^ E k 1 ; J s , k 1 ± = x ^ H k 1
E sca , k ± = x ^ [ E k 1 ± 2 ± η 0 2 H k 1 ± ( E k 1 2 ± η 0 2 H k 1 ) exp j k d ] H sca , k ± = y ^ [ ± E k 1 ± 2 η 0 + H k 1 ± 2 + ( E k 1 2 η 0 H k 1 2 ) exp j k d ]
E o , k ± = x ^ [ E k 1 ± 2 η r 2 H k 1 ± + ( E k 1 2 ± η r 2 H k 1 ) exp j k r d ] H o , k ± = y ^ [ E k 1 ± 2 η r + H k 1 ± 2 + ( ± E k 1 2 η r + H k 1 2 ) exp j k r d ]
[ M s , k + J s , k + M s , k J s , k ] = t [ M i + η 0 η r J i + M i η 0 η r J i ] + C = [ M s , k 1 + J s , k 1 + M s , k 1 J s , k 1 ]
C = = [ 0 0 𝒫 η 0 + 𝒫 r η r 2 Y = 𝒫 + 𝒫 r 2 Y = 0 0 𝒫 + 𝒫 r 2 Z = η 0 𝒫 + η r 𝒫 r 2 Z = 𝒫 η 0 + 𝒫 r η r 2 Y = 𝒫 + 𝒫 r 2 Y = 0 0 𝒫 + 𝒫 r 2 Z = η 0 𝒫 + η r 𝒫 r 2 Z = 0 0 ]
[ Λ = 1 0 0 Λ = 2 ] = [ r 𝒫 r 0 0 0 0 r 𝒫 r 0 0 0 0 r 𝒫 0 0 0 0 r 𝒫 ]
E x ( x , y ) = exp x 2 + y 2 w 2
z ( x , y ) = 10 λ + λ cos ( k 40 x ) cos ( k 40 y )
E x ( x , y ) = exp ( x x o f f ) 2 + y 2 w 2
z ( x , y ) = R 2 x 2 y 2 ( R h )
F R n 1 = 2 R = 6.8

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