Abstract

Finite-difference time-domain (FDTD) algorithm with a new method of plane wave excitation is used to investigate the RCS (Radar Cross Section) characteristics of targets over layered half space. Compare with the traditional excitation plane wave method, the calculation memory and time requirement is greatly decreased. The FDTD calculation is performed with a plane wave incidence, and the RCS of far field is obtained by extrapolating the currently calculated data on the output boundary. However, methods available for extrapolating have to evaluate the half space Green function. In this paper, a new method which avoids using the complex and time-consuming half space Green function is proposed. Numerical results show that this method is in good agreement with classic algorithm and it can be used in the fast calculation of scattering and radiation of targets over layered half space.

© 2014 Optical Society of America

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References

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  1. Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
    [CrossRef]
  2. X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
    [CrossRef]
  3. J. Li, L. X. Guo, Y. C. Jiao, and K. Li, “Investigation on wide-band scattering of a 2-D target above 1-D randomly rough surface by FDTD method,” Opt. Express 19(2), 1091–1100 (2011).
    [CrossRef] [PubMed]
  4. H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).
  5. L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
    [CrossRef]
  6. R. M. Shubair and Y. L. Chow, “A closed-form solution of vertical dipole antennas above a dielectric half-space,” IEEE Trans. Antenn. Propag. 41(12), 1737–1741 (1993).
    [CrossRef]
  7. R. E. Arias and A. A. Maradudin, “Scattering of a surface plasmon polariton by a localized dielectric surface defect,” Opt. Express 21(8), 9734–9756 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  11. E. Danicki, “Green’s function for anisotropic dielectric halfspace,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 643–643 (1988).
    [CrossRef] [PubMed]
  12. E. Pan, “Three-dimensional Green's functions in an anisotropic half-space with general boundary conditions,” J APPL MECH-T ASME 70(1), 101–110 (2003).
    [CrossRef]
  13. J. T. Johnson, “A study of the four-path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30(2), 130–134 (2001).
    [CrossRef]
  14. P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
    [CrossRef]
  15. S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
    [CrossRef]
  16. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  17. J. A. Kong, Electromagnetic Wave Theory (John Wiley, 1986).
  18. J. R. Wait, Electromagnetic Wave Theory (Harper & Row, 1985).
  19. L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
    [CrossRef]
  20. Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
    [CrossRef]

2014

2013

R. E. Arias and A. A. Maradudin, “Scattering of a surface plasmon polariton by a localized dielectric surface defect,” Opt. Express 21(8), 9734–9756 (2013).
[CrossRef] [PubMed]

L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
[CrossRef]

2012

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

2011

J. Li, L. X. Guo, Y. C. Jiao, and K. Li, “Investigation on wide-band scattering of a 2-D target above 1-D randomly rough surface by FDTD method,” Opt. Express 19(2), 1091–1100 (2011).
[CrossRef] [PubMed]

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

2010

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

2006

L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
[CrossRef]

2005

S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
[CrossRef]

2003

E. Pan, “Three-dimensional Green's functions in an anisotropic half-space with general boundary conditions,” J APPL MECH-T ASME 70(1), 101–110 (2003).
[CrossRef]

2001

J. T. Johnson, “A study of the four-path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30(2), 130–134 (2001).
[CrossRef]

1999

T. J. Cui and W. C. Chew, “Fast evaluation of sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans-Geo & Remo 37(2), 887–910 (1999).

1996

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

1993

R. M. Shubair and Y. L. Chow, “A closed-form solution of vertical dipole antennas above a dielectric half-space,” IEEE Trans. Antenn. Propag. 41(12), 1737–1741 (1993).
[CrossRef]

1988

E. Danicki, “Green’s function for anisotropic dielectric halfspace,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 643–643 (1988).
[CrossRef] [PubMed]

Arias, R. E.

Baron, J. E.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

Cao, L.

L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
[CrossRef]

Chen, R.

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

Chen, R. S.

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

Chew, W. C.

T. J. Cui and W. C. Chew, “Fast evaluation of sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans-Geo & Remo 37(2), 887–910 (1999).

Chow, Y. L.

R. M. Shubair and Y. L. Chow, “A closed-form solution of vertical dipole antennas above a dielectric half-space,” IEEE Trans. Antenn. Propag. 41(12), 1737–1741 (1993).
[CrossRef]

Cui, T. J.

T. J. Cui and W. C. Chew, “Fast evaluation of sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans-Geo & Remo 37(2), 887–910 (1999).

Danicki, E.

E. Danicki, “Green’s function for anisotropic dielectric halfspace,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 643–643 (1988).
[CrossRef] [PubMed]

Ding, D. Z.

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

Ding, J. J.

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

Ge, D. B.

L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
[CrossRef]

Guo, L. X.

Gurrola, E. M.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

Hu, X. Q.

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

Hu, Y. Q.

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

Jiang, Z. N.

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

Jiao, Y. C.

Johnson, J. T.

J. T. Johnson, “A study of the four-path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30(2), 130–134 (2001).
[CrossRef]

Karami, H. R.

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

Ketterson, J.

L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
[CrossRef]

Kosmas, P.

S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
[CrossRef]

Li, J.

Li, K.

Luan, L.

L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
[CrossRef]

Maradudin, A. A.

Moini, R.

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

Pan, E.

E. Pan, “Three-dimensional Green's functions in an anisotropic half-space with general boundary conditions,” J APPL MECH-T ASME 70(1), 101–110 (2003).
[CrossRef]

Rappaport, C. M.

S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
[CrossRef]

Sadeghi, S. H. H.

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

Sheng, Y. J.

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

Sheshyekani, K.

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

Shubair, R. M.

R. M. Shubair and Y. L. Chow, “A closed-form solution of vertical dipole antennas above a dielectric half-space,” IEEE Trans. Antenn. Propag. 41(12), 1737–1741 (1993).
[CrossRef]

Sievert, P.

L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
[CrossRef]

Simpson, R. A.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

Tyler, G. L.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

Wei, B.

L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
[CrossRef]

Winton, S. C.

S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
[CrossRef]

Wong, P. B.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

Xu, R. W.

Xu, Y.

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

Zhang, C.

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

Zhu, M. M.

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

IEEE T Electromagn C

H. R. Karami, R. Moini, S. H. H. Sadeghi, and K. Sheshyekani, “Transient response of nonlinearly loaded antennas above a lossy dielectric half-space: A modified MoM-AOM approach,” IEEE T Electromagn C 54(4), 922–930 (2012).

IEEE Trans-Geo & Remo

T. J. Cui and W. C. Chew, “Fast evaluation of sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans-Geo & Remo 37(2), 887–910 (1999).

IEEE Trans. Antenn. Propag.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, “A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surface,” IEEE Trans. Antenn. Propag. 44(4), 504–514 (1996).
[CrossRef]

S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antenn. Propag. 53(5), 1721–1728 (2005).
[CrossRef]

R. M. Shubair and Y. L. Chow, “A closed-form solution of vertical dipole antennas above a dielectric half-space,” IEEE Trans. Antenn. Propag. 41(12), 1737–1741 (1993).
[CrossRef]

Z. N. Jiang, Y. Xu, Y. J. Sheng, and M. M. Zhu, “Efficient analyzing EM scattering of objects above a lossy half-space by the combined MLQR/MLSSM,” IEEE Trans. Antenn. Propag. 59(12), 4609–4614 (2011).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

E. Danicki, “Green’s function for anisotropic dielectric halfspace,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 643–643 (1988).
[CrossRef] [PubMed]

IET Microw Antenna Propag.

Y. Q. Hu, J. J. Ding, D. Z. Ding, and R. S. Chen, “Analysis of electromagnetic scattering from dielectric objects above a lossy half-space by multiresolution preconditioned multilevel fast multiple algorithm,” IET Microw Antenna Propag. 4(2), 232–239 (2010).
[CrossRef]

J APPL MECH-T ASME

E. Pan, “Three-dimensional Green's functions in an anisotropic half-space with general boundary conditions,” J APPL MECH-T ASME 70(1), 101–110 (2003).
[CrossRef]

Microw. Opt. Technol. Lett.

J. T. Johnson, “A study of the four-path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30(2), 130–134 (2001).
[CrossRef]

X. Q. Hu, C. Zhang, Y. Xu, D. Z. Ding, and R. Chen, “An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space,” Microw. Opt. Technol. Lett. 54(3), 573–577 (2012).
[CrossRef]

New J. Phys.

L. Luan, P. Sievert, and J. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half-space,” New J. Phys. 8(11), 264 (2006).
[CrossRef]

Opt. Express

Waves Random Complex

L. Cao, B. Wei, and D. B. Ge, “Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space,” Waves Random Complex 23(4), 446–460 (2013).
[CrossRef]

Other

K. Li, Electromagnetic Fields in Stratified Media (ZJU Press and Springer, 2009).

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

J. A. Kong, Electromagnetic Wave Theory (John Wiley, 1986).

J. R. Wait, Electromagnetic Wave Theory (Harper & Row, 1985).

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Figures (17)

Fig. 1
Fig. 1

The scattering problem of targets over a half space.

Fig. 2
Fig. 2

The traditional FDTD computational domain.

Fig. 3
Fig. 3

The new FDTD computational region.

Fig. 4
Fig. 4

Modeling diagram of Tomahawk cruise missile.

Fig. 5
Fig. 5

The total and scattered fields region.

Fig. 6
Fig. 6

Introduction of incident wave .

Fig. 7
Fig. 7

Introduction of multi-direction incident wave.

Fig. 8
Fig. 8

Far radiation of a vertical electric dipole over half space.

Fig. 9
Fig. 9

Variation of reflected wave polarizing angle with incident angle.

Fig. 10
Fig. 10

Near field distribution pattern of free space.

Fig. 11
Fig. 11

Near field distribution pattern of half space.

Fig. 12
Fig. 12

Comparison of Green function method and proposed method.

Fig. 13
Fig. 13

RCS of the cylinder over a lossy half space.

Fig. 14
Fig. 14

Schematic diagram of a missile over a layered half space.

Fig. 15
Fig. 15

RCS of xoz plane.

Fig. 16
Fig. 16

RCS of yoz plane.

Fig. 17
Fig. 17

Variation of reflection coefficient with incident angle.

Tables (1)

Tables Icon

Table 1 The Memory Usage and Time Step for Missile

Equations (10)

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

J= e n × Η i J m = e n × E i
E rh = e rh exp[ i(ωtπ/2 + ψ rh ) ] E rv = e rv exp[ i(ωtπ/2 + ψ rv ) ]
ψ r = ψ rv ψ rh e rh =| R TM | E 0 cosα e rv =| R TE | E 0 sinα
α= 1 2 tan 1 ( 2 e h e v cos ψ r e h 2 e v 2 )
E θ ( P )={ [ 1exp( ikΔ ) R TM ]sin θ 0 cosθcos( φ 0 φ ) [ 1+exp( ikΔ ) R TM ]cos θ 0 sinθ } iωμIl exp( ikr ) 4πr
E φ ( P )=sin θ 0 sin( φ 0 φ )[ 1+exp( ikΔ ) R TE ]iωμIl exp( ikr ) 4πr
exp( ikr ) 4πr exp[ ik( r 0 r r ^ 0 ) ] 4π r 0
Δ= h( 1+cos2θ ) / cosθ
E φ ( P )=ikIl{ [ 1exp( ikΔ ) R TM ]sin θ 0 cosθcos( φ 0 φ ) [ 1+exp( ikΔ ) R TM ]cos θ 0 sinθ } exp( ikr ) 4πr
E θ ( P )=ikIlsin θ 0 sin( φ 0 φ )[ 1+exp( ikΔ ) R TE ] exp( ikr ) 4πr

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