Abstract

A fast, exact numerical method based on the method of moments (MM) is developed to calculate the scattering from an object below a randomly rough surface. Déchamps et al. [J. Opt. Soc. Am. A 23, 359 (2006) ] have recently developed the PILE (propagation-inside-layer expansion) method for a stack of two one-dimensional rough interfaces separating homogeneous media. From the inversion of the impedance matrix by block (in which two impedance matrices of each interface and two coupling matrices are involved), this method allows one to calculate separately and exactly the multiple-scattering contributions inside the layer in which the inverses of the impedance matrices of each interface are involved. Our purpose here is to apply this method for an object below a rough surface. In addition, to invert a matrix of large size, the forward-backward spectral acceleration (FB-SA) approach of complexity O(N) (N is the number of unknowns on the interface) proposed by Chou and Johnson [Radio Sci. 33, 1277 (1998) ] is applied. The new method, PILE combined with FB-SA, is tested on perfectly conducting circular and elliptic cylinders located below a dielectric rough interface obeying a Gaussian process with Gaussian and exponential height autocorrelation functions.

© 2008 Optical Society of America

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References

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  1. G. Videen and D. Hgo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70-78 (1997).
    [CrossRef]
  2. T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. I. TM case,” J. Opt. Soc. Am. A 6, 1270-1280 (1989).
    [CrossRef]
  3. T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. II. TE case,” J. Opt. Soc. Am. A 8, 1986-1990 (1991).
    [CrossRef]
  4. A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from dielectric structures with two rough boundaries using the small perturbation method,” IEEE Trans. Geosci. Remote Sens. 44, 2102-2114 (2006).
    [CrossRef]
  5. Y. Altuncu, A. Yapar, and I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44, 1435-1443 (2006).
    [CrossRef]
  6. D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
    [CrossRef]
  7. A. Soubret, G. Berginc, and C. Bourrely, “Backscattering enhancement of an electromagnetic wave scattered by two-dimensional rough layers,” J. Opt. Soc. Am. A 18, 2778-2788 (2001).
    [CrossRef]
  8. I. M. Fuks and A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253-272 (2000).
    [CrossRef]
  9. Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315-335 (1999).
    [CrossRef]
  10. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations, Series on Remote Sensing (Wiley, 2001).
    [CrossRef]
  11. P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates,” Appl. Opt. 33, 512-523 (1994).
    [CrossRef] [PubMed]
  12. P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158-162 (1998).
    [CrossRef]
  13. A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A 12, 1298-1309 (1995).
    [CrossRef]
  14. A. Madrazo and M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859-1866 (1997).
    [CrossRef]
  15. X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207-227 (2003).
    [CrossRef]
  16. Chih-Hao Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
    [CrossRef]
  17. H. Ye and Y.-Q. Jin, “A hybrid analytic-numerical algorithm of scattering from an object above a rough surface,” IEEE Trans. Geosci. Remote Sens. 45, 1174-1180 (2007).
    [CrossRef]
  18. M. El-Shenawee, “Polarimetric scattering from two-layered two dimensional random rough surfaces with and without buried objects,” IEEE Trans. Geosci. Remote Sens. 42, 67-76 (2001).
    [CrossRef]
  19. M. El-Shenawee, C. Rappaport, and M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077-3084 (2001).
    [CrossRef]
  20. J. T. Johnson and R. J. Burkholder, “A study of scattering from an object below a rough surface,” IEEE Trans. Geosci. Remote Sens. 42, 59-66 (2004).
    [CrossRef]
  21. L. Tsang, C. H. Chang, and H. Sangani, “A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large scale random rough surface problems: TM case,” Electron. Lett. 29, 1666-1667 (1993).
    [CrossRef]
  22. L. Tsang, C. H. Chang, H. Sangani, A. Ishimaru, and P. Phu, “A banded matrix iterative approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 29, 1185-1200 (1993).
    [CrossRef]
  23. D. Holliday, L. L. DeRaad Jr., and G. J. St-Cyr, “Forward-backward: a new method for computing low-grazing angle scattering,” IEEE Trans. Antennas Propag. 44, 1199-1206 (1995).
    [CrossRef]
  24. H. T. Chou and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from rough surfaces with the forward-backward method,” Radio Sci. 33, 1277-1287 (1998).
    [CrossRef]
  25. D. Torrungrueng, J. T. Johnson, and H. T. Chou, “Some issues related to the novel spectral acceleration method for the fast computation of radiation/scattering from one-dimensional extremely large scale quasi-planar structures,” Radio Sci. 37, 1-20 (2002).
    [CrossRef]
  26. N. Déchamps, N. De Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23, 359-369 (2006).
    [CrossRef]
  27. N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to an updated BMIA/CAG approach,” IEEE Trans. Antennas Propag. 55, 2790-2802 (2007).
    [CrossRef]
  28. N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: propagation-inside-layer expansion method combined to the forward-backward novel spectral acceleration,” IEEE Trans. Antennas Propag. 55, 3576-3586 (2007).
    [CrossRef]
  29. Eric I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1988).
    [CrossRef]
  30. A. Iodice, “Forward-backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antennas Propag. 50, 901-911 (2002).
    [CrossRef]
  31. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: extended PILE method combined to FB-SA,” submitted to Waves Random Media 31 January 2008, TWRM-2008-007 (2008).

2008

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: extended PILE method combined to FB-SA,” submitted to Waves Random Media 31 January 2008, TWRM-2008-007 (2008).

2007

H. Ye and Y.-Q. Jin, “A hybrid analytic-numerical algorithm of scattering from an object above a rough surface,” IEEE Trans. Geosci. Remote Sens. 45, 1174-1180 (2007).
[CrossRef]

N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to an updated BMIA/CAG approach,” IEEE Trans. Antennas Propag. 55, 2790-2802 (2007).
[CrossRef]

N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: propagation-inside-layer expansion method combined to the forward-backward novel spectral acceleration,” IEEE Trans. Antennas Propag. 55, 3576-3586 (2007).
[CrossRef]

2006

Chih-Hao Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from dielectric structures with two rough boundaries using the small perturbation method,” IEEE Trans. Geosci. Remote Sens. 44, 2102-2114 (2006).
[CrossRef]

Y. Altuncu, A. Yapar, and I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44, 1435-1443 (2006).
[CrossRef]

N. Déchamps, N. De Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23, 359-369 (2006).
[CrossRef]

2004

J. T. Johnson and R. J. Burkholder, “A study of scattering from an object below a rough surface,” IEEE Trans. Geosci. Remote Sens. 42, 59-66 (2004).
[CrossRef]

2003

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207-227 (2003).
[CrossRef]

2002

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

D. Torrungrueng, J. T. Johnson, and H. T. Chou, “Some issues related to the novel spectral acceleration method for the fast computation of radiation/scattering from one-dimensional extremely large scale quasi-planar structures,” Radio Sci. 37, 1-20 (2002).
[CrossRef]

A. Iodice, “Forward-backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antennas Propag. 50, 901-911 (2002).
[CrossRef]

2001

2000

I. M. Fuks and A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253-272 (2000).
[CrossRef]

1999

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315-335 (1999).
[CrossRef]

1998

H. T. Chou and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from rough surfaces with the forward-backward method,” Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158-162 (1998).
[CrossRef]

1997

1995

A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A 12, 1298-1309 (1995).
[CrossRef]

D. Holliday, L. L. DeRaad Jr., and G. J. St-Cyr, “Forward-backward: a new method for computing low-grazing angle scattering,” IEEE Trans. Antennas Propag. 44, 1199-1206 (1995).
[CrossRef]

1994

1993

L. Tsang, C. H. Chang, and H. Sangani, “A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large scale random rough surface problems: TM case,” Electron. Lett. 29, 1666-1667 (1993).
[CrossRef]

L. Tsang, C. H. Chang, H. Sangani, A. Ishimaru, and P. Phu, “A banded matrix iterative approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 29, 1185-1200 (1993).
[CrossRef]

1991

1989

1988

Eric I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

Appl. Opt.

Electron. Lett.

L. Tsang, C. H. Chang, and H. Sangani, “A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large scale random rough surface problems: TM case,” Electron. Lett. 29, 1666-1667 (1993).
[CrossRef]

IEEE Trans. Antennas Propag.

D. Holliday, L. L. DeRaad Jr., and G. J. St-Cyr, “Forward-backward: a new method for computing low-grazing angle scattering,” IEEE Trans. Antennas Propag. 44, 1199-1206 (1995).
[CrossRef]

N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to an updated BMIA/CAG approach,” IEEE Trans. Antennas Propag. 55, 2790-2802 (2007).
[CrossRef]

N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: propagation-inside-layer expansion method combined to the forward-backward novel spectral acceleration,” IEEE Trans. Antennas Propag. 55, 3576-3586 (2007).
[CrossRef]

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

Chih-Hao Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

A. Iodice, “Forward-backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antennas Propag. 50, 901-911 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

H. Ye and Y.-Q. Jin, “A hybrid analytic-numerical algorithm of scattering from an object above a rough surface,” IEEE Trans. Geosci. Remote Sens. 45, 1174-1180 (2007).
[CrossRef]

M. El-Shenawee, “Polarimetric scattering from two-layered two dimensional random rough surfaces with and without buried objects,” IEEE Trans. Geosci. Remote Sens. 42, 67-76 (2001).
[CrossRef]

J. T. Johnson and R. J. Burkholder, “A study of scattering from an object below a rough surface,” IEEE Trans. Geosci. Remote Sens. 42, 59-66 (2004).
[CrossRef]

A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from dielectric structures with two rough boundaries using the small perturbation method,” IEEE Trans. Geosci. Remote Sens. 44, 2102-2114 (2006).
[CrossRef]

Y. Altuncu, A. Yapar, and I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44, 1435-1443 (2006).
[CrossRef]

J. Acoust. Soc. Am.

Eric I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

J. Electromagn. Waves Appl.

L. Tsang, C. H. Chang, H. Sangani, A. Ishimaru, and P. Phu, “A banded matrix iterative approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 29, 1185-1200 (1993).
[CrossRef]

J. Opt. Soc. Am. A

A. Soubret, G. Berginc, and C. Bourrely, “Backscattering enhancement of an electromagnetic wave scattered by two-dimensional rough layers,” J. Opt. Soc. Am. A 18, 2778-2788 (2001).
[CrossRef]

M. El-Shenawee, C. Rappaport, and M. Silevitch, “Monte Carlo simulations of electromagnetic wave scattering from a random rough surface with three-dimensional penetrable buried object: mine detection application using the steepest-descent fast multipole method,” J. Opt. Soc. Am. A 18, 3077-3084 (2001).
[CrossRef]

N. Déchamps, N. De Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23, 359-369 (2006).
[CrossRef]

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158-162 (1998).
[CrossRef]

G. Videen and D. Hgo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70-78 (1997).
[CrossRef]

A. Madrazo and M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859-1866 (1997).
[CrossRef]

T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. I. TM case,” J. Opt. Soc. Am. A 6, 1270-1280 (1989).
[CrossRef]

T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. II. TE case,” J. Opt. Soc. Am. A 8, 1986-1990 (1991).
[CrossRef]

A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A 12, 1298-1309 (1995).
[CrossRef]

Prog. Electromagn. Res.

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207-227 (2003).
[CrossRef]

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315-335 (1999).
[CrossRef]

Radio Sci.

H. T. Chou and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from rough surfaces with the forward-backward method,” Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

D. Torrungrueng, J. T. Johnson, and H. T. Chou, “Some issues related to the novel spectral acceleration method for the fast computation of radiation/scattering from one-dimensional extremely large scale quasi-planar structures,” Radio Sci. 37, 1-20 (2002).
[CrossRef]

Waves Random Media

I. M. Fuks and A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253-272 (2000).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: extended PILE method combined to FB-SA,” submitted to Waves Random Media 31 January 2008, TWRM-2008-007 (2008).

Other

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations, Series on Remote Sensing (Wiley, 2001).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Top, illustration of the integration contours of the 2D Green function C g , and of that used for the SA algorithm C δ . Bottom, physical interpretation of C δ in the spatial domain.

Fig. 3
Fig. 3

Illustration of steps for the product B ¯ f v 2 (left) and B ¯ b v 2 (right), where v 2 = v 2 f + v 2 b . First, the elements of domain (a) are multiplied by v 2 exactly. Then elements (b) and (c) are multiplied by v 2 with those of (b) using the SA algorithm and those of (c) exactly as for (a).

Fig. 4
Fig. 4

Comparison of the scattering coefficient (without object) in dB scale with that obtained from a direct LU inversion versus the scattering angle θ s . Top, TE case; bottom, TM case. In the legends, the order P FB and the RRE in linear scale of the scattering coefficients are given. The parameters are the same as in Table 1 with ϵ r 2 = 2 + 0.01 j , θ i = 0 ° and σ z = 2 λ 0 [(a) case].

Fig. 5
Fig. 5

Comparison of the field ψ + and its normal derivative ψ + n + (without object) on the surface computed from FB-SA with those obtained from a direct LU inversion versus the normalized abscissa x λ 0 and for the TE case. The parameters are the same as in Fig. 4 with x d 0 = 3 L c and the order P FB is taken from Table 1.

Fig. 6
Fig. 6

Modulus ψ + of the rough surface versus the normalized abscissa x λ 0 for the TE case. θ i = 0 ° , L c = 2 λ 0 , σ z = λ 0 , ϵ r 2 = 2 + 0.01 j , N + = 1200 , L + = 120 λ 0 , g = L + 6 , N = 126 , h c = 4 λ 0 , and a = 2 λ 0 . Top, PILE method. Middle, PILE+FB method with P FB = 7 . Bottom, PILE + FB - SA method with x d 0 = 3 L c . In each subfigure, the order of PILE and the corresponding RRE are given in the legend. In addition, the results computed from a direct LU inversion are plotted.

Fig. 7
Fig. 7

Modulus of the radiated field computed from the fields on the rough surface and the object versus the normalized abscissa x λ 0 and the normalized height h λ 0 for the TE polarization and for different orders P PILE . The parameters are the same as in Fig. 6, but σ z = 0.5 λ 0 , L + = 80 λ 0 , θ i = 30 ° , and g = L + 4 .

Fig. 8
Fig. 8

Comparison of the scattering coefficient in dB scale with that obtained from a direct LU inversion versus the scattering angle θ s . The parameters are the same as in Fig. 6.

Fig. 9
Fig. 9

Comparison of the RRE over the scattering coefficient versus the normalized RMS height σ z λ 0 for the TE polarization. The order P PILE is obtained from Table 2, from which the (a) case is considered.

Fig. 10
Fig. 10

Same variation as in Fig. 9 but for the TM polarization.

Fig. 11
Fig. 11

RRE over the scattering coefficient of PILE + FB - SA versus the normalized distance x d 0 L c for the TE and TM polarizations. The parameters are the same as in Fig. 6. The horizontal lines indicate the values of RRE of PILE+FB obtained from Figs. 9, 10 with σ z = λ 0 .

Fig. 12
Fig. 12

CPU time versus the number of samples N + on the rough surface. The parameters are the same as in Fig. 6 with { P FB = 7 , P PILE = 3 } and { P FB = 6 , P PILE = 3 } for the TE and TM polarizations (Tables 1, 2), respectively. The number of unknowns is 2 N + + N = 2 N + + 126 .

Tables (3)

Tables Icon

Table 1 Order P FB for a Single Rough Dielectric Surface (without Object) and for the TE and TM Polarizations

Tables Icon

Table 2 Order P PILE for a Circular Cylinder below a Rough Dielectric Surface and for the TE and TM Polarizations

Tables Icon

Table 3 Order P PILE for an Elliptic Cylinder below a Rough Dielectric Surface and for the TE and TM Polarizations

Equations (37)

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

ψ i ( r ) = exp ( j k i r ) exp ( ( x + z tan θ i ) 2 g 2 ) exp [ j w ( r ) k i r ] ,
Z ¯ X = s ,
X T = [ X + T X T ] ,
s T = [ s + T s T ] = [ s + T 0 T ] ,
Z ¯ = [ Z ¯ + Z ¯ Z ¯ ± Z ¯ ] .
X + = [ p = 0 p = P PILE M ¯ c p ] Z ¯ + 1 s + = p = 0 p = P PILE Y + ( p ) ,
{ Y + ( 0 ) = Z ¯ + 1 s + for p = 0 Y + ( p ) = M ¯ c Y + ( p 1 ) for p > 0 ,
M ¯ c = Z ¯ + 1 Z ¯ Z ¯ 1 Z ¯ ± .
X = Z ¯ 1 Z ¯ ± X + .
Z ¯ + = [ A ¯ B ¯ C ¯ D ¯ ] ,
{ A ¯ d u 1 f + B ¯ d u 2 f = v 1 A ¯ f u 1 B ¯ f u 2 C ¯ d u 1 f + D ¯ d u 2 f = v 2 C ¯ f u 1 D ¯ f u 2 ,
{ A ¯ d u 1 b + B ¯ d u 2 b = A ¯ b u 1 B ¯ b u 2 C ¯ d u 1 b + D ¯ d u 2 b = C ¯ b u 1 D ¯ b u 2 .
{ A d m , m u 1 f m + B d m , m u 2 f m = v 1 m n = 1 n = m 1 ( A f m , n u 1 n + B f m , n u 2 n ) C d m , m u 1 f m + D d m , m u 2 f m = v 2 m n = 1 n = m 1 ( C f m , n u 1 n + D f m , n u 2 n ) .
g 1 ( r + m , r + n ) = j 4 π C g exp { j K 1 [ ( x + m x + n ) cos ϕ + ( z + m z + n ) sin ϕ ] } d ϕ ,
E f m , ( s ) = j Δ x + 4 π n = 1 m N s 1 u 2 n C δ exp { j K 1 [ ( x + m x + n ) cos ϕ + ( z + m z + n ) sin ϕ ] } d ϕ = j Δ x + 4 π C δ F m ( ϕ ) exp ( j K 1 z + m sin ϕ ) d ϕ = j Δ x + 4 π exp ( j δ ) p = Q p = + Q F m ( ϕ p ) exp ( j K 1 z + m sin ϕ p ) Δ ϕ
F m ( ϕ ) = n = 1 m N s 1 u 2 n exp { j K 1 [ ( x + m x + n ) cos ϕ z + n sin ϕ ] } .
F m ( ϕ ) = F m 1 ( ϕ ) exp ( j K 1 Δ x + cos ϕ ) + u 2 m N s 1 exp { j K 1 [ ( N s + 1 ) Δ x + cos ϕ z + m N s 1 sin ϕ ] } .
g 1 ( r + m , r + n ) n + = K 1 4 π C exp { j K 1 [ ( x + m x + n ) cos ϕ + ( z + m z + n ) sin ϕ ] } ( γ + n cos ϕ sin ϕ ) d ϕ ,
F m ( ϕ ) = n = 1 m N s 1 u 1 n exp { j K 1 [ ( x + m x + n ) cos ϕ + ( z + m z + n ) sin ϕ ] } × ( γ + n cos ϕ sin ϕ ) ,
F m ( ϕ ) = F m 1 ( ϕ ) exp ( j K 1 Δ x + cos ϕ ) + ( γ + m N s 1 cos ϕ sin ϕ ) u 1 m N s 1 exp { j K 1 [ ( N s + 1 ) Δ x + cos ϕ z + m N s 1 sin ϕ ] } .
ϕ max = min ( ϕ s , max 2 + ϕ s , max 2 4 + b s K 1 r d 0 tan δ 0 ; π 2 ) b s = 6 ,
tan δ = min ( 4 a s K 1 r d 0 ϕ s , max 2 ; 1 ) a s = 5 ,
ϕ s , max = arctan [ z + max z + min x d 0 ] ,
r d 0 = x d 0 2 + ( z + max z + min ) 2 ,
{ 8 N + N + 4 N 2 ( matrix - vector products ) + [ 28 ( 2 Q + 1 ) ( N + N s ) + 4 N + N s ] P FB ( inversion of Z ¯ + ) } P PILE + [ 28 ( 2 Q + 1 ) ( N + N s ) + 4 N + N s ] P FB ( order 0 , inversion of Z ¯ + ) + ( 2 N ) 3 3 ( initialization : inversion of Z ¯ )
RRE : r e = norm ( X X LU ) norm ( X LU ) ,
σ s ( θ i , θ s ) = ψ 2 8 π K 0 g cos θ i [ 1 1 + 2 tan 2 θ i 2 K 0 2 g 2 cos 2 θ i ] ,
ψ = Σ + { ψ + n + 1 + γ + 2 j K 0 ψ + [ γ + sin θ s cos θ s ] } e j k s r d x + ,
M ¯ FB = ( Z ¯ + , d + Z ¯ + , f ) 1 Z ¯ + , f ( Z ¯ + , d + Z ¯ + , b ) 1 Z ¯ + , b
ψ rad ( r ) = p = ± s p Σ p [ ψ p ( r p ) g p ( r p , r ) n p g p ( r p , r ) ψ p ( r p ) n p ] d Σ p ,
Z ¯ + = [ A ¯ + B ¯ + C ¯ + ρ 21 D ¯ + ] , Z ¯ = [ A ¯ B ¯ C ¯ ρ 32 D ¯ ] ,
Z ¯ ± = [ A ¯ ± ρ 21 B ¯ ± 0 ¯ 0 ¯ ] , Z ¯ = [ 0 ¯ 0 ¯ A ¯ B ¯ ] ,
A + m , n = { j K 1 Δ x + 4 H 1 ( 1 ) ( K 1 r + n r + m ) r + n r m + [ γ + n ( x + n x + m ) ( z + n z + m ) ] for m n , + 1 2 Δ x + 4 π ( γ + m ) 1 + ( γ + m ) 2 for m = n ,
B + m , n = j Δ x + α + n 4 { 1 + 2 j π ln ( 0.164 K 1 α + m Δ x + ) for n = m H 0 ( 1 ) ( K 1 r + n r + m ) for n m ,
A ± m , n = j K 2 Δ x + 4 H 1 ( 1 ) ( K 2 r + n r m ) r + n r m [ γ + n ( x + n x m ) ( z + n z m ) ] .
B ± m , n = j α + n Δ x + 4 H 0 ( 1 ) ( K 2 r + n r m ) .
{ TE case : Z ¯ = B ¯ , Z ¯ = [ 0 ¯ B ¯ ] , X ψ n TM case : Z ¯ = A ¯ , Z ¯ = [ 0 ¯ A ¯ ] , X ψ ] .

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