Abstract

In this Letter, a fast red rigorous numerical method, based on the method of moments, is developed to calculate the scattering from an object above a rough surface for three-dimensional problems (3D). G. Kubické has recently developed the extended propagation-inside-layer expansion (E-PILE) method to calculate the scattering from an object above a rough surface for two-dimensional problems. This method allows us to calculate separately and exactly the interactions between the object and the rough surface. The purpose of this paper is to extend the E-PILE method to a 3D problem. In addition, to invert a matrix of large size, the forward-backward (FB) method is applied to calculate the local interactions on the rough surface.

© 2012 Optical Society of America

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References

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  1. N. Déchamps, N. De Beaucoudrey, C. Bourlier, and S. Toutain, J. Opt. Soc. Am. A 23, 359 (2006).
    [CrossRef]
  2. C. Bourlier, G. Kubické, and N. Déchamps, J. Opt. Soc. Am. A 25, 891 (2008).
    [CrossRef]
  3. G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).
  4. G. Kubické and C. Bourlier, IEEE Trans. Antennas Propag. 59, 189 (2011).
    [CrossRef]
  5. J. T. Johnson, IEEE Trans. Antennas Propag. 50, 1361 (2002).
    [CrossRef]
  6. H. Ye and Y.-Q. Jin, Radio Science 43, RS3005 (2008).
    [CrossRef]
  7. B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
    [CrossRef]
  8. L.-X. Guo, J. Li, and H. Zeng, J. Opt. Soc. Am. A 26, 2383 (2009).
    [CrossRef]
  9. W.-J. Ji and C.-M. Tong, Prog. Electromagn. Res. 33, 317 (2011).
    [CrossRef]
  10. D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

2011 (2)

G. Kubické and C. Bourlier, IEEE Trans. Antennas Propag. 59, 189 (2011).
[CrossRef]

W.-J. Ji and C.-M. Tong, Prog. Electromagn. Res. 33, 317 (2011).
[CrossRef]

2009 (2)

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

L.-X. Guo, J. Li, and H. Zeng, J. Opt. Soc. Am. A 26, 2383 (2009).
[CrossRef]

2008 (3)

C. Bourlier, G. Kubické, and N. Déchamps, J. Opt. Soc. Am. A 25, 891 (2008).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).

H. Ye and Y.-Q. Jin, Radio Science 43, RS3005 (2008).
[CrossRef]

2006 (1)

2002 (1)

J. T. Johnson, IEEE Trans. Antennas Propag. 50, 1361 (2002).
[CrossRef]

1995 (1)

D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

Bourlier, C.

G. Kubické and C. Bourlier, IEEE Trans. Antennas Propag. 59, 189 (2011).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, J. Opt. Soc. Am. A 25, 891 (2008).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).

N. Déchamps, N. De Beaucoudrey, C. Bourlier, and S. Toutain, J. Opt. Soc. Am. A 23, 359 (2006).
[CrossRef]

Cui, T.

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

De Beaucoudrey, N.

Déchamps, N.

DeRaad, L. L.

D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

Guan, B.

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

Guo, L.-X.

Holliday, D.

D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

Ji, W.-J.

W.-J. Ji and C.-M. Tong, Prog. Electromagn. Res. 33, 317 (2011).
[CrossRef]

Jin, Y.-Q.

H. Ye and Y.-Q. Jin, Radio Science 43, RS3005 (2008).
[CrossRef]

Johnson, J. T.

J. T. Johnson, IEEE Trans. Antennas Propag. 50, 1361 (2002).
[CrossRef]

Kubické, G.

G. Kubické and C. Bourlier, IEEE Trans. Antennas Propag. 59, 189 (2011).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, J. Opt. Soc. Am. A 25, 891 (2008).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).

Li, J.

Saillard, J.

G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).

St-Cyr, G. J.

D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

Tong, C.-M.

W.-J. Ji and C.-M. Tong, Prog. Electromagn. Res. 33, 317 (2011).
[CrossRef]

Toutain, S.

Ye, H.

H. Ye and Y.-Q. Jin, Radio Science 43, RS3005 (2008).
[CrossRef]

Zeng, H.

Zhang, J.

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

Zhou, X.

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

G. Kubické and C. Bourlier, IEEE Trans. Antennas Propag. 59, 189 (2011).
[CrossRef]

J. T. Johnson, IEEE Trans. Antennas Propag. 50, 1361 (2002).
[CrossRef]

D. Holliday, L. L. DeRaad, and G. J. St-Cyr, IEEE Trans. Antennas Propag. 44, 1199 (1995).

IEEE Trans. Geosci. Remote Sens. (1)

B. Guan, J. Zhang, X. Zhou, and T. Cui, IEEE Trans. Geosci. Remote Sens. 47, 3399 (2009).
[CrossRef]

J. Opt. Soc. Am. A (3)

Prog. Electromagn. Res. (1)

W.-J. Ji and C.-M. Tong, Prog. Electromagn. Res. 33, 317 (2011).
[CrossRef]

Radio Science (1)

H. Ye and Y.-Q. Jin, Radio Science 43, RS3005 (2008).
[CrossRef]

Waves Random Complex Media (1)

G. Kubické, C. Bourlier, and J. Saillard, Waves Random Complex Media 18, 495 (2008).

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Figures (4)

Fig. 1.
Fig. 1.

Forward and backward processes.

Fig. 2.
Fig. 2.

Convergence of E-PILE for the case of plate S1 above a rough surface S2. The parameters are S1=2λ0×2λ0, S2=6λ0×6λ0, the distance between them D=4λ0, RMS height σz=0.2λ0, correlation lengths Lcx=Lcy=1λ0, and the incidence angle θi=0°.

Fig. 3.
Fig. 3.

Convergence of E-PILE+FB for the same scenario as in Fig. 2.

Fig. 4.
Fig. 4.

Comparison of the computing time of the two methods versus the length of the rough surface.

Equations (12)

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Ei(R)=e^ieiki·R,Hi(R)=1η0k^i×Ei(R),
n^1×Hi(R)+n^1×S1J1(R1)×R1G(R1,R)dSlocal interactions+n^1×S2J2(R2)×R2G(R2,R)dScoupling interactions=12J1(R),
n^2×Hi(R)+n^2×S2J2(R2)×R2G(R2,R)dSlocal interactions+n^2×S1J1(R1)×R1G(R1,R)dScoupling interactions=12J2(R),
Z¯X=b,
XjT=[J(Rj1)J(RjNj)Scattererj],
bjT=[n^jHi(Rj1)n^jHi(RjNj)Scattererj].
Z¯=[Z¯1Z¯21Z¯12Z¯2].
X1=[p=0p=PE-PILEM¯c,1p]Z¯11(b1Z¯21Z¯21b2)=p=0p=PE-PILEX¯1(p),
{X1(0)=Z¯11(b1Z¯21Z¯21b2)forp=0X1(p)=M¯c,1X¯1(p1)forp>0.
M¯c,1=Z¯11Z¯21Z¯21Z¯12.
Z¯dUf=b2Z¯f(Ub+Uf),
Z¯dUb=Z¯b(Ub+Uf).

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