Abstract

The 1-D range profiles are suitable features for target identification and target discrimination because they provide discriminative information on the geometry of the target. To resolve features of the buried target, the contribution from individual scattering centers of the buried target in the range profiles need to be identified. Thus, the study of complex scattering mechanisms from which the range profiles are produced is of great importance. In order to clearly establish the relationship between the range profile characteristics and the complicated electromagnetic (EM) scattering mechanisms, such as reflections and diffractions, a buried cuboid possessing straight edges is chosen as the buried target in this paper. By performing an inverse discrete Fourier transform (IDFT) on the wideband backscattered field data computed with an accurate and fast EM method, the 1-D range profiles of the buried cuboid is successfully simulated. The simulated range profiles provide information about the position and scattering strength of the cuboid’s scattering centers along the range direction. Meanwhile, a predicted distribution of the scattering centers is quantitatively calculated for the buried cuboid based on the ray path computation. Good agreement has been found between simulated and predicted locations of the range profiles. Validation for amplitudes of the range profiles is further provided in the research. Both the peak amplitudes and locations of the range profiles could be understood and analyzed based on the knowledge of the scattering mechanisms. The formation of the 1-D range profiles has been revealed clearly from the full analysis of the scattering mechanisms and contributions. The problem has been solved for both near and far field regions. Finally, the buried depth and the characteristic size of the object are reasonably deduced from the simulated range profiles.

© 2011 OSA

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  1. S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).
  2. S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
    [CrossRef]
  3. S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
    [CrossRef]
  4. N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
    [CrossRef]
  5. V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
    [CrossRef]
  6. F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
    [CrossRef]
  7. K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
    [CrossRef]
  8. S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
    [CrossRef]
  9. Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
    [CrossRef]
  10. L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
    [CrossRef]
  11. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  12. T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
    [CrossRef]

2010 (1)

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

2009 (1)

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

2007 (1)

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

2006 (1)

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

2003 (2)

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

2001 (1)

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

1999 (1)

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

1997 (1)

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

1996 (1)

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

1995 (1)

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Aydiner, A. A.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Boix, R. R.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Carin, L.

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Chen, H.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Chew, W. C.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Cui, T. J.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Deng, F.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Frezza, F.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Geng, N.

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

He, S.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

He, S. Y.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Hu, W.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Hu, W. D.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Kim, H. T.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Kim, K. T.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Le, F. H.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Losada, V.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Martinelli, P.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Medina, F.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Pajewski, L.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Ressler, M. A.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Schettini, G.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Seo, D. K.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Sturgess, K.

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

Vitebskiy, S.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Xiao, B. X.

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

Ye, X. B.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Yu, W.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Yu, W. X.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Zhang, Y. H.

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Zhu, G.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Zhu, G. Q.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

Zhuang, L.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

IEE Proc., Radar Sonar Navig. (1)

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (1)

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

IEEE Trans. Antenn. Propag. (3)

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

IEEE Trans. Geosci. Rem. Sens. (5)

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Other (1)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

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Figures (15)

Fig. 1
Fig. 1

Coordinate system of a 3-D dielectric object buried in a half-space.

Fig. 2
Fig. 2

The lateral distribution of far region range profiles along x axis.

Fig. 3
Fig. 3

Range profiles of the buried object along θ = ϕ = 0.

Fig. 4
Fig. 4

The equivalent multilayered media.

Fig. 5
Fig. 5

Range profiles of the buried object along θ = π /4 , ϕ = 0.

Fig. 6
Fig. 6

Ray paths of the buried object.

Fig. 7
Fig. 7

Range profiles of the same object in free space along θ = sin 1 ( sin ( π / 4 ) / 2 ) , ϕ = 0.

Fig. 8
Fig. 8

Range profiles of the buried object when the dipole locates at ( 0 m , 0 m , 2 m ) .

Fig. 9
Fig. 9

Range profiles of the buried object when the dipole locates at ( 4 m , 0 m , 2 m ) .

Fig. 10
Fig. 10

Range profiles of the buried object when the dipole locates at ( 0 m , 4 m , 2 m ) .

Fig. 11
Fig. 11

Ray paths of the buried object.

Fig. 12
Fig. 12

The lateral distribution of near region range profiles along x ' axis.

Fig. 13
Fig. 13

Range profiles of the buried object when the dipole locates at ( 4.5255 m , 4.5255 m , 2 m ) .

Fig. 14
Fig. 14

Ray paths of the buried object.

Fig. 15
Fig. 15

Range profiles of the buried object when the dipole locates at ( 4 m , 0 m , 2 m ) .

Tables (4)

Tables Icon

Table 1 Simulated and Expected Range Locations of a Buried Object (Unit: m)

Tables Icon

Table 2 Simulated and Expected Range Locations of Fig. 8, Fig. 9 and Fig. 10 (Unit: m)

Tables Icon

Table 3 Simulated and Expected Range Locations of Fig. 13 (Unit: m)

Tables Icon

Table 4 Simulated and expected range locations of Fig. 15 (Unit: m)

Equations (13)

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

τ = ( b + c ) / c 1 + ( a + d ) / c 2
τ o = ( R T + R R ) / c 2
Δ R 2 = ( ( τ τ o ) c 2 ) / 2
Δ R 1 / Δ R 2 = c 1 / c 2
Δ R 1 = Δ R 2 / ( ε 1 / ε 2 ) = ( ( ε 1 / ε 2 ) ( b + c ) + a + d R T R R ) / ( 2 ε 1 / ε 2 )
R P = | Г 43 T M R ˜ 32 T M R 34 T M R ˜ 32 T M Г 34 T M e j 2 k 3 h Г 43 T M R ˜ 32 T M Г 34 T M e j 2 k 3 h |      = | R 34 T M R 32 T M + R 21 T M e j 2 k 2 L z 1 + R 32 T M R 21 T M e j 2 k 2 L z |
Δ R ( path 1 ) = ( a + ε 1 b R ) / ε 1
Δ R ( path 3 ) = ( a + ε 1 ( b + c ) R ) / ε 1
a = R + ( L x / 2 h tan θ ) sin θ b = h / cos θ , c = h , sin θ = ε 1 sin θ
E 1 i n c ( r , r ) = η 2 k 2 G ¯ e e 12 ( r , r ) e ^ x
Δ R ( path 1 ) = ( a + ε 1 b R ) / ε 1
Δ R ( path 3 ) = ( a + ε 1 ( b + c ) R ) / ε 1
a sin θ 1 + b sin θ 1 = L x / 2 + L b = h / cos θ 1 , a = H / cos θ 1 , c = h , sin θ = ε 1 sin θ

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