Abstract

The theoretical solution for scattering by an arbitrary configuration of closely spaced parallel infinite cylinders buried in a lossy half space is presented in this paper. The refractive index and permeability of the half space and cylinders are complex in general. Each cylinder is radially stratified with a distinct complex refractive index and permeability. The incident radiation is an arbitrarily polarized plane wave propagating in the plane normal to the axes of the cylinders. Analytic solutions are derived for the electric and magnetic fields and the Poynting vector of backscattered radiation emerging from the half space. Numerical examples are presented to illustrate the application of the scattering solution to calculate backscattering from a lossy half space containing multiple homogeneous and radially stratified cylinders at various depths and different angles of incidence.

© 2013 Optical Society of America

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References

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  1. T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. 1. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [CrossRef]
  2. 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]
  3. T. C. Rao and R. Barakat, “Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization,” Appl. Opt 3, 1023–1047 (1994).
    [CrossRef]
  4. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinder in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2456 (1996).
    [CrossRef]
  5. S. C. Lee and J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
    [CrossRef]
  6. S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350–1361 (1999).
    [CrossRef]
  7. S. J. Radzevocois and J. L. Daniels, “Ground penetrating radar polarization and scattering from cylinders,” J. Appl. Geophys. 45, 111–125 (2000).
    [CrossRef]
  8. M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
    [CrossRef]
  9. H. Jia and K. Yasumoto, “Scattering and absorption characteristics of multilayered gratings embedded in a dielectric slab,” Int. J. Infrared Millim. Waves 26, 1111–1126 (2005).
    [CrossRef]
  10. S. C. Lee, “Optical extinction by closely spaced parallel cylinders inside a finite dielectric slab,” J. Opt. Soc. Am. A 23, 2219–2232 (2006).
    [CrossRef]
  11. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
    [CrossRef]
  12. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
    [CrossRef]
  13. S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
    [CrossRef]
  14. J. D. Kanellopoulos and N. E. Buris, “Scattering from conducting cylinders embedded in a lossy medium,” Int. J. Electron. 57, 391–401 (1984).
    [CrossRef]
  15. F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
    [CrossRef]
  16. S. C. Lee, “Scattering by a radially stratified infinite cylinder buried in an absorbing half space,” J. Opt. Soc. Am. A 30, 565–572 (2013).
    [CrossRef]
  17. S. C. Lee, “Scattering by closely spaced parallel nonhomogeneous cylinders in an absorbing medium,” J. Opt. Soc. Am. A 28, 1812–1819 (2011).
    [CrossRef]
  18. User’s Manual: FORTRAN Subroutines for Mathematical Applications (IMSL, Inc., 1991).

2013

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

S. C. Lee, “Scattering by a radially stratified infinite cylinder buried in an absorbing half space,” J. Opt. Soc. Am. A 30, 565–572 (2013).
[CrossRef]

2011

2010

2009

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

2006

2005

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

H. Jia and K. Yasumoto, “Scattering and absorption characteristics of multilayered gratings embedded in a dielectric slab,” Int. J. Infrared Millim. Waves 26, 1111–1126 (2005).
[CrossRef]

2000

S. J. Radzevocois and J. L. Daniels, “Ground penetrating radar polarization and scattering from cylinders,” J. Appl. Geophys. 45, 111–125 (2000).
[CrossRef]

1999

1998

1996

1994

T. C. Rao and R. Barakat, “Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization,” Appl. Opt 3, 1023–1047 (1994).
[CrossRef]

1991

1989

1984

J. D. Kanellopoulos and N. E. Buris, “Scattering from conducting cylinders embedded in a lossy medium,” Int. J. Electron. 57, 391–401 (1984).
[CrossRef]

1981

S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
[CrossRef]

Ali, S. M.

S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
[CrossRef]

Barakat, R.

Borghi, R.

Buris, N. E.

J. D. Kanellopoulos and N. E. Buris, “Scattering from conducting cylinders embedded in a lossy medium,” Int. J. Electron. 57, 391–401 (1984).
[CrossRef]

Daniels, J. L.

S. J. Radzevocois and J. L. Daniels, “Ground penetrating radar polarization and scattering from cylinders,” J. Appl. Geophys. 45, 111–125 (2000).
[CrossRef]

di Vico, M.

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Frezza, F.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinder in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2456 (1996).
[CrossRef]

Gori, F.

Grzesik, J. A.

Jia, H.

H. Jia and K. Yasumoto, “Scattering and absorption characteristics of multilayered gratings embedded in a dielectric slab,” Int. J. Infrared Millim. Waves 26, 1111–1126 (2005).
[CrossRef]

Kanellopoulos, J. D.

J. D. Kanellopoulos and N. E. Buris, “Scattering from conducting cylinders embedded in a lossy medium,” Int. J. Electron. 57, 391–401 (1984).
[CrossRef]

Lee, S. C.

Mahmoud, S. F.

S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
[CrossRef]

Pajewski, L.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Ponti, C.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

Radzevocois, S. J.

S. J. Radzevocois and J. L. Daniels, “Ground penetrating radar polarization and scattering from cylinders,” J. Appl. Geophys. 45, 111–125 (2000).
[CrossRef]

Rao, T. C.

Santarsiero, M.

Schettini, G.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinder in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2456 (1996).
[CrossRef]

Tedeschi, N.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

Wait, J. R.

S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
[CrossRef]

Yasumoto, K.

H. Jia and K. Yasumoto, “Scattering and absorption characteristics of multilayered gratings embedded in a dielectric slab,” Int. J. Infrared Millim. Waves 26, 1111–1126 (2005).
[CrossRef]

Appl. Opt

T. C. Rao and R. Barakat, “Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization,” Appl. Opt 3, 1023–1047 (1994).
[CrossRef]

IEEE Geosci. Remote Sens. Lett.

F. Frezza, G. Schettini, L. Pajewski, C. Ponti, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[CrossRef]

IEEE Trans. Antennas Propag.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Int. J. Electron.

J. D. Kanellopoulos and N. E. Buris, “Scattering from conducting cylinders embedded in a lossy medium,” Int. J. Electron. 57, 391–401 (1984).
[CrossRef]

Int. J. Infrared Millim. Waves

H. Jia and K. Yasumoto, “Scattering and absorption characteristics of multilayered gratings embedded in a dielectric slab,” Int. J. Infrared Millim. Waves 26, 1111–1126 (2005).
[CrossRef]

J. Appl. Geophys.

S. J. Radzevocois and J. L. Daniels, “Ground penetrating radar polarization and scattering from cylinders,” J. Appl. Geophys. 45, 111–125 (2000).
[CrossRef]

J. Opt. Soc. Am. A

S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350–1361 (1999).
[CrossRef]

S. C. Lee and J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[CrossRef]

T. C. Rao and R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. 1. 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]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinder in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2456 (1996).
[CrossRef]

S. C. Lee, “Optical extinction by closely spaced parallel cylinders inside a finite dielectric slab,” J. Opt. Soc. Am. A 23, 2219–2232 (2006).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[CrossRef]

S. C. Lee, “Scattering by closely spaced parallel nonhomogeneous cylinders in an absorbing medium,” J. Opt. Soc. Am. A 28, 1812–1819 (2011).
[CrossRef]

S. C. Lee, “Scattering by a radially stratified infinite cylinder buried in an absorbing half space,” J. Opt. Soc. Am. A 30, 565–572 (2013).
[CrossRef]

Radio Sci.

S. F. Mahmoud, S. M. Ali, and J. R. Wait, “Electromagnetic scattering from buried cylindrical inhomogeneity inside a lossy earth,” Radio Sci. 16, 1285–1298 (1981).
[CrossRef]

Other

User’s Manual: FORTRAN Subroutines for Mathematical Applications (IMSL, Inc., 1991).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic diagram showing an arbitrary configuration of cylinders buried in a lossy half space.

Fig. 2.
Fig. 2.

Configuration of cylinders for numerical illustration.

Fig. 3.
Fig. 3.

Backscattering from homogeneous cylinders buried in a lossless and lossy half space: (a) TM mode and (b) TE mode.

Fig. 4.
Fig. 4.

Effect of angle of incidence on backscattering: (a) TM mode, (b) TE mode, and (c) phase difference.

Fig. 5.
Fig. 5.

Backscattering from hollow magnetic cylinders buried at different depths in a lossy magnetic half space: (a) TM mode, (b) TE mode, and (c) phase difference.

Equations (46)

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(EjHj)=(Ej+Hj+)+kjN(EkjsHkjs)+k=1N(Ekjr+Hkjr+)+(EjscaHjsca),
[E⃗1iH⃗1i/m¯1]=ik1[αvi(sinθ1e⃗x+cosθ1e⃗y)+αuie⃗zαui(sinθ1e⃗x+cosθ1e⃗y)+αvie⃗z]exp(ik⃗iR⃗P),
[E⃗1rH⃗1r/m¯1]=ik1[αvr(sinθ1e⃗xcosθ1e⃗y)+αure⃗zαur(sinθ1e⃗xcosθ1e⃗y)+αvre⃗z]exp(ik⃗rR⃗P),
[E⃗2+H⃗2+/m¯2]=ik2[αv+(sinθ2e⃗x+cosθ2e⃗y)+αu+e⃗zαu+(sinθ2e⃗x+cosθ2e⃗y)+αv+e⃗z]exp(ik⃗+R⃗P),
k⃗i=k1(cosθ1e⃗xsinθ1e⃗y),
k⃗r=k1(cosθ1e⃗x+sinθ1e⃗y),
k⃗+=k2(cosθ2e⃗xsinθ2e⃗y),
(E⃗j+H⃗j+/m¯2)=k2εjn=(i)nexp[in(γjP+θ2)][ink2RjPJn(αv+αu+)e⃗jR+Jn(αv+αu+)e⃗jγ+iJn(αu+αv+)e⃗z],
(E⃗kscaH⃗ksca/m¯2)=k2s=(i)sexp(isγkP)[isk2RkPHs(aksbks)e⃗kR+Hs(aksbks)e⃗kγiHs(bksaks)e⃗z],
(E⃗kjscaH⃗kjsca/m¯2)=k2n=s=(i)nexp(inγjP)Gksjn[ink2RjPJn(aksbks)e⃗jR+Jn(aksbks)e⃗jγiJn(bksaks)e⃗z],
Gksjn=(i)snexp[i(sn)γkj]Hsn(k2Rjk).
exp(inγkP)Hn(k2RkP)=(i)nπexp[insin1(ky/k2)]k22ky2exp(ik22ky2|xkP|ikyykP)dky.
[E⃗ksH⃗ks/m¯2]=ik2[Yk(kye⃗x+k22ky2e⃗y)k2Xke⃗zXk(kye⃗x+k22ky2e⃗y)k2Yke⃗z]exp[ik22ky2(xPxk)ikyykP]dky,
(XkYk)=k2πs=(1)sexp[issin1(ky/k2)]k22ky2(bksaks).
[E⃗kr+H⃗kr+/m¯2]=ik2[Ykr+(kye⃗xk22ky2e⃗y)k2Xkr+e⃗zXkr+(kye⃗xk22ky2e⃗y)k2Ykr+e⃗z]exp[ik22ky2(xP+xk)ikyykP]dky,
[E⃗ktH⃗kt/m¯1]=ik1k2[Ykt(kye⃗x+γ1e⃗y)/k1Xkte⃗zXkt(kye⃗x+γ1e⃗y)/k1Ykte⃗z]exp(iγ1xPik22ky2xkikyykP)dky,
exp[ik22ky2(xP+xk)ikyykP]=exp(i2xkk22ky2)s=(i)sexp(isγkP)Js(ksRkP),
(E⃗kjr+H⃗kjr+/m¯2)=k2n,s=(i)nexp(inγjP)·[ink2RjPJn(aksRv,ksv,jnbksRu,ksu,jn)e⃗jR+Jn(aksRv,ksv,jnbksRu,ksu,jn)e⃗jγiJn(bksRu,ksu,jnaksRv,ksv,jn)e⃗z],
Rψ,ksψ,jn=(1)sπr21ψexp[i(n+s)sin1(ky/k2)]k22ky2exp(ix¯jkk22ky2ikyykj)dky
k22ky2={ρexp(iϑ/2),Re(k22)>|ky|2k2rk2iexp(iπ/2),Re(k22)=±kyiρexp(iϑ/2),Re(k22)<|ky|,
ρ=[Re(k22)ky2]2+(2k2rk2i)2,
ϑ=tan1[2k2rk2i/|Re(k22)ky2|].
Rψ,ksψ,jnafψ(ky)exp{x¯jkρcos(ϑ/2)i[x¯jkρsin(ϑ/2)kyykj]}dky+exp(2k2rk2ix¯jk)0a[fψ(ky)exp(ikyykj)+fψ(ky)exp(ikyykj)]dky+afψ(ky)exp{x¯jkρsin(ϑ/2)i[x¯jkρcos(ϑ/2)+kyykj]}dky,
(E⃗jqincH⃗jqinc/m¯jq)=ikjqn=(i)nexp(inγjP)(nkjqRjPJnAjn(q)e⃗jR+iJnAjn(q)e⃗jγ+JnBjn(q)e⃗znkjqRjPJnBjn(q)e⃗jRiJnBjn(q)e⃗jγ+JnAjn(q)e⃗z),
(E⃗jqscaH⃗jqsca/m¯jq)=ikjqn=(i)nexp(inγjP)(nkjqRjPHnajn(q)e⃗jRiHnajn(q)e⃗jγHnbjn(q)e⃗znkjqRjPHnbjn(q)e⃗jR+iHnbjn(q)e⃗jγHnajn(q)e⃗z),
k=1Ns={δjkδns+[(1δjk)Gksjn+Ru,ksu,jn]bjno,I}bks=αu+εjexp(inθ2)bjno,I,
k=1Ns={δjkδns+[(1δjk)Gksjn+Rv,ksv,jn]ajno,II}aks=αv+εjexp(inθ2)ajno,II,
Cjnψ={αψ+εjexp(inθ2)k=1Ns=[(1δjk)Gksjn+Rψ,ksψ,jn]cks}Cjno,ψ,
Cjnu=[bjn(Q)Bjn(Q)bjn(2)Bjn(2)Bjn(1)]T,
Cjnv=[ajn(Q)Ajn(Q)ajn(2)Ajn(2)Ajn(1)]T,
E⃗t=iπk1j=1N[(kye⃗x+γ1e⃗y)Tvjk1Tuje⃗z]exp[iRP(γ1cosγPkysinγP)]dky,
H⃗t=m¯1iπk1j=1N[Tuj(kye⃗x+γ1e⃗y)+Tvjk1e⃗z]exp[iRP(γ1cosγPkysinγP)]dky
(TujTvj)=k2exp(ik22ky2xj+ikyyj)k22ky2n=(1)nexp[insin1(ky/k2)](t21ubjnt21vajn).
γ1={k12ky2k1|ky|iky2k12,k1<|ky|.
ky=k1sinγP,γ1=k1cosγP.
[E⃗tH⃗t/m¯1]=2k1iπRPexp(ik1RP)[Tv(sinγPe⃗x+cosγPe⃗y)+Tue⃗zTu(sinγPe⃗xcosγPe⃗y)+Tve⃗z],
Tujo=2k2cosγPexp(ixjk22k12sin2γP+ik1yjsinγP)k22k12sin2γP(μ2/μ1)k1cosγPn=(1)nexp[insin1(k1k2sinγP)]bjn,
Tvjo=2k2cosγPexp(ixjk22k12sin2γP+ik1yjsinγP)k1μ2/(k2μ1)k22k12sin2γPk2cosγPn=(1)nexp[insin1(k1k2sinγP)]ajn.
k22k12sin2γP={ρ1exp(iϑ1/2),Re(k22)>k12sin2γPi2k2rk2i,Re(k22)=±k1|sinγP|iρ1exp(iϑ1/2),Re(k22)<k12sin2γP,
ρ1=[Re(k22)k12sin2γP]2+(2k2rk2i)2,
ϑ1=tan1[2k2rk2i/|Re(k22)k12sin2γP|].
S⃗t=2Soπk1RP|Tu|2+|Tv|2|αui|2+|αvi|2e⃗R,
e⃗R=cosγPe⃗x+sinγPe⃗y,
So=cok128πm¯1(|αui|2+|αvi|2).
[I,Q,U,V]=[|Tu|2+|Tv|2,|Tu|2|Tv|2,2Re(TuTv*),2Im(TuTv*)].
δs=cos1[Re(TuTv*)/|TuTv|]

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