Abstract

The complex Poynting theorem has been used to study power flow and energy storage for the case in which a plane wave (polarization wherein the electric field is in the plane of incidence) is scattered from a generally lossy, anisotropic, non-Hermitian diffraction grating. The full electromagnetic fields of the diffraction grating system were specified, and, in applying the complex Poynting theorem to the grating system, a full calculation of the diffraction efficiency, the electromagnetic (electric and magnetic) energy, and the real, reactive, dissipative, and evanescent power of the grating was made. A step profile grating was used to test numerical examples, and, in all cases considered, the complex Poynting theorem was obeyed to a high degree of numerical accuracy. In the study the effects that anisotropy and lossiness of the grating system had on the complex power of the system were illustrated. A comparison of the complex power that resulted from scattering from diffraction gratings composed of Hermitian and non-Hermitian anisotropic materials was numerically studied.

© 1999 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  2. K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  11. M. G. Moharam, T. K. Gaylord, “Comments on analyses of reflection gratings,” J. Opt. Soc. Am. 73, 399–401 (1983).
    [Crossref]
  12. J. M. Jarem, “Rigorous coupled wave theory solution of phi-periodic circular cylindrical dielectric systems,” J. Electromagn. Waves Appl. 11, 197–213 (1997).
    [Crossref]
  13. J. M. Jarem, “Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems,” Prog. Electromagn. Res. 19, 109–128 (1998).
    [Crossref]
  14. J. M. Jarem, “Rigorous coupled-wave-theory analysis of dipole scattering from a three-dimensional, inhomogeneous, spherical dielectric and permeable system,” IEEE Trans. Microwave Theory Tech. 45, 1193–1203 (1997).
    [Crossref]
  15. J. M. Jarem, “A rigorous coupled wave theory and crossed diffraction grating analysis of radiation and scattering from three-dimensional inhomogeneous objects,” IEEE Trans. Antennas Propag. 46, 740–741 (1998).
    [Crossref]
  16. J. M. Jarem, P. Banerjee, “An exact, dynamical analysis of the Kukhtarev equations in photorefractive barium ti-tanate using rigorous wave coupled wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831 (1996).
    [Crossref]
  17. P. St. J. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
    [Crossref]
  18. R. Petit, G. Tayeb, “On the use of the energy balance criteria as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1987).
    [Crossref]
  19. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [Crossref]
  20. E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
    [Crossref]
  21. E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
    [Crossref]
  22. E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
    [Crossref]
  23. E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” Prog. Opt. 31, 141–187 (1993).
  24. B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
    [Crossref]
  25. R. F. Harrington, “Complex power,” in Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 1–10.
  26. H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New York, 1984), Sec. 11.1.
  27. J. Yamakita, K. Rokushima, “Modal expansion for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. SPIE503, 239–243 (1984), Fig. 5.
    [Crossref]
  28. P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, UK, 1990), p. 127.

1998 (2)

J. M. Jarem, “Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems,” Prog. Electromagn. Res. 19, 109–128 (1998).
[Crossref]

J. M. Jarem, “A rigorous coupled wave theory and crossed diffraction grating analysis of radiation and scattering from three-dimensional inhomogeneous objects,” IEEE Trans. Antennas Propag. 46, 740–741 (1998).
[Crossref]

1997 (3)

J. M. Jarem, “Rigorous coupled-wave-theory analysis of dipole scattering from a three-dimensional, inhomogeneous, spherical dielectric and permeable system,” IEEE Trans. Microwave Theory Tech. 45, 1193–1203 (1997).
[Crossref]

J. M. Jarem, “Rigorous coupled wave theory solution of phi-periodic circular cylindrical dielectric systems,” J. Electromagn. Waves Appl. 11, 197–213 (1997).
[Crossref]

B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
[Crossref]

1996 (1)

1995 (2)

1993 (1)

E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” Prog. Opt. 31, 141–187 (1993).

1990 (3)

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
[Crossref]

1987 (2)

1984 (1)

1983 (4)

1981 (3)

Adams, J. L.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Andrewarthar, J. R.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Banerjee, P.

Botten, C.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Butcher, P. N.

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, UK, 1990), p. 127.

Cotter, D.

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, UK, 1990), p. 127.

Craig, M. S.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Feit, M. D.

B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
[Crossref]

Gaylord, T. K.

Glytsis, E. N.

Grann, E. B.

Harrington, R. F.

R. F. Harrington, “Complex power,” in Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 1–10.

Haus, H.

H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New York, 1984), Sec. 11.1.

Jarem, J. M.

J. M. Jarem, “A rigorous coupled wave theory and crossed diffraction grating analysis of radiation and scattering from three-dimensional inhomogeneous objects,” IEEE Trans. Antennas Propag. 46, 740–741 (1998).
[Crossref]

J. M. Jarem, “Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems,” Prog. Electromagn. Res. 19, 109–128 (1998).
[Crossref]

J. M. Jarem, “Rigorous coupled-wave-theory analysis of dipole scattering from a three-dimensional, inhomogeneous, spherical dielectric and permeable system,” IEEE Trans. Microwave Theory Tech. 45, 1193–1203 (1997).
[Crossref]

J. M. Jarem, “Rigorous coupled wave theory solution of phi-periodic circular cylindrical dielectric systems,” J. Electromagn. Waves Appl. 11, 197–213 (1997).
[Crossref]

J. M. Jarem, P. Banerjee, “An exact, dynamical analysis of the Kukhtarev equations in photorefractive barium ti-tanate using rigorous wave coupled wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831 (1996).
[Crossref]

Li, L.

B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
[Crossref]

Marom, E.

Maystre, D.

E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[Crossref]

McPhedran, R. C.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Moharam, M. G.

Petit, R.

R. Petit, G. Tayeb, “On the use of the energy balance criteria as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1987).
[Crossref]

Pommet, D. A.

Popov, E.

E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” Prog. Opt. 31, 141–187 (1993).

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[Crossref]

Rokushima, K.

K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
[Crossref]

J. Yamakita, K. Rokushima, “Modal expansion for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. SPIE503, 239–243 (1984), Fig. 5.
[Crossref]

Russell, P. St. J.

Shore, B. W.

B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
[Crossref]

Tayeb, G.

R. Petit, G. Tayeb, “On the use of the energy balance criteria as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1987).
[Crossref]

Tsonev, L.

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
[Crossref]

Yamakita, J.

K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
[Crossref]

J. Yamakita, K. Rokushima, “Modal expansion for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. SPIE503, 239–243 (1984), Fig. 5.
[Crossref]

Zylberberg, Z.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

J. M. Jarem, “A rigorous coupled wave theory and crossed diffraction grating analysis of radiation and scattering from three-dimensional inhomogeneous objects,” IEEE Trans. Antennas Propag. 46, 740–741 (1998).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

J. M. Jarem, “Rigorous coupled-wave-theory analysis of dipole scattering from a three-dimensional, inhomogeneous, spherical dielectric and permeable system,” IEEE Trans. Microwave Theory Tech. 45, 1193–1203 (1997).
[Crossref]

J. Electromagn. Waves Appl. (1)

J. M. Jarem, “Rigorous coupled wave theory solution of phi-periodic circular cylindrical dielectric systems,” J. Electromagn. Waves Appl. 11, 197–213 (1997).
[Crossref]

J. Mod. Opt. (3)

B. W. Shore, L. Li, M. D. Feit, “Poynting vectors and electric field distributions in simple dielectric gratings,” J. Mod. Opt. 44, 69–81 (1997).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[Crossref]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[Crossref]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewarthar, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[Crossref]

Prog. Electromagn. Res. (1)

J. M. Jarem, “Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems,” Prog. Electromagn. Res. 19, 109–128 (1998).
[Crossref]

Prog. Opt. (1)

E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” Prog. Opt. 31, 141–187 (1993).

Surf. Sci. (1)

E. Popov, L. Tsonev, D. Maystre, “Total absorption of light by metallic gratings and energy flow distribution,” Surf. Sci. 230, 290–294 (1990).
[Crossref]

Other (6)

R. F. Harrington, “Complex power,” in Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 1–10.

H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New York, 1984), Sec. 11.1.

J. Yamakita, K. Rokushima, “Modal expansion for dielectric gratings with rectangular grooves,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. SPIE503, 239–243 (1984), Fig. 5.
[Crossref]

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, UK, 1990), p. 127.

R. Petit, G. Tayeb, “On the use of the energy balance criteria as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1987).
[Crossref]

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

(a) Geometry of the E-mode diffraction grating system. (b) Complex Poynting box used for calculations.

Fig. 2
Fig. 2

(a) Transmitted diffraction efficiency (DE) of a lossless and a lossy step diffraction grating. (b)–(e) Plots of the real and the imaginary parts of the normalized complex power PIN and PBOX as computed by Eq. (56a) of the complex Poynting theorem for [(b), (c)] the lossless and [(d), (e)] the lossy cases. (f ), (g) Plots of the evanescent power as computed by Eqs. (67, 68, 69) for (f ) the lossless and (g) the lossy cases.

Fig. 3
Fig. 3

(a), (b) Plots of the real and the imaginary parts, respectively, of the normalized complex power PIN and PBOX as computed by Eq. (56a) of the complex Poynting theorem. (c) Plot of the evanescent power as calculated by Eqs. (67)–(69). (d) Plots of PdiffRWEM and PdiffR [calculated in Eqs. (60)–(62)] for Hermitian and non-Hermitian step diffraction gratings. (e) Plots of PdiffIWEM and PdiffI as calculated in Eqs. (63)–(65) for Hermitian and non-Hermitian step diffraction gratings.

Equations (85)

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×E=-jμ(η0H),
×(η0H)=jε¯¯E,
ε¯¯=εxxεxy0εyxεyy000εzz.
E=i=-[Sxi(y)aˆx+Syi(y)aˆy]exp(-jkxi x),
Uη0H=i=-Uzi(y)exp(-jkxi x)aˆz,
kxi=kx0-iKx,kx0=μ1ε1 sin(θ),
Kx=2π/Λ,Λ=k0Λ˜,
ε(x)=i=-ε˘i exp(jiKxx),
Sy̲=εyy¯¯-1(Kx¯¯Uz̲-εyx¯¯Sx̲),
Ve(y)̲y=A¯¯Ve(y)̲,
Ve̲=Sx̲Uz̲,A¯¯=a11¯¯a12¯¯a21¯¯a22¯¯,
a11¯¯=j(Kx¯¯εyy¯¯-1εyx¯¯),a12¯¯=j(-Kx¯¯εyy¯¯-1Kx¯¯+I¯¯),
a21¯¯=j(εxx¯¯-εxy¯¯εyy¯¯-1εyx¯¯),
a22¯¯=j(εxy¯¯εyy¯¯-1Kx¯¯),
E(2)=n=1NTCnE ne=m=-MTMTn=1NTCn(Sxinaˆx+Syinaˆy)×exp(qn y)exp(-jkxi x),
U(2)=η0H(2)=n=1NTCnUne=m=-MTMTn=1NTCn(Uzinaˆz)exp(qn y)exp(-jkxi x),
Uz(1)=η0Hz(1)=i=-[E0δ i,0 exp(jky1i y)+ri ×exp(-jky1i y)]exp(-jkxi x),
Ex(1)=1ε1i=-{ky1i[E0δi,0 exp(jky1i y)-ri exp(-jky1i y)]}exp(-jkxi x),
Ey(1)=1ε1i=-{kxi[E0δ i,0 exp(jky1i y)+ri exp(-jky1i y)]}exp(-jkxix),
Uz(3)=η0 Hz(3)=i=-{ti exp[jky3i(y+L)]}exp(-jkxi x),
Ex(3)=1ε3i=-{ky3iti exp[jky3i(y+L)]}exp(-jkxix),
Ey(3)=1ε3i=-{kxiti exp[jky3i(y+L)]}exp(-jkxi x),
kyri=(μrεr-kxi2)1/2μrεr>kxi-j(kxi2-μrεr)1/2kxi>μrεr ,r=1, 3.
n=1NTCnky1iε1Uzin+Sxin=2E0ky1iε1δi0,
n=1NTCnexp(-qnL)Sxin-ky3iε3Uzin=0,
PfINu=PfOUTu+PDEu+PDMu+j(-PWEu+PWMu),
PfINu=ΔSE(1)×U(1)*y=0+·(-aˆy)dS,
PfOUTu=ΔSE(3)×U(3)*y=-L-·(-aˆy)dS,
PDEu=ΔVE(2)·[ε¯¯E(2)]*dV,
PDMu=ΔVU(2)·[μU(2)]*dV,
PWEu=ΔVE(2)·[ε¯¯E(2)]*dV,
PWMu=ΔVU(2)·[μU(2)]*dV.
E(2)·ε¯¯E(2)*=Ex(2)εxx(x)Ex(2)*+Ex(2)εxy(x)Ey(2)*+Ey(2)εyx(x)Ex(2)*+Ey(2)εyy(x)Ey(2)*.
V(x, y)=i,nCnVi,n exp(qny)exp(-jkxix)
ε(x)=iε˘i exp(jiKx x)
W(x, y)=i,nCnWi,n exp(qn y)exp(-jkxi x)
P=ΔVV(x, y)ε(x)W(x, y)*dV=ΔVi,nCnVin exp(qny)exp(-jkxix)×iε˘i exp(jiKxx)×i,nCnWin exp(qny)exp(-jkxix)*dxdydz.
ΔVV(x, y)ε(x)W(x, y)*dV
=ΔzΛn,nCnCn *Iyn,ni,iε˘i-iVinWin *,
Iyn,n=-L-0+ exp[(qn+qn *)y]d y.
μ(x)μ=i=-μ˘i exp(jiκx),
μ˘i=(μ-jμ)δi,0.
PWMu=ΔzΛμn,nCnCn *Iyn,ni[Uzin(2)Uzin(2)*],
PDMu=ΔzΛμn,nCnCn *Iyn,ni[U zin(2)U zin(2)*].
PfINu=ΔzΛε1iky1i(E0δi,0-ri)(E0δi,0+ri)*.
Pincu=ΔzΛε1|E0|2ky10.
PINPfINuPincu=iky1i(E0δi,0-ri)×(E0δi,0+ri)*/(ky10|E0|2).
Prefu=ΔSEref(1)×Uref(1)*y=0+·aˆydS=ΔzΛε1iky1iriri*,
PfOUTuPtransu=ΔSE(3)×U(3)*y=-L-·(-aˆy)dS
=ΔzΛε3iky3ititi*.
Tiky1i[E0δ i,0-ri][E0δi,0+ri]*.
T=iky1i[E02δi,0+(-ri+ri*)E0δi,0-riri*],
T=ky10[E02-2 j Im(r0)E0-r0r0*]-i,i0ky1iriri*,
T=ky10 E02-2jky10 Im(r0)E0-iky1iriri*.
PfINu=ΔzΛε1iky1i E0 E0*δi,0-ΔzΛε1iky1iriri*-ΔzΛε1[2jky10 Im(r0)E0].
PfINu=Pincu-Prefu-ΔzΛε1[2 jky10 Im(r0)E0].
Pincu-Prefu-ΔzΛε1[2 jky10 Im(r0)E0]
=PfOUTu+PDEu+PDMu+j(-PWEu+PWMu).
Pincu=Prefu+ΔzΛε1[2 jky10 Im(r0)E0]+PfOUTu+PDEu+PDMu+j(-PWEu+PWMu).
PincPincu/Pincu=1,
Pref=Prefu/Pincu=1ky10iky1iriri*,
POUTPtrans=PfOUTu/Pincu=ε1ε31ky10iky3ititi*,
PDEPDEu/Pincu,PDMPDMu/Pincu,
PWEPWEu/Pincu,PWMPWMu/Pincu,
PIN=POUT+PDE+PDM+j(-PWE+PWM)PBOX
1=Pinc=Pref+2j Im(r0)+Ptrans+PDE+PDM+j(-PWE+PWM).
1=Pinc=Re(Pref)+Re(Ptrans)+Re(PDE)+Re(PDM)+Re[j(-PWE+PWM)].
0=Im(Pref)+Im(Ptrans)+2 Im(r0)
+Im(PDE)+Im(PDM)+Im[j(-PWE+PWM)].
Re[j(-PWE+PWM)]=Pinc-Re(Pref)-Re(Ptrans)-Re(PDE)-Re(PDM).
PdiffRWEMRe[j(-PWE+PWM)]
PdiffRPinc-Re(Pref)-Re(Ptrans)-Re(PDE)-Re(PDM),
PdiffRWEM=PdiffR.
PdiffIWEMIm[j(-PWE+PWM)],
PdiffI-Im(Pref+Ptrans+PDE+PDM)-2 Im(r0).
PdiffIWEM=PdiffI.
Im(Pref)+Im(Ptrans)=-2 Im(r0)-Im(PDE)-Im(PDM)-Im[j(-PWE+PWM)].
PevanIm(Pref)+Im(Ptrans)
Pevandiff-2 Im(r0)-Im(PDE)-Im(PDM)-Im[j(-PWE+PWM)],
Pevan=P evandiff.
Pref=Prefu /Pincu=1ky10iky1i riri*,
ky1i=(μ1ε1-kxi2)1/2μ1ε1>kxi-j(kxi2-μ1ε1)1/2kxi>μ1ε1.
PdiffRWEM=Re[j(-PWE+PWM)]=PdiffR=0,
PdiffRPinc-Re(Pref)-Re(Ptrans)-Re(PDE)-Re(PDM)=0,
Pinc=Re(Pref)+Re(Ptrans)+Re(PDE)+Re(PDM).

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