Abstract

Scattering of s-polarized electromagnetic planes waves from a film, with a shallow random rough one-dimensional surface, bounded by vacuum and a perfect conductor is calculated. An integral equation that relates the amplitude of the scattered field to the incident wave is found by use of the Rayleigh hypothesis. The integral equation is solved numerically and by use of the perturbation theory, up to the fourth order in the surface profile function. In the angular dependence of the incoherent part of the differential reflection coefficient, the backscattering peak and two additional satellite peaks are observed, owing to two guided waves supported by the film. Analysis of the perturbation solution reveals that the background scattering exhibits minima and maxima as functions of the thickness. By studying the behavior of the scattering as a function of the absorption index of the film, it is shown that the amplitudes of the peaks are low when k ∼ 10-2 and high when k ∼ 10-4.

© 2000 Optical Society of America

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  1. J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
    [CrossRef]
  2. A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on a perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
    [CrossRef]
  3. T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
    [CrossRef]
  4. A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface,” Opt. Commun. 72, 279–285 (1989).
    [CrossRef]
  5. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  6. D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
    [CrossRef]
  7. N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
    [CrossRef]
  8. M. Saillard, J. A. DeSanto, “A coordinate-spectral method for rough surfaces scattering,” Waves Random Media 3, 135–150 (1996).
    [CrossRef]
  9. R. García-Llamas, “Scattering of electromagnetic plane waves from rough periodic multilayered film,” J. Opt. Soc. Am. B 11, 618–623 (1994).
    [CrossRef]
  10. R. García-Llamas, L. E. Regalado, “Effects of rough interfaces in a multilayer stack,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1298–1312 (1994).
    [CrossRef]
  11. R. García-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with shallow random rough interfaces between dissimilar media,” Opt. Commun. 142, 167–172 (1997).
    [CrossRef]
  12. R. García-Llamas, L. E. Regalado, “Transmitted scattered light from a thin film with shallow random rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
    [CrossRef]
  13. R. García-Llamas, L. E. Regalado, C. Amra, “Scattering of light from a two-layer system with a rough surface,” J. Opt. Soc. Am. A 16, 2713–2719 (1999).
    [CrossRef]

1999 (1)

1997 (2)

R. García-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with shallow random rough interfaces between dissimilar media,” Opt. Commun. 142, 167–172 (1997).
[CrossRef]

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on a perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

1996 (2)

M. Saillard, J. A. DeSanto, “A coordinate-spectral method for rough surfaces scattering,” Waves Random Media 3, 135–150 (1996).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Transmitted scattered light from a thin film with shallow random rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
[CrossRef]

1994 (3)

R. García-Llamas, “Scattering of electromagnetic plane waves from rough periodic multilayered film,” J. Opt. Soc. Am. B 11, 618–623 (1994).
[CrossRef]

T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

1991 (2)

D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

1990 (1)

1989 (1)

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

Amra, C.

Bruce, N. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

Dainty, J. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

DeSanto, J. A.

M. Saillard, J. A. DeSanto, “A coordinate-spectral method for rough surfaces scattering,” Waves Random Media 3, 135–150 (1996).
[CrossRef]

Freilikher, V. D.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

García-Llamas, R.

R. García-Llamas, L. E. Regalado, C. Amra, “Scattering of light from a two-layer system with a rough surface,” J. Opt. Soc. Am. A 16, 2713–2719 (1999).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with shallow random rough interfaces between dissimilar media,” Opt. Commun. 142, 167–172 (1997).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Transmitted scattered light from a thin film with shallow random rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
[CrossRef]

R. García-Llamas, “Scattering of electromagnetic plane waves from rough periodic multilayered film,” J. Opt. Soc. Am. B 11, 618–623 (1994).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Effects of rough interfaces in a multilayer stack,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1298–1312 (1994).
[CrossRef]

Ingers, J.

D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
[CrossRef]

Lu, J. Q.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

Madrazo, A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on a perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Maradudin, A. A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on a perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

Maystre, D.

D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

Michel, T. R.

Pustilnik, M.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

Regalado, L. E.

R. García-Llamas, L. E. Regalado, C. Amra, “Scattering of light from a two-layer system with a rough surface,” J. Opt. Soc. Am. A 16, 2713–2719 (1999).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with shallow random rough interfaces between dissimilar media,” Opt. Commun. 142, 167–172 (1997).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Transmitted scattered light from a thin film with shallow random rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
[CrossRef]

R. García-Llamas, L. E. Regalado, “Effects of rough interfaces in a multilayer stack,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1298–1312 (1994).
[CrossRef]

Saillard, M.

M. Saillard, J. A. DeSanto, “A coordinate-spectral method for rough surfaces scattering,” Waves Random Media 3, 135–150 (1996).
[CrossRef]

D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

Yurkevich, I.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

R. García-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with shallow random rough interfaces between dissimilar media,” Opt. Commun. 142, 167–172 (1997).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on a perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Phys. Rev. B (1)

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Optics 43, 435–452 (1996); J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metallic films with a random rough surface,” Phys. Rev. B 50, 17100–17115 (1995).
[CrossRef]

Waves Random Media (2)

D. Maystre, M. Saillard, J. Ingers, “Scattering by one or two randomly rough surfaces,” Waves Random Media 3, S143–S155 (1991).
[CrossRef]

M. Saillard, J. A. DeSanto, “A coordinate-spectral method for rough surfaces scattering,” Waves Random Media 3, 135–150 (1996).
[CrossRef]

Other (1)

R. García-Llamas, L. E. Regalado, “Effects of rough interfaces in a multilayer stack,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1298–1312 (1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of the system. The media are vacuum, z < 0, a film, 0 < z < d 2, and a perfect conductor in the region z > d 2.

Fig. 2
Fig. 2

Incoherent part of the mean DRC versus the angle of the scattering for an angle of incidence θ i = 0°, the rms height h 1 = 15 nm, and d 2 = 500 nm. The solid curve corresponds to the exact calculation and the dashed one to the perturbation solution.

Fig. 3
Fig. 3

Same as Fig. 2, except d 2 = 493 nm. The solid curve corresponds to numerical calculation and the dashed one to the perturbation solution.

Fig. 4
Fig. 4

Same as Fig. 2, except d 2 = 556 nm. This thickness value is close to that given in Eq. (18) (578.8 nm with m = 2), producing minimum scattering. The angular positions of the resonances are now θ± = ±14.45°.

Fig. 5
Fig. 5

Incoherent part of the mean DRC versus the angle of scattering for different values of the absorption index of the film, k = 2.3 × 10-p , θ i = 0°, and d 2 = 493 nm. The continuous curve corresponds to the numerical solution and the dotted curve to the perturbation one when p = 4; the dashed curve corresponds to the numerical calculation and the three-dotted–dashed curve to the perturbation one when p = 3. The previous curves were calculated with the latter value. The dashed–dotted curve corresponds to numerical solution and the long-dashed curve to the perturbation one when p = 2.

Fig. 6
Fig. 6

Incoherent part of the mean DRC versus the angle of scattering for an angle of incidence θ i = 0°, d 2 = 500 nm, and two heights: h 1 = 30, 60 nm. The solid curve (numerical) and the dashed one (perturbation) correspond to the first height, whereas the dashed–dotted curve corresponds to the last height.

Fig. 7
Fig. 7

Incoherent part of the mean DRC is plotted as a function of the scattering angle for two different incidence angles, θ i = 0°, the solid curve, and θ i = 5°, the dashed one. A square roughness spectrum was used. Two well-defined satellite peaks at θ s = ±17.7° when θ i = 0° are observed, whereas when light is incident at 5°, the satellite peaks are θ s = 12.3° and θ s = -22.8°.

Equations (36)

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

f1y=0,
fyfy=g¯y,
gα=h12T1/2πexp-α2T12/4,
gα=Soif kmin|α|kmax0elsewhere,
U1y, z=expiαiy+β1αiz+-dαrαexpiαy-β1αz,
U2y, z=-dα(aαexpiαy+β2αz+bαexpiαy-β2αz),
αi=ω/cε1 sin θi,
βjα=ω/c2εj-α21/2,
-dαG2bγ, αrα=H2bγ, αi,
G2bγ, α=I1-β1α+β2γβ1α+β2γ exp+iβ2γd2-I1-β1α-β2γβ1α-β2γ exp-iβ2γd2,
H2bγ, αi=I1+β1αi-β2γβ1αi-β2γ exp+iβ2γd2-I1+β1αi+β2γβ1αi+β2γ exp-iβ2γd2,
I1±β1α±β2γ=12π-+ expiα-γy×exp±iβ1α±β2γf1ydy,
rα=q=0 rqαh1q,
G2bγ, α=p=0ipp! h1pGpγ, αFγ-αp,  Gpγ, α=-β1α+β2γpβ1α+β2γ exp+iβ2γd2--β1α-β2γpβ1α-β2γ exp-iβ2γd2,
H2bγ, αi=p=0ipp! h1pHpγ, αiFγ-αip,  Hpγ, αi=+β1αi-β2γpβ1αi-β2γ exp+iβ2γd2-+β1αi+β2γpβ1αi+β2γ exp-iβ2γd2,
Fγ-αp=12π-+ expiα-γyf1pydy
p=0k-1pipp!  dαGpγ, αFγ-αprk-pα=ikk! Hpγ, αiFγ-αip.
r0α=δα-αiH0α, αiG0α, α,
r1α=i E1α, αiG0α, α Fα-αi1,
E1α, αiG0α, α=1+r0αi * exp+iβ2αd2-exp-iβ2αd2β1α+β2α-1 exp+iβ2αd2-β1α-β2α-1 exp-iβ2αd2.
d2=λ2m+24ε2
d2=λ2m+14ε2,
r2α=+1G0α, α  dγG1α, γE1γ, αiG0γ, γ Fα-γ1Fγ-αi1-12E2α, αiG0α, α  dγFγ1Fα-αi-γ1.
r3α=-iG0α, α  dγ G1α, γG0γ, γ× dηG1γ, ηE1η, αiG0η, η Fα-γ1Fγ-η1Fη-αi1+i2G0α, α  dγ G1α, γG0γ, γ E2γ, αi× dηFα-γ1Fη1Fη-αi-η1+i2G0α, α× dγG2α, γE1γ, αiG0γ, γ  dηFη1Fα-γ-η1Fγ-αi1-iE3α, αi6G0α, α  dγ  dηFγ1Fη1Fα-αi-γ-η1,
Epα, αi=Hpα, αi-r0αiGpα, αi.
dRdθsincoh=cos2 θscos θi|rα|2-|rα|2α=ω/csin θs.
dRdθsincoh=cos2 θscos θiI1,1+I2,2+I3,1,
I1,1=r1αr1*α=E1α, αiG0α, α2g|α-αi|.
I2,2=r2αr2*α-r2αr2α*=+1|G0α, α|2  dγG1α, γG1*α, γ×E1γ, αiG0γ, γE1*γ, αiG0*γ, γ g|α-γ|g|γ-αi|+1|G0α, α|2  dγG1α, γG1*α, α+αi-γ×E1γ, αiG0γ, γE1*α+αi-γ, αiG0*α+αi-γ, α+αi-γ×g|α-γ|g|γ-αi|-2 E2*α, αi|G0α, α|2× dγG1α, γE1γ, αiG0γ, γ g|α-γ|g|γ-αi|+12|E2α, αi|2|G0α, α|2  dγg|γ|g|α-αi-γ|.
I3,1=r1*αr3α=E1*α, αi|G0α, α|2 g|α-αi|G1α, αiG0αi, αi× dηG1αi, ηE1η, αiG0η, η g|αi-η|+E1*α, αi|G0α, α|2 g|α-αi|× dγ G1α, γG0γ, γ G1γ, γ-α+αi×E1γ-α+αi, αiG0γ-α+αi, γ-α+αi g|α-γ|+E1*α, αi|G0α, α|2 g|α-αi|× dγ G1α, γG0γ, γ G1γ, αE1α, αiG0α, α g|α-γ|-E1*α, αi2|G0α, α|2 g|α-αi|G1α, αiG0αi, αi E2αi, αi-E1*α, αi|G0α, α|2 g|α-αi|  dγ G1α, γG0γ, γ E2×γ, αiγ, αig|α-γ|-E1*α, αi|G0α, α|2 g|α-αi|× dγ G2α, γE1γ, αiG0γ, γ g|-γ+αi|-E1*α, αi2|G0α, α|2 g|α-αi|G2α, αE1α, αiG0α, α+E1*α, αi2|G0α, α|2 E3α, αig|α-αi|.
I1±β1α±β2γ=1/Mm=1M expiα-γym±β1α±β2γf1ym,
fym=h1/al=-M/2M/2-1 Fl expiαlym.
Fl=2πagαl1/2×N0, 1+iN0, 1/2,l0, M/2N0, 1,l=0, M/2,
αn=2π/λε1 sin θi+Δαn,
n=-NN G2bαm, αnrαn=H2bαm, αo,
sin θ±=-sin θi±c/ωα1ω; d2-α2ω; d2,

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