Abstract

The scattered light from a two-layer system with a shallow, random, one-dimensional rough surface bounded by semi-infinite dissimilar optical media is calculated. The systems is composed of metallic and weak absorbent dielectric films between glass and vacuum. The dielectric constant and the thickness of the dielectric film are chosen in such a way that in the absence of roughness the system supports eight transverse magnetic (TM) guided modes, whose wave numbers are q1(TM)(λ), q2(TM)(λ),, q8(TM) (λ), or nine transverse electric (TE) guided modes, whose wave numbers are q1(TE)(λ), q2(TE)(λ),, q9(TE)(λ), at the wavelength λ. The Rayleigh hypothesis is used to obtain an integral equation relating the amplitudes of the reflected fields to the incident wave. The scattering integral is solved both by perturbation and numerically. Results are obtained by assuming a Gaussian roughness spectrum for the surface, and the formalism is applied to simulate the scattering from the system in the attenuated total reflection configuration, allowing the excitation of guided waves. The angular dependence of the scattering shows four peaks, in addition to the backscattering effect. The angular positions of these peaks are given by (2π/λ)n1 sinθk(t)=±qk(t), with k=7, 8 when t={p, TM} or k=8, 9, when t={s, TE}; they are also independent of the angle of incidence and are due to single-scattering effects.

© 1999 Optical Society of America

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References

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  1. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11, 197–210 (1994).
    [CrossRef]
  2. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiments,” J. Opt. Soc. Am. 71, 1115–123 (1981).
    [CrossRef]
  3. O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
    [CrossRef]
  4. J. M. Elson, “Multilayer-coated optics: guided-wave coupling and scattering by means of interface random roughness,” J. Opt. Soc. Am. A 12, 729–742 (1995).
    [CrossRef]
  5. 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]
  6. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
    [CrossRef]
  7. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
    [CrossRef]
  8. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
    [CrossRef]
  9. A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
    [CrossRef]
  10. H. Ogura, Z. L. Wang, “Surface-plasmon mode on a random rough metal surface: enhanced backscattering and localization,” Phys. Rev. B 53, 10358–10371 (1996).
    [CrossRef]
  11. J. Q. Lu, A. A. Maradudin, T. Michel, “Enhanced backscattering from a rough dielectric film on a reflecting substrate,” J. Opt. Soc. Am. B 8, 311–318 (1991).
    [CrossRef]
  12. R. Garcı́a-Llamas, “Scattering of electromagnetic plane waves from a rough periodic multilayer film,” J. Opt. Soc. Am. B 11, 618–623 (1994).
    [CrossRef]
  13. R. Garcı́a-Llamas, L. E. Regalado, “Effects of rough interface in a multilayer stack,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1298–1312 (1994).
    [CrossRef]
  14. R. Garcı́a-Llamas, L. E. Regalado, “Transmitted scattered light from a thin film with shallow random rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
    [CrossRef]
  15. R. Garcı́a-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with a shallow random rough surface between dissimilar media. I. Theory,” Opt. Commun. 142, 167–172 (1997).
    [CrossRef]

1997

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

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

1996

H. Ogura, Z. L. Wang, “Surface-plasmon mode on a random rough metal surface: enhanced backscattering and localization,” Phys. Rev. B 53, 10358–10371 (1996).
[CrossRef]

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

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

1995

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. M. Elson, “Multilayer-coated optics: guided-wave coupling and scattering by means of interface random roughness,” J. Opt. Soc. Am. A 12, 729–742 (1995).
[CrossRef]

1994

C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11, 197–210 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
[CrossRef]

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

1991

1989

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]

1981

Amra, C.

Bousquet, P.

Elson, J. M.

Flory, F.

Freilikher, V. D.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

Garci´a-Llamas, R.

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

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

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

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

Kienzle, O.

O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
[CrossRef]

Lu, J. Q.

Lu, Jun Q.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[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 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 perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

J. Q. Lu, A. A. Maradudin, T. Michel, “Enhanced backscattering from a rough dielectric film on a reflecting substrate,” J. Opt. Soc. Am. B 8, 311–318 (1991).
[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]

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.

Ogura, H.

H. Ogura, Z. L. Wang, “Surface-plasmon mode on a random rough metal surface: enhanced backscattering and localization,” Phys. Rev. B 53, 10358–10371 (1996).
[CrossRef]

Pustilnick, M.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

Regalado, L. E.

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

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

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

Roche, P.

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

Staub, J.

O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
[CrossRef]

Tschudi, T.

O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
[CrossRef]

Wang, Z. L.

H. Ogura, Z. L. Wang, “Surface-plasmon mode on a random rough metal surface: enhanced backscattering and localization,” Phys. Rev. B 53, 10358–10371 (1996).
[CrossRef]

Yurkevich, I.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

R. Garcı́a-Llamas, L. E. Regalado, “Scattering of light from a thin metallic film with a shallow random rough surface between dissimilar media. I. Theory,” 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 perfectly conducting substrate,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Phys. Rev. B

H. Ogura, Z. L. Wang, “Surface-plasmon mode on a random rough metal surface: enhanced backscattering and localization,” Phys. Rev. B 53, 10358–10371 (1996).
[CrossRef]

O. Kienzle, J. Staub, T. Tschudi, “Light scattering from transparent substrate: theory and experiment,” Phys. Rev. B 50, 1848–1860 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surfaces,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnick, I. Yurkevich, “Scattering of electromagnetic wave from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15360 (1994).
[CrossRef]

Other

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

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Figures (4)

Fig. 1
Fig. 1

The thin film is growing to positive values of z, the dielectric film/vacuum interface roughness variation is along only the y axis, and the glass/metallic film and metallic film/dielectric film interfaces are smooth and planar. The incidence plane is x=0.

Fig. 2
Fig. 2

(a) Reflection as a function of the angle of incidence for the system depicted in Fig. 1, where a smooth and planar interface is assumed, the dashed curve corresponds to s polarization, and the solid curve corresponds to p polarization. (b) Angular position of the minima of the attenuated total reflection curve plotted as a function of the thickness of the dielectric film. The circles correspond to GM(TM), and the squares correspond to GM(TE). The vertical dotted line is a reference for a thickness equal to 1230 nm.

Fig. 3
Fig. 3

Incoherent part of the mean DRC as a function of the scattered angle θs, for p polarization of the incident light, for four values of θi (0°, 30°, θ8(p), θ7(p)); both perturbation (dashed curves) and numeric (solid curves) solutions are presented. The studied system is depicted in Fig. 1, and the parameters of the system are given in the text. All curves show the well-known backscattering peak, appearing for every angle of incidence and located at θs=-θi.

Fig. 4
Fig. 4

Incoherent part of the mean DRC as a function of the scattered angle θs, for s polarization of the incident light, for four values of θi (0°, 30°, θ9(s), θ8(s)); both perturbation (dashed curves) and numerical (solid curves) solutions are presented. The difference with Fig. 3 is in the angular position of the peaks.

Equations (34)

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f3(y)=0,
f3(y)f3(y)=h32 exp[-(y-y)2/σ32],
G(|α|)=σ32π exp(-σ32α2/4);
U1(y, z)=exp{i[αiy+β1(αi)z]}+-dα r(α)exp{i[αy-β1(α)z]},
U2(y, z)=-dα(a2(α)exp{i[αy+β2(α)z]}+b2(α)exp{i[αy-β2(α)z]}).
U4(y, z)=-dα t(α)exp{i[αy+β4(α)z]}.
-dα G(l+1)(γ, α)r(α)=H(l+1)(γ, αi),
G(3)(γ, α)=1β3(α) (G(2)(α, α)D(3)(γ, α)×exp[+iβ3(α)d3]I(3){+[β3(α)+β4(γ)]}+G(2)(α, α)C(3)(γ, α)exp[-iβ3(α)d3]×I(3){-[β3(α)-β4(γ)]}),
H(3)(γ, αi)=1β3(αi) (H(2)(αi, αi)D(3)(γ, αi)×exp[+iβ3(αi)d3]I(3){+[β3(αi)+β4(γ)]}+H(2)(αi, αi)C(3)(γ, αi)exp[-iβ3(αi)d3]×I(3){-[β3(αi)-β4(γ)]}),
G(2)(α, α)=1β2(α) {D(1)(α, α)D(2)(α, α)×exp[+iβ2(α)d2]+C(1)(α, α)×C(2)(α, α)exp[-iβ2(α)d2]},
H(2)(αi, αi)=1β2(αi) {C(1)(αi, αi)D(2)(αi, αi)×exp[+iβ2(αi)d2]+D(1)(αi, αi)C(2)(αi, αi)×exp[-iβ2(αi)d2]},
G(2)(α, α)=1β2(α) {D(1)(α, α)C(2)(α, α)×exp[+iβ2(α)d2]+C(1)(α, α)D(2)(α, α)×exp[-iβ2(α)d2]},
H(2)(αi, αi)=1β2(αi) {C(1)(αi, αi)C(2)(αi, αi)×exp[+iβ2(αi)d2]+D(1)(αi, αi)D(2)(αi, αi)×exp[-iβ2(αi)d2]},
D(j)(γ, α)=αwj+γwj+1 α-γβj(α)+βj+1(γ)+βj(α)wj-βj+1(γ)wj+1,
C(j)(γ, α)=αwj+γwj+1 α-γβj(α)-βj+1(γ)+βj(α)wj+βj+1(γ)wj+1,
I(3){±[β3(α)±β4(γ)]}=12π -+ exp{i(α-γ)y±i[β3(α)±β4(γ)]f3(y)}dy.
r(α)=r(0)(α)+r(1)(α)+r(2)(α)+ .
G(3)(γ, α)=p=0 ipp! Fγ-α[f3p(y)]G3a(p)(γ, α),
G3a(p)(γ, α)=1β3(α) (G(2)(α, α)D(3)(γ, α)×exp[+iβ3(α)d3]{+[β3(α)+β4(γ)]}p+G(2)(α, α)C(3)(γ, α)×exp[-iβ3(α)d3]{-[β3(α)-β4(γ)]}p),
H(3)(γ, αi)=p=0 ipp! Fγ-αi[f3p(y)]H3a(p)(γ, αi),
H3a(p)(αi, γ)=1β3(αi) (H(2)(αi, αi)D(3)(γ, αi)×exp[+iβ3(αi)d3]×{+[β3(αi)+β4(γ)]}p+H(2)(αi, αi)C(3)(γ, αi)×exp[-iβ3(αi)d3]×{-[β3(αi)-β4(γ)]}p),
Fγ-α[f3p(y)]=12π -+ ×exp[i(α-γ)y]f3p(y)dy
p=0k ipp! dα Fγ-α[f3p(y)]G3a(p)(γ, α)r(k-p)(α)
=ikk! Fγ-αi[f3k(y)]H3a(k)(γ, αi).
r(0)(γ)=δγ-αiH3a(0)(γ, αi)/G3a(0)(γ, γ).
r(1)(γ)=iO(γ, αi)Fγ-αi[f3(y)],
dRdθsincoh=cos2 θscos θi [|r(γ)|2-|r(γ)|2]γ=(ω/c)1 sin θs,
dRdθscoher=(cos θi)|r(α)|αi=-(ω/c)1 sin θi2.
dRdθsincoh=2πh3λ2 cos2 θscos θi |O(γ, αi)|2×12π 2πσ3λexp[-(γ-αi)2σ32/4];
f3(ym)=(h3/a)l=-M/2M/2-1Fl exp(iαlym),
Fl=2πaGl[N(0, 1)+iN(0, 1)]/2l0, M/2N(0, 1)l=0, M/2,
I(3)×{±[β3(α)±β4(γ)]}
=1a m=1M exp(i{(α-γ)ym±[β3(α)±β4(γ)]f3(ym)})Δy.
n=-NN-1G(3)(αm, αn)r(αn)=H(3)(αm, αi),

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