Abstract

We present a multilayer modal method to investigate electromagnetic scattering from a lossy inhomogeneous aperture in a perfectly conducting thick screen. We consider inhomogeneous apertures that consist of two homogeneous zones filled with different isotropic materials that are characterized by complex refraction indices. The shape of the boundary between adjacent homogeneous zones is rather arbitrary, which makes it possible to model a great number of interesting geometries, such as slanted slabs or wedges. We illustrate the behavior of eigenvalues (solutions of the transcendental equation that has to be solved whenever a modal method is applied) as the imaginary part of the complex refraction index is varied. The method is used to study the electromagnetic response of a thick aperture with a sloping internal border. Calculation of power losses at the aperture walls and curves of scattered intensity versus observation angle are shown for different polarizations of the incident beam.

© 1998 Optical Society of America

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References

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  1. M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
    [CrossRef]
  2. R. A. Depine, D. C. Skigin, “Multilayer modal method for diffraction from dielectric inhomogeneous apertures,” J. Opt. Soc. Am. A 15, 675–683 (1998).
    [CrossRef]
  3. G. A. Schiavone, K. O’Neill, K. D. Paulsen, “Scattering from groove patterns in a perfectly conducting surface,” J. Opt. Soc. Am. A 14, 2212–2222 (1997).
    [CrossRef]
  4. O. Mata-Mendez, J. Sumaya-Martinez, “Scattering of TE-polarized waves by a finite grating: giant resonant enhancement of the electric field within the grooves,” J. Opt. Soc. Am. A 14, 2203–2211 (1997).
    [CrossRef]
  5. J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
    [CrossRef]
  6. T. J. Park, H. J. Eom, K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905–911 (1993).
    [CrossRef]
  7. R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
    [CrossRef]
  8. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  9. D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
    [CrossRef]
  10. D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
    [CrossRef]
  11. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  12. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  13. D. C. Skigin, R. A. Depine, “The R-matrix method applied to the SIBC for diffraction gratings of arbitrary profile,” Optik (Stuttgart) 105, 165–174 (1997).
  14. H. Lochbihler, R. A. Depine, “Diffraction from highly conducting wire gratings,” Appl. Opt. 32, 3459–3465 (1993).
    [CrossRef] [PubMed]
  15. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  16. D. J. Zvijak, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  17. J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  18. L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
    [CrossRef]
  19. L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
    [CrossRef]
  20. G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering 3, R. Kingslake, ed. (Academic, New York, 1966), p. 309.
  21. L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
    [CrossRef]
  22. M. F. Otero, R. G. Rojas, “Two-dimensional Green’s function for a wedge with impedance faces,” IEEE Trans. Antennas Propag. 45, 1799–1809 (1997).
    [CrossRef]
  23. A. R. Lopez, “Application of wedge diffraction theory to estimating power density at airport humped runways,” IEEE Trans. Antennas Propag. 35, 708–714 (1987).
    [CrossRef]

1998 (1)

1997 (6)

O. Mata-Mendez, J. Sumaya-Martinez, “Scattering of TE-polarized waves by a finite grating: giant resonant enhancement of the electric field within the grooves,” J. Opt. Soc. Am. A 14, 2203–2211 (1997).
[CrossRef]

G. A. Schiavone, K. O’Neill, K. D. Paulsen, “Scattering from groove patterns in a perfectly conducting surface,” J. Opt. Soc. Am. A 14, 2212–2222 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “The R-matrix method applied to the SIBC for diffraction gratings of arbitrary profile,” Optik (Stuttgart) 105, 165–174 (1997).

L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
[CrossRef]

M. F. Otero, R. G. Rojas, “Two-dimensional Green’s function for a wedge with impedance faces,” IEEE Trans. Antennas Propag. 45, 1799–1809 (1997).
[CrossRef]

1996 (2)

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
[CrossRef]

1994 (1)

1993 (4)

1987 (1)

A. R. Lopez, “Application of wedge diffraction theory to estimating power density at airport humped runways,” IEEE Trans. Antennas Propag. 35, 708–714 (1987).
[CrossRef]

1983 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

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

1977 (1)

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

1976 (2)

D. J. Zvijak, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Adams, J. L.

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

Andrewartha, J. R.

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

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

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

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

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

De Zutter, D.

L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
[CrossRef]

Depine, R. A.

R. A. Depine, D. C. Skigin, “Multilayer modal method for diffraction from dielectric inhomogeneous apertures,” J. Opt. Soc. Am. A 15, 675–683 (1998).
[CrossRef]

D. C. Skigin, R. A. Depine, “The R-matrix method applied to the SIBC for diffraction gratings of arbitrary profile,” Optik (Stuttgart) 105, 165–174 (1997).

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[CrossRef]

H. Lochbihler, R. A. Depine, “Diffraction from highly conducting wire gratings,” Appl. Opt. 32, 3459–3465 (1993).
[CrossRef] [PubMed]

Eom, H. J.

Fox, J. R.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Hass, G.

G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering 3, R. Kingslake, ed. (Academic, New York, 1966), p. 309.

Knockaert, L.

L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
[CrossRef]

Kuittinen, M.

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

Light, J. C.

D. J. Zvijak, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Lochbihler, H.

Lopez, A. R.

A. R. Lopez, “Application of wedge diffraction theory to estimating power density at airport humped runways,” IEEE Trans. Antennas Propag. 35, 708–714 (1987).
[CrossRef]

Mata-Mendez, O.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

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

O’Neill, K.

Olyslager, F.

L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
[CrossRef]

Otero, M. F.

M. F. Otero, R. G. Rojas, “Two-dimensional Green’s function for a wedge with impedance faces,” IEEE Trans. Antennas Propag. 45, 1799–1809 (1997).
[CrossRef]

Park, T. J.

Paulsen, K. D.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Rojas, R. G.

M. F. Otero, R. G. Rojas, “Two-dimensional Green’s function for a wedge with impedance faces,” IEEE Trans. Antennas Propag. 45, 1799–1809 (1997).
[CrossRef]

Schiavone, G. A.

Skigin, D. C.

R. A. Depine, D. C. Skigin, “Multilayer modal method for diffraction from dielectric inhomogeneous apertures,” J. Opt. Soc. Am. A 15, 675–683 (1998).
[CrossRef]

D. C. Skigin, R. A. Depine, “The R-matrix method applied to the SIBC for diffraction gratings of arbitrary profile,” Optik (Stuttgart) 105, 165–174 (1997).

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[CrossRef]

Sumaya-Martinez, J.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Turunen, J.

Walker, R. B.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Yoshitomi, K.

Zvijak, D. J.

D. J. Zvijak, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Appl. Opt. (1)

Chem. Phys. (1)

D. J. Zvijak, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Comput. Phys. Commun. (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

L. Knockaert, F. Olyslager, D. De Zutter, “The diaphanous wedge,” IEEE Trans. Antennas Propag. 45, 1374–1381 (1997).
[CrossRef]

M. F. Otero, R. G. Rojas, “Two-dimensional Green’s function for a wedge with impedance faces,” IEEE Trans. Antennas Propag. 45, 1799–1809 (1997).
[CrossRef]

A. R. Lopez, “Application of wedge diffraction theory to estimating power density at airport humped runways,” IEEE Trans. Antennas Propag. 35, 708–714 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Chem. Phys. (1)

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Mod. Opt. (2)

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Acta (3)

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

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Opt. Commun. (1)

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

Optik (Stuttgart) (1)

D. C. Skigin, R. A. Depine, “The R-matrix method applied to the SIBC for diffraction gratings of arbitrary profile,” Optik (Stuttgart) 105, 165–174 (1997).

Other (1)

G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering 3, R. Kingslake, ed. (Academic, New York, 1966), p. 309.

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Figures (10)

Fig. 1
Fig. 1

Configuration of the problem.

Fig. 2
Fig. 2

The multilayer approximation.

Fig. 3
Fig. 3

Im(umq) versus Re(umq) for a layer with L/λ=3.2, Δ1/λ=1.6, ν1=1, and several values of ν2 corresponding to aluminum at different wavelengths. (a) s polarization, (b) p polarization.

Fig. 4
Fig. 4

Im[(υmq)2] versus Re[(υmq)2] for a layer with L/λ=3.2, Δ1/λ=1.6, ν1=1, and several values of ν2 corresponding to a lossy dielectric. (a) s polarization, (b) p polarization.

Fig. 5
Fig. 5

Transmitted intensity versus sin θ for a structure with a vertical interface between zones 1 and 2, for several values of ν2. L/λ=3.2, Δ1/λ=1.6, ν1=νr=νt=1, w/λ=10, θ0=0°, and s polarization.

Fig. 6
Fig. 6

Transmitted intensity versus sin θ for a structure with a vertical interface between zones 1 and 2, for several angles of incidence. L/λ=3.2, Δ1/λ=1.6, ν1=νr=νt=1, ν2=2 +i0.05, w/λ=10, and s polarization.

Fig. 7
Fig. 7

Structure considered for Tables 3 and 4 and Figs. 8 and 9.

Fig. 8
Fig. 8

Transmitted intensity versus sin θ for the structure of Fig. 7, for several values of ν2. L/λ=3.2, Δ1/λ=1.6, ν1=νr=νt=1, w/λ=10, θ0=0°, and s polarization.

Fig. 9
Fig. 9

Transmitted intensity versus sin θ for the structure of Fig. 7, for several values of the width c. L/λ=3.2, Δ1/λ=1.6, ν1=νr=νt=1, ν2=2+i0.1, w/λ=10, θ0=0°, and s polarization.

Fig. 10
Fig. 10

Structure considered for Table 6.

Tables (6)

Tables Icon

Table 1 Power Rates for a Structure with a Vertical Interface between Zones 1 and 2, for Several Values of ν2a

Tables Icon

Table 2 Power Rates for a Structure with a Vertical Interface between Zones 1 and 2, for Several Values of θ0a

Tables Icon

Table 3 Power Rates for the Structure of Fig. 7, for Several Values of ca

Tables Icon

Table 4 Power Rates for a Structure with a Vertical Interface between Zones 1 and 2, for Several Values of w and x0a

Tables Icon

Table 5 Power Rates for a Structure with a Vertical Interface between Zones 1 and 2, for Several Values of ν2a

Tables Icon

Table 6 Power Rates for the Structure of Fig. 10, for Several Values of c, with the Incident Beam Centered at the Center of Zone 2a

Equations (16)

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

fI(x, y)=-krkrA(α)exp[i(αx-βry)]dα+-Rq(α)exp[i(αx+βry)]dα,
fIII(x, y)=-Tq(α)exp[i(ax-βty)]dα,
βl=(kl2-α2)1/2i(α2-kl2)1/2ifkl2>α2ifkl2<α2,l=r,t,
A(α)=1σπexp-(α-α0)2σ2exp[i(α-α0)b],
fjq(x, y)=m=1Um,jq(x)wm,jq(y)rectx-xminL,
wm,jq(y)=am,jq cos(vm,jqy)+bm,jq sin(vm,jqy),
Um,js(x)=sin[um,1,js(x-xmin)]forxmin<x<x1,j,sin(um,1,jsΔ1j)cos[um,2,js(x-x1,j)]+um,1,jsum,2,jscos(um,1,jsΔ1j)sin[um,2,js(x-x1,j)]forx1,j<x<xmax,
Um,jp(x)=cos[um,1,jp(x-xmin)]forxmin<x<x1,j,cos(um,1,jpΔ1j)cos[um,2,jp(x-x1,j)]-um,1,jpum,2,jp21sin(um,1,jpΔ1j)sin[um,2,jp(x-x1,j)]forx1,j<x<xmax,
um,n,jq=kn2-(vm,jq)2,n=1, 2,
1um,1,jscos(um,2,jsΔ2j)sin(um,1,jsΔ1j)
+1um,2,jssin(um,2,jsΔ2j)cos(um,1,jsΔ1j)=0,
um,2,jp2sin(um,2,jpΔ2j)cos(um,1,jpΔ1j)
+um,1,jp1cos(um,2,jpΔ2j)sin(um,1,jpΔ1j)=0.
umq(p.c.)=mπΔ1.
prefl+ptrans+pabs=1,
It(α)=|Tq(α)|2βt

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