Abstract

The reflection of s- and p-polarized light from an N-layer system of inhomogeneous ultrathin dielectric films upon absorbing homogeneous substrates is investigated. The first-order approximate expressions for differential reflectance and changes in the ellipsometric parameters that are caused by a multilayer system are obtained in the long-wavelength limit. The possibilities of using these formulas for resolving the inverse problem for inhomogeneous ultrathin films are discussed. A number of novel options are developed for simultaneously determining the dielectric constant and thickness of a homogeneous ultrathin film by differential reflectance and ellipsometric measurements.

© 2003 Optical Society of America

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References

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  1. P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, Germany, 1912).
  2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. L. Ward, The Optical Constants of Bulk Materials and Films (IOP Publishing, Bristol, UK, 1988).
  4. G. Bauer and W. Richter, eds., Optical Characterization of Epitaxial Semiconductor Layers (Springer-Verlag, Berlin, 1996).
  5. A. N. Saxena, “Changes in the phase and amplitude of polarized light reflected from a film-covered surface and their relations with the film thickness,” J. Opt. Soc. Am. 55, 1061-1067 (1965).
    [CrossRef]
  6. J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
    [CrossRef]
  7. P. V. Adamson, “Differential reflection spectroscopy of surface layers on thick transparent substrates with normally incident light,” Opt. Spectrosc. 80, 459-468 (1996).
  8. G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969).
  9. B. Sheldon, J. S. Haggerty, and A. G. Emslie, “Exact computation of the reflectance of a surface layer of arbitrary refractive-index profile and an approximate solution of the inverse problem,” J. Opt. Soc. Am. 72, 1049-1055 (1982).
    [CrossRef]
  10. F. Abeles, “Optical properties of thin absorbing films,” J. Opt. Soc. Am. 47, 473-482 (1957).
    [CrossRef]
  11. W. J. Plieth and K. Naegele, “U¨ber die Bestimmung der Optischen Konstanten Du¨nnster Oberfla¨chenschichten und das Problem der Schichtdicke,” Surf. Sci. 64, 484-496 (1977).
    [CrossRef]
  12. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, The Netherlands, 1987).
  13. A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
    [CrossRef]
  14. S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858-1861 (2000).
    [CrossRef]
  15. R. J. Archer and G. W. Gobeli, “Measurement of oxygen adsorption on silicon by ellipsometry,” J. Phys. Chem. Solids 26, 343-351 (1965).
    [CrossRef]
  16. W.-K. Paik and J. O’M. Bockris, “Exact ellipsometric measurement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,” Surf. Sci. 28, 61-68 (1971).
    [CrossRef]
  17. R. C. O’Handley, “Obtaining three surface-film parameters from two ellipsometric measurements,” Surf. Sci. 46, 24-42 (1974).
    [CrossRef]
  18. B. D. Cahan, “Implications of three parameter solutions to the three-layer model,” Surf. Sci. 56, 354–372 (1976).
    [CrossRef]

2000

S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858-1861 (2000).
[CrossRef]

1996

P. V. Adamson, “Differential reflection spectroscopy of surface layers on thick transparent substrates with normally incident light,” Opt. Spectrosc. 80, 459-468 (1996).

1982

1979

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
[CrossRef]

1977

W. J. Plieth and K. Naegele, “U¨ber die Bestimmung der Optischen Konstanten Du¨nnster Oberfla¨chenschichten und das Problem der Schichtdicke,” Surf. Sci. 64, 484-496 (1977).
[CrossRef]

1976

B. D. Cahan, “Implications of three parameter solutions to the three-layer model,” Surf. Sci. 56, 354–372 (1976).
[CrossRef]

1974

R. C. O’Handley, “Obtaining three surface-film parameters from two ellipsometric measurements,” Surf. Sci. 46, 24-42 (1974).
[CrossRef]

1971

W.-K. Paik and J. O’M. Bockris, “Exact ellipsometric measurement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,” Surf. Sci. 28, 61-68 (1971).
[CrossRef]

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
[CrossRef]

1965

1957

Abeles, F.

Adamson, P. V.

P. V. Adamson, “Differential reflection spectroscopy of surface layers on thick transparent substrates with normally incident light,” Opt. Spectrosc. 80, 459-468 (1996).

Archer, R. J.

R. J. Archer and G. W. Gobeli, “Measurement of oxygen adsorption on silicon by ellipsometry,” J. Phys. Chem. Solids 26, 343-351 (1965).
[CrossRef]

Aspnes, D. E.

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
[CrossRef]

Bagchi, A.

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
[CrossRef]

Barrera, R. G.

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
[CrossRef]

Bockris, J. O’M.

W.-K. Paik and J. O’M. Bockris, “Exact ellipsometric measurement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,” Surf. Sci. 28, 61-68 (1971).
[CrossRef]

Cahan, B. D.

B. D. Cahan, “Implications of three parameter solutions to the three-layer model,” Surf. Sci. 56, 354–372 (1976).
[CrossRef]

Emslie, A. G.

Gobeli, G. W.

R. J. Archer and G. W. Gobeli, “Measurement of oxygen adsorption on silicon by ellipsometry,” J. Phys. Chem. Solids 26, 343-351 (1965).
[CrossRef]

Haggerty, J. S.

McIntyre, J. P. E.

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
[CrossRef]

Naegele, K.

W. J. Plieth and K. Naegele, “U¨ber die Bestimmung der Optischen Konstanten Du¨nnster Oberfla¨chenschichten und das Problem der Schichtdicke,” Surf. Sci. 64, 484-496 (1977).
[CrossRef]

O’Handley, R. C.

R. C. O’Handley, “Obtaining three surface-film parameters from two ellipsometric measurements,” Surf. Sci. 46, 24-42 (1974).
[CrossRef]

Paik, W.-K.

W.-K. Paik and J. O’M. Bockris, “Exact ellipsometric measurement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,” Surf. Sci. 28, 61-68 (1971).
[CrossRef]

Plieth, W. J.

W. J. Plieth and K. Naegele, “U¨ber die Bestimmung der Optischen Konstanten Du¨nnster Oberfla¨chenschichten und das Problem der Schichtdicke,” Surf. Sci. 64, 484-496 (1977).
[CrossRef]

Rajagopal, A. K.

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
[CrossRef]

Saxena, A. N.

Sheldon, B.

Sihvola, A. H.

S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858-1861 (2000).
[CrossRef]

Tretyakov, S. A.

S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858-1861 (2000).
[CrossRef]

IEEE Trans. Antennas Propag.

S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858-1861 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Chem. Solids

R. J. Archer and G. W. Gobeli, “Measurement of oxygen adsorption on silicon by ellipsometry,” J. Phys. Chem. Solids 26, 343-351 (1965).
[CrossRef]

Opt. Spectrosc.

P. V. Adamson, “Differential reflection spectroscopy of surface layers on thick transparent substrates with normally incident light,” Opt. Spectrosc. 80, 459-468 (1996).

Phys. Rev. B

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824-4838 (1979).
[CrossRef]

Surf. Sci.

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
[CrossRef]

W.-K. Paik and J. O’M. Bockris, “Exact ellipsometric measurement of thickness and optical properties of a thin light-absorbing film without auxiliary measurements,” Surf. Sci. 28, 61-68 (1971).
[CrossRef]

R. C. O’Handley, “Obtaining three surface-film parameters from two ellipsometric measurements,” Surf. Sci. 46, 24-42 (1974).
[CrossRef]

B. D. Cahan, “Implications of three parameter solutions to the three-layer model,” Surf. Sci. 56, 354–372 (1976).
[CrossRef]

W. J. Plieth and K. Naegele, “U¨ber die Bestimmung der Optischen Konstanten Du¨nnster Oberfla¨chenschichten und das Problem der Schichtdicke,” Surf. Sci. 64, 484-496 (1977).
[CrossRef]

Other

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, The Netherlands, 1987).

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969).

P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, Germany, 1912).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

L. Ward, The Optical Constants of Bulk Materials and Films (IOP Publishing, Bristol, UK, 1988).

G. Bauer and W. Richter, eds., Optical Characterization of Epitaxial Semiconductor Layers (Springer-Verlag, Berlin, 1996).

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Figures (12)

Fig. 1
Fig. 1

Relative errors of approximate formulas (8) (dashed curves) and (2) (solid, dashed–dotted, and dashed–dot-dotted curves) as functions (a) of wavelength λ if φa=45° and (b) of angle of incidence φa if λ=1000 for a single film with d1=1 (curves 1 and 2), a three-film system with d1=d2=d3=1 (3), and a two-film system with d1=d2=1 (4) at na=1; ns=4; ks=1; n01=3 (1), 1.5 (2), 1 (3), 4 (4); nd1=3 (1), 4 (2), 1.5 (3, 4); n02=nd1 (3, 4); nd2=5 (3), 4 (4); n03=nd2 (3); and nd3=2 (3). Profiles n1(z) are described by Eq. (6) with g=1 (2), 2 (3), 0.25 (4); profiles n2(z) are described by Eq. (7) with g=1 (3) 4 (4); and profiles n3(z) are described by Eq. (8) (3). The quantities λ and di are measured in arbitrary common units.

Fig. 2
Fig. 2

Relative errors of approximate formulas (1) (dashed curves) and (2) (solid curves) as functions (a) of ks if ns=2 and (b) of ns if ks=2 for a single film with d1/λ=10-3 (curve 1), 2×10-3 (4); a three-film system with d1/λ=d2/λ=d3/λ=10-3 (2); and a two-film system with d1/λ=d2/λ=2×10-3 (3) at φa=75°; na=1; n01=1 (3), 1.5 (2), 3 (4), 4 (1); nd1=1.5 (1), 2.5 (2), 3 (4), 4 (3); n02=nd1 (2, 3); nd2=2 (3), 4 (2); n03=nd2 (2); and nd3=1.5 (2). Profiles n1(z) are described by Eq. (6) with g=4 (1), 3 (2), 1 (3); n2(z) by Eq. (7) with g=0.5 (2), and by Eq. (8) (3), and n3(z) by Eq. (12) (2).

Fig. 3
Fig. 3

(a) Differential reflectivities (ΔR1/R0)(s) (dotted curves) and (ΔR1/R0)(p) (solid, dashed–dotted, and dashed–dot-dotted curves) and (b) the relation (ΔR1/R0)(p)/(ΔR1/R0)(s) (solid, dashed–dotted, and dashed–dot-dotted curves) as functions of angle of incidence φa for a single inhomogeneous layer with a linear profile [Eq. (6)] if g=1, na=1, d1/λ=2×10-3, n01=1.5, nd1=ns=4, ks=0.5 (solid and dotted curves), ks=1 (dashed–dotted curves), and ks=3 (dashed–dot-dotted curves). Dashed curves correspond to the calculation by approximate formulas (1) and (2).

Fig. 4
Fig. 4

Relative errors of approximate Eqs. (11) (solid and dashed curves) and (12) (dashed–dotted, dashed–dot-dotted, and dotted curves) as functions (a) of wavelength λ and (b) of angle of incidence φa for na=1, ns=4, and ks=0.5. (a) φa=50°, d1=3, n01=1.5, nd1=4 (solid, dashed, dashed–dot-dotted, and dotted curves) and nd1=1.5 (dashed–dotted curve). Profiles n1(z) are described by Eq. (6) with g=1 (solid and dashed–dot-dotted curves) and g=0.5 (dashed and dotted curves). The quantities λ and d1 are measured in arbitrary common units. (b) A single film with d1/λ=2×10-3 (solid and dashed–dot-dotted curves), a three-film system (dotted curve) with d1/λ=d2/λ=d3/λ=10-3, and a two-film system (dashed–dotted curve) with d1/λ=10-3 and d2/λ=2×10-3; n01=1.5; nd1=4 (solid, dashed–dot-dotted, and dotted curves), and nd1=1.5 (dashed–dotted curve); n02=nd1; nd2=1.5 (dotted curve) and nd2=4 (dashed–dotted curve); n03=nd2 and nd3=2.5 (dotted curve). Profiles n1(z) are described by Eq. (6) with g=1 (solid and dashed–dot-dotted curves) and g=2 (dotted curve), n2(z) by Eq. (7) with g=0.5 (dotted curve) and g=4 (dashed–dotted curve), and n3(z) by Eq. (8) (dotted curve).

Fig. 5
Fig. 5

Relative errors of approximate Eqs. (11) (dashed, short-dashed, and dotted curves) and (12) (solid, dashed–dotted, and dashed–dot-dotted curves) as functions (a) of ks if ns=2.5 and (b) of ns if ks=1 for a single film (solid and dashed curves) with d1/λ=3×10-3, a two-film system (dashed–dotted and short-dashed curves) with d1/λ=2×10-3 and d2/λ=10-3, and a three-film system (dashed–dot-dotted and dotted curves) with d1/λ=d2/λ=d3/λ=10-3; φa=45°; na=1; n01=1.5 (solid, dashed, dashed–dot-dotted, and dotted curves), and n01=2.5 (dashed–dotted and short-dashed curves); nd1=1.5 (dashed–dotted and short-dashed curves), nd1=3 (solid and dashed curves), and nd1=4 (dashed–dot-dotted and dotted curves); n02=nd1; nd2=1.5 (dashed–dot-dotted and dotted curves) and nd2=2.5 (dashed–dotted and short-dashed curves); n03=nd3=1.5 (dashed–dot-dotted and dotted curves). Profiles n1(z) are described by Eq. (6) with g=3 (dashed–dot-dotted and dotted curves), by Eq. (7) with g=1 (dashed–dotted and short-dashed curves), and by Eq. (8) (solid and dashed curves), n2(z) by Eq. (7) with g=1 (dashed–dot-dotted and dotted curves) and by Eq. (8) (dashed–dotted and short-dashed curves), and n3(z) by Eq. (8) (dashed–dotted and dotted curves).

Fig. 6
Fig. 6

Ellipsometric quantities δΔ and δΨ as functions of angle of incidence φa for na=1, d1/λ=2×10-3; ns=4 (solid, dashed–dotted, and short-dashed curves), 1.5 (dashed and dashed–dot-dotted curves); ks=0.1 (dashed–dot-dotted curves), 0.5 (solid curve), 1 (dashed–dotted curve), 3 (dashed curve), 6 (short-dashed curves); n01=1 (dashed and dashed–dot-dotted curves), 1.5 (the other curves); nd1=1.5 (dashed and dashed–dot-dotted curves), 4 (the other curves). Profiles n1(z) are described by Eq. (6) with g=1. Dotted curves correspond to the calculation by approximate formulas (9).

Fig. 7
Fig. 7

Ellipsometric quantities δΔ (solid, dashed–dotted, and dashed–dot-dotted curves) and δΨ (dashed, short-dashed, and dotted curves) as functions of ks for na=1, φa=50°, d1/λ=10-3, n01=1.5, nd1=3.5, ns=1.5 (solid and dashed curves) and ns=3.5 (the other curves). Profiles n1(z) are described by Eq. (6) with g=5 (dashed–dotted and short-dashed curves) and g=1 (the other curves).

Fig. 8
Fig. 8

Relative error of approximate formula (14) as a function of d1/λ for φa=60°; na=1; ns=4 (curve 1), 1.5 (2); ks=0.5; n1=1.5 (1), 2 (2); μ=0 (solid curves), 5% (dashed curves); μ is the relative error of t1.

Fig. 9
Fig. 9

Relative error of approximate formula (15) as a function of ks for φa=55°, d1/λ=2×10-3; na=1; ns=4 (solid, dashed–dotted, and dashed–dot-dotted curves), 1.5 (dashed, short-dashed, and dotted curves); n1=1.5 (solid, dashed–dotted, and dashed–dot-dotted), 2 (dashed, short-dashed, and dotted curves); η=0 (solid and dashed curves), 5% (short-dashed and dashed–dotted curves), -5% (dashed–dot-dotted and dotted curves). η is the relative error of t2.

Fig. 10
Fig. 10

Relative errors of approximate formula (17) as functions (a) of d1/λ if φa=40° and (b) of φa if d1/λ=2×10-3 for ns=4 (solid, dashed, dashed–dot-dotted, and short-dashed curves), 1.5 (dashed-dotted and dotted curves); ks=2 (short-dashed curve), 0.5 (the other curves); n1=1.5 (solid, dashed, and short-dashed curves), 2 (dashed–dotted and dotted curves), 3.5 (dashed–dot-dotted curves); γ=0 (solid, dashed–dotted, and dashed–dot-dotted curves), 5% (dashed and dotted curves). γ is the relative error δΔ/(ΔR1/R0)(p).

Fig. 11
Fig. 11

Relative error of approximate formula (17) as a function of ks for na=1; φa=55°; d1/λ=4×10-3; ns=4 (curve 1), 1.5 (2, 3, 4); n1=1.5 (1), 2 (2), 2.5 (3), 3 (4); μ=0 (solid curves), -5% (dashed curve 1), 5% (dashed curve 2). μ is the relative error δΔ/(ΔR1/R0)(p).

Fig. 12
Fig. 12

Relative error of approximate formula (17) as a function of ns for na=1; φa=55°; d1/λ=4×10-3; ks=0.5 (solid and dashed curves), 1 (dashed–dotted curve), 2 (dashed–dot-dotted curve); n1=1.5 (solid and dashed curves), 3 (dashed–dotted and dashed–dot-dotted curves); μ=5% (dashed curve), 0 (the other curves).

Equations (43)

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ΔRNR0(s)8πnacos φaDξs×i=1N(εa-εti)(di/λ),
εti=di-10di εi(z)dz,
D=[(εs-εa)2+ξs2]-1,
ΔRNR0(p)8πnacos φaξsK i=1N(εa-εti)[1-sin2 φa(1+εaεti-1pi)](di/λ),
K=[1-2εaα sin2 φa]/{[εa(1-εaα sin2 φa)-εscos2 φa]2+ξs2[cos2 φa-εa2εs-1α sin2 φa]2},
εni-1=di-10di εi-1(z)dz,
pi=(1-εa/εni)/(1-εa/εti),
α=εs/|ε^s|2,|ε^s|2=εs2+ξs2.
φBi=arctan[εti/(εapi)]1/2,
ΔRNR0Bs(p)8πεs2[ξs(εs-εa)(εs+εa)1/2]-1i=1Nεti-1(εti-εa)(εspi-εti)(di/λ).
(ΔR1/R0)Bs(p)/[(ΔR1/R0)(s)(φa=φBs)]
εs2(εt1-εsp1)(εs-εa)/(εaεt1ξs2).
ni(z)=n0i+(ndi-n0i)(z/di)g,
ni(z)=n0indi[ndig-(ndig-n0ig)(z/di)]-1/g,
ni(z)=n0i(ndi/n0i)z/din0iexp[ln(ndi/n0i)(z/di)],
δΨ=(aΨ/2)sin 2Ψ0,δΔtan(δΔ)aΔ,
r^N(σ)r^0(σ)(1+i4πPa(σ){[Pa(σ)]2-[P^s(σ)]2}-1×i=1N{ci(σ)-[P^s(σ)]2bi(σ)}(di/λ)),
aΨ=4πnaξscos φasin2 φaA1-1i=1N(εti-εa)×εti-1(a2iM1-a1iM2)(di/λ),
aΔ=4πnacos φasin2 φaA1-1i=1N(εti-εa)×εti-1(a1iM1+ξs2a2iM2)(di/λ),
tan(δΔ)=4πnaεscos φasin2 φa[(εs-εa)(εscos2 φa-εasin2 φa)]-1i=1N(εa+εs-εti-εaεsεni-1)(di/λ).
ε1εa(K/D)sin2 φa[(K/D)cos2 φa-t1]-1.
ε1εa(K/D)sin2 φacos φa[(K/D)cos3 φa-t2]-1,
ε1(2εsf1+εs2-ξs2)/(f1+εs),
ε1[[M1(εs2-ξs2)-2M2εsξs2+f2εa]/(M1εs-M2ξs2+f2cotan2 φa),
f2=2ξsKA1[δΔ/(ΔR1/R0)(p)],
ε1[2M1εs+M2(εs2-ξs2)-f3εa]/(M1+M2εs-f3cotan2 φa),
f3=4KA1sin-1 2Ψ0[δΨ/(ΔR1/R0)(p)]
εasin2 φa(p){A2(ΔR1/R0)(s)
+2 cos φa(s)DA1ξs[cos φa(Δ)sin2 φa(Δ)]-1δΔ}
=A3{[D cos φa(s)/K cos φa(p)](ΔR1/R0)(p)
-cos2 φa(p)(ΔR1/R0)(s)},
A4{2 cos φa(s)[cos φa(Δ)sin2 φa(Δ)]-1DA1ξsδΔ
+A2(ΔR1/R0)(s)}
=A3{4 cos φa(s)[cos φa(Ψ)sin2 φa(Ψ)sin 2Ψ0]-1DA1δΨ
-A5(ΔR1/R0)(s)},
A2=M1εs-ξs2M2,
A3=2εsξs2M2-(εs2-ξs2)M1,
A4=2εsM1+(εs2-ξs2)M2,
A5=M1+εsM2,
εt1εa+(A4c1-A3c2)(A2A4+A3A5)-1,
εn1-1εa-1[1-(A5c1+A2c2)(A2A4+A3A5)-1],
c1=A1[4πnacos φa(Δ)sin2 φa(Δ)]-1(λ/d1)δΔ,
c2=A1[2πnacos φa(Ψ)sin2 φa(Ψ)sin 2Ψ0ξs]-1(λ/d1)δΨ.

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