Abstract

We report on a procedure of generalized ellipsometry for the determination of the optical constants of stratified samples that present a weak in-plane anisotropy. We first derive the analytical expressions of the Jones reflection matrix and of the detected intensity for a rotating-polarizer ellipsometer configuration. These expressions show that the in-plane birefringence and the orientation of the principal axes of the dielectric tensor can be obtained from a measurement of the normalized off-diagonal terms of the reflection matrix as a function of the sample azimuth, followed by a Fourier analysis and a wavelength-by-wavelength inversion of these experimental data. We apply this method to the optical characterization of Langmuir–Blodgett molecular films deposited on silicon substrates. The obtained results show that even for transparent and ultrathin films it is possible to accurately determine very weak in-plane birefringence and axis orientations.

© 1998 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  3. D. den Engelsen, “Ellipsometry of anisotropic films,” J. Opt. Soc. Am. 61, 1460–1466 (1971).
    [CrossRef]
  4. R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
    [CrossRef]
  5. D. J. De Smet, “Ellipsometry of anisotropic thin films,” J. Opt. Soc. Am. 64, 631–638 (1974).
    [CrossRef]
  6. A. Y. Tronin, A. F. Konstantinova, “Ellipsometric study of the optical anisotropy of lead arachidate Langmuir films,” Thin Solid Films 177, 305–314 (1989).
    [CrossRef]
  7. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
    [CrossRef]
  8. M. Schubert, B. Rheinländer, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, B. Johs, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930–1940 (1996).
    [CrossRef]
  9. D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
    [CrossRef]
  10. M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films, 313–314, 323–332 (1998).
    [CrossRef]
  11. H. Kuhn, D. Möbius, H. Bücher, Physical Methods of Chemistry (Wiley, New York, 1972), Vol. 1, Chap. 7.
  12. G. G. Roberts, Langmuir–Blodgett Films (Plenum, New York, 1990).
  13. A. Ulman, Ultrathin Organic Films (Academic, San Diego, Calif., 1991).
  14. Y. Ishino, H. Ishida, “Spectral simulation of uniaxially oriented monolayers in the infrared,” Langmuir 4, 1341–1346 (1988).
    [CrossRef]
  15. P. A. Chollet, “Determination by infrared absorption of the orientation of molecules in monomolecular layers,” Thin Solid Films 52, 343–360 (1978).
    [CrossRef]
  16. D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
    [CrossRef]
  17. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  18. H. Wöhler, M. Fritsch, G. Haas, D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A 8, 536–540 (1991).
    [CrossRef]
  19. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  20. H. Coudrile, M. Steers, J. B. Theeten, “Dispositif électronique pour l’analyse et le calcul des coefficients de Fourier d’une fonction périodique, et ellipsomètre comportant un tel dispositif,” French patent8020838 (September29, 1980).
  21. E. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), pp. 547–552 and 749–752.
  22. D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
    [CrossRef]

1998

D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
[CrossRef]

M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films, 313–314, 323–332 (1998).
[CrossRef]

1996

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[CrossRef]

M. Schubert, B. Rheinländer, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, B. Johs, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930–1940 (1996).
[CrossRef]

1992

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

1991

1989

A. Y. Tronin, A. F. Konstantinova, “Ellipsometric study of the optical anisotropy of lead arachidate Langmuir films,” Thin Solid Films 177, 305–314 (1989).
[CrossRef]

1988

Y. Ishino, H. Ishida, “Spectral simulation of uniaxially oriented monolayers in the infrared,” Langmuir 4, 1341–1346 (1988).
[CrossRef]

1978

P. A. Chollet, “Determination by infrared absorption of the orientation of molecules in monomolecular layers,” Thin Solid Films 52, 343–360 (1978).
[CrossRef]

1974

1972

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
[CrossRef]

1971

1941

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
[CrossRef]

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

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
[CrossRef]

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

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

Blaudez, D.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

Bücher, H.

H. Kuhn, D. Möbius, H. Bücher, Physical Methods of Chemistry (Wiley, New York, 1972), Vol. 1, Chap. 7.

Buffeteau, T.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

Castaings, N.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

Chollet, P. A.

P. A. Chollet, “Determination by infrared absorption of the orientation of molecules in monomolecular layers,” Thin Solid Films 52, 343–360 (1978).
[CrossRef]

Coudrile, H.

H. Coudrile, M. Steers, J. B. Theeten, “Dispositif électronique pour l’analyse et le calcul des coefficients de Fourier d’une fonction périodique, et ellipsomètre comportant un tel dispositif,” French patent8020838 (September29, 1980).

Cramer, C.

De Smet, D. J.

den Engelsen, D.

Desbat, B.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

DeVries, M. J.

D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
[CrossRef]

Escafre, N.

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

Fritsch, M.

Haas, G.

Herzinger, C. M.

Ishida, H.

Y. Ishino, H. Ishida, “Spectral simulation of uniaxially oriented monolayers in the infrared,” Langmuir 4, 1341–1346 (1988).
[CrossRef]

Ishino, Y.

Y. Ishino, H. Ishida, “Spectral simulation of uniaxially oriented monolayers in the infrared,” Langmuir 4, 1341–1346 (1988).
[CrossRef]

Johs, B.

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
[CrossRef]

Konstantinova, A. F.

A. Y. Tronin, A. F. Konstantinova, “Ellipsometric study of the optical anisotropy of lead arachidate Langmuir films,” Thin Solid Films 177, 305–314 (1989).
[CrossRef]

Kuhn, H.

H. Kuhn, D. Möbius, H. Bücher, Physical Methods of Chemistry (Wiley, New York, 1972), Vol. 1, Chap. 7.

Mlynski, D. A.

Möbius, D.

H. Kuhn, D. Möbius, H. Bücher, Physical Methods of Chemistry (Wiley, New York, 1972), Vol. 1, Chap. 7.

Palik, E.

E. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), pp. 547–552 and 749–752.

Rheinländer, B.

Roberts, G. G.

G. G. Roberts, Langmuir–Blodgett Films (Plenum, New York, 1990).

Schmiedel, H.

Schubert, M.

Steers, M.

H. Coudrile, M. Steers, J. B. Theeten, “Dispositif électronique pour l’analyse et le calcul des coefficients de Fourier d’une fonction périodique, et ellipsomètre comportant un tel dispositif,” French patent8020838 (September29, 1980).

Theeten, J. B.

H. Coudrile, M. Steers, J. B. Theeten, “Dispositif électronique pour l’analyse et le calcul des coefficients de Fourier d’une fonction périodique, et ellipsomètre comportant un tel dispositif,” French patent8020838 (September29, 1980).

Thompson, D. W.

D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
[CrossRef]

Tiwald, T. E.

D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
[CrossRef]

Tronin, A. Y.

A. Y. Tronin, A. F. Konstantinova, “Ellipsometric study of the optical anisotropy of lead arachidate Langmuir films,” Thin Solid Films 177, 305–314 (1989).
[CrossRef]

Turlet, J.-M.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

Ulman, A.

A. Ulman, Ultrathin Organic Films (Academic, San Diego, Calif., 1991).

Wöhler, H.

Woollam, J. A.

J. Chem. Phys.

D. Blaudez, T. Buffeteau, N. Castaings, B. Desbat, J.-M. Turlet, “Organization in pure and alternate deuterated cadmium arachidate monolayers on solid substrates and at the air/water interface by conventional and differential Fourier transform infrared spectroscopies,” J. Chem. Phys. 104, 9983–9993 (1996).
[CrossRef]

J. Opt. Soc. Am.

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

J. Opt. Soc. Am.

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Langmuir

Y. Ishino, H. Ishida, “Spectral simulation of uniaxially oriented monolayers in the infrared,” Langmuir 4, 1341–1346 (1988).
[CrossRef]

Phys. Rev. B

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Thin Solid Films

D. W. Thompson, M. J. DeVries, T. E. Tiwald, J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films, 313–314, 341–346 (1998).
[CrossRef]

Thin Solid Films

P. A. Chollet, “Determination by infrared absorption of the orientation of molecules in monomolecular layers,” Thin Solid Films 52, 343–360 (1978).
[CrossRef]

Thin Solid Films

D. Blaudez, T. Buffeteau, B. Desbat, N. Escafre, J.-M. Turlet, “Inplane organization of Langmuir–Blodgett monolayers from FTIR,” Thin Solid Films 210/211, 648–651 (1992).
[CrossRef]

M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films, 313–314, 323–332 (1998).
[CrossRef]

A. Y. Tronin, A. F. Konstantinova, “Ellipsometric study of the optical anisotropy of lead arachidate Langmuir films,” Thin Solid Films 177, 305–314 (1989).
[CrossRef]

Other

H. Coudrile, M. Steers, J. B. Theeten, “Dispositif électronique pour l’analyse et le calcul des coefficients de Fourier d’une fonction périodique, et ellipsomètre comportant un tel dispositif,” French patent8020838 (September29, 1980).

E. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), pp. 547–552 and 749–752.

H. Kuhn, D. Möbius, H. Bücher, Physical Methods of Chemistry (Wiley, New York, 1972), Vol. 1, Chap. 7.

G. G. Roberts, Langmuir–Blodgett Films (Plenum, New York, 1990).

A. Ulman, Ultrathin Organic Films (Academic, San Diego, Calif., 1991).

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

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Figures (9)

Fig. 1
Fig. 1

Schematic of the Cartesian laboratory frame (x, y, z) and of the plane of incidence (hatched area). The azimuth of sample φ is shown as positive.

Fig. 2
Fig. 2

Schematic of the principal axes (a, b, c) of the dielectric tensor of a film with its main principal axis c normal to the surface and with in-plane birefringence Δ n = n a - n b . Here φ ( Δ ) is shown as positive.

Fig. 3
Fig. 3

Index ellipsoid of the studied film with its main principal axis c tilted at an angle θ relative to the surface normal z and with in-plane birefringence Δ n . Here θ and φ ( θ ) are shown as positive.

Fig. 4
Fig. 4

Contour plots of the errors δ(θ) (solid curves) and δ ( Δ n ) (dashed curves) introduced by the first-order expansions used in the method.

Fig. 5
Fig. 5

a, Measured quantities β p (circles) and β s (squares) as a function of the sample azimuth φ at a wavelength λ = 0.28   µ m and at an angle of incidence ϕ i = 70 ° for a LB film constituted of 19 cadmium behenate monolayers deposited on silicon. Solid curves are corresponding harmonic fits. b, Semilogarithmic representation of the Fourier analysis of the behavior of β p (circles) and β s (squares) versus φ.

Fig. 6
Fig. 6

a, Amplitudes A p ( θ ) (circles) and A s ( θ ) (squares) of the first Fourier component of β p , s ( ϕ ) as a function of the wavelength. Solid curves serve as a guide for the eye. b, Phase φ ( θ ) of the first Fourier component of β p ( φ ) (circles) and of β s ( φ ) (squares) as a function of the wavelength.

Fig. 7
Fig. 7

a, Amplitudes A p ( Δ ) (circles) and A s ( Δ ) (squares) of the second Fourier component of β p , s ( φ ) as a function of the wavelength. Solid curves serve as a guide for the eye. b, Phase φ ( Δ ) of the second Fourier component of β p ( φ ) (circles) and from analysis of β s ( φ ) (squares) as a function of the wavelength.

Fig. 8
Fig. 8

a, Tilt angle θ of the main principal axis c obtained from A p ( θ ) (circles) and A s ( θ ) (squares) as a function of λ for 19 monolayers of cadmium behenate deposited on silicon. b, In-plane birefringence Δ n obtained from A p ( Δ ) (circles) and A s ( Δ ) (squares) as a function of λ for 19 monolayers of cadmium behenate deposited on silicon.

Fig. 9
Fig. 9

a, Tilt angle θ obtained from A p ( θ ) (circles) and A s ( θ ) (squares) as a function of λ for 9 monolayers of cadmium behenate deposited on silicon. b, In-plane birefringence Δ n obtained from A p ( Δ ) (circles) and A s ( Δ ) (squares) as a function of λ for 9 monolayers of cadmium behenate deposited on silicon.

Equations (165)

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

[ ( ipi ) ] = n o 2 0 0 0 n o 2 0 0 0 n e 2 ,
n a = n o + Δ n / 2 ,
n b = n o - Δ n / 2 .
[ ] = [ P ( Δ ) ] ( n o + Δ n / 2 ) 2 0 0 0 ( n o - Δ n / 2 ) 2 0 0 0 n e 2 × [ P ( Δ ) ] - 1 ,
P ( Δ ) = cos ( φ - φ ( Δ ) ) sin ( φ - φ ( Δ ) ) 0 - sin ( φ - φ ( Δ ) ) cos ( φ - φ ( Δ ) ) 0 0 0 1 .
[ ] = [ P ( θ ) ] [ ] [ P ( θ ) ] - 1 ,
P ( θ ) = 1 0 - θ   cos ( φ - φ ( θ ) ) 0 1 θ   sin ( φ - φ ( θ ) ) θ   cos ( φ - φ ( θ ) ) - θ   sin ( φ - φ ( θ ) ) 1 .
[ ] = [ ( ipi ) ] + ( n e 2 - n o 2 ) × θ × [ ( θ ) ] + n o × Δ n × [ ( Δ ) ] ,
[ ( θ ) ] = 0 0 - cos ( φ - φ ( θ ) ) 0 0 sin ( φ - φ ( θ ) ) - cos ( φ - φ ( θ ) ) sin ( φ - φ ( θ ) ) 0 ,
[ ( Δ ) ] = cos ( 2 φ - 2 φ ( Δ ) ) - sin ( 2 φ - 2 φ ( Δ ) ) 0 - sin ( 2 φ - 2 φ ( Δ ) ) - cos ( 2 φ - 2 φ ( Δ ) ) 0 0 0 0 .
d Ψ ( z ) = - ik 0 Δ Ψ ( z ) dz ,
Ψ ( z ) = E x H y E y - H x
Δ = Δ ( ipi ) + η ( θ ) Δ ( θ ) + η ( Δ ) Δ ( Δ ) ,
η ( θ ) = sin ( ϕ i ) × ( 1 - n o 2 / n e 2 ) × θ ,
η ( Δ ) = n o Δ n
Δ ( ipi ) = 0 A 0 0 B 0 0 0 0 0 0 1 0 0 C 0 ,
Δ ( θ ) = cos ( φ - φ ( θ ) ) 0 - sin ( φ - φ ( θ ) ) 0 0 cos ( φ - φ ( θ ) ) 0 0 0 0 0 0 0 - sin ( φ - φ ( θ ) ) 0 0 ,
Δ ( Δ ) = 0 0 0 0 cos ( 2 φ - 2 φ ( Δ ) ) 0 - sin ( 2 φ - 2 φ ( Δ ) ) 0 0 0 0 0 - sin ( 2 φ - 2 φ ( Δ ) ) 0 - cos ( 2 φ - 2 φ ( Δ ) ) 0 ,
A = 1 - sin ( ϕ i ) n e 2 ,
B = n o 2 ,
C = n o 2 - sin ( ϕ i ) 2 ,
Ψ ( z 0 + d ) = L ( d ) Ψ ( z 0 ) ,
L = L ( ipi ) + η ( θ ) L ( θ ) + η ( Δ ) L ( Δ ) ,
L ( ipi ) = L [ 1,1 ] ( ipi ) L [ 1,2 ] ( ipi ) 0 0 L [ 2,1 ] ( ipi ) L [ 2,2 ] ( ipi ) 0 0 0 0 L [ 3,3 ] ( ipi ) L [ 3,4 ] ( ipi ) 0 0 L [ 4,3 ] ( ipi ) L [ 4,4 ] ( ipi ) ,
L [ 1,1 ] ( ipi ) = cos ( k 0 d AB ) ,
L [ 1,2 ] ( ipi ) = - i A / B   sin ( k 0 d AB ) ,
L [ 2,1 ] ( ipi ) = - i B / A   sin ( k 0 d AB ) ,
L [ 2,2 ] ( ipi ) = L [ 1,1 ] ( ipi ) ,
L [ 3,3 ] ( ipi ) = cos ( k 0 d C ) ,
L [ 3,4 ] ( ipi ) = - i 1 / C   sin ( k 0 d C ) ,
L [ 4,3 ] ( ipi ) = - i C   sin ( k 0 d C ) ,
L [ 4,4 ] ( ipi ) = L [ 3,3 ] ( ipi )
L ( θ ) = L ( θ 1 )   cos ( φ - φ ( θ ) ) + L ( θ 2 )   sin ( φ - φ ( θ ) ) ,
L ( θ 1 ) = k 0 d L [ 1,1 ] ( θ 1 ) L [ 1,2 ] ( θ 1 ) 0 0 L [ 2,1 ] ( θ 1 ) L [ 2,2 ] ( θ 1 ) 0 0 0 0 0 0 0 0 0 0
L [ 1,1 ] ( θ 1 ) = - i   cos ( k 0 d AB ) ,
L [ 1,2 ] ( θ 1 ) = - A / B   sin ( k 0 d AB ) ,
L [ 2,1 ] ( θ 1 ) = - B / A   sin ( k 0 d AB ) ,
L [ 2,2 ] ( θ 1 ) = L [ 1,1 ] ( θ 1 )
L ( θ 2 ) = 1 AB - C 0 0 L [ 1,3 ] ( θ 2 ) L [ 1,4 ] ( θ 2 ) 0 0 L [ 2,3 ] ( θ 2 ) L [ 2,4 ] ( θ 2 ) L [ 3,1 ] ( θ 2 ) L [ 3,2 ] ( θ 2 ) 0 0 L [ 4,1 ] ( θ 2 ) L [ 4,2 ] ( θ 2 ) 0 0
L [ 1,3 ] ( θ 2 ) = i AB   sin ( k 0 d AB ) - i C   sin ( k 0 d C ) ,
L [ 1,4 ] ( θ 2 ) = - cos ( k 0 d AB ) + cos ( k 0 d C ) ,
L [ 2,3 ] ( θ 2 ) = BL [ 1,4 ] ( θ 2 ) ,
L [ 2,4 ] ( θ 2 ) = i B / A   sin ( k 0 d AB ) - iB 1 / C   sin ( k 0 d C ) ,
L [ 3,1 ] ( θ 2 ) = L [ 2,4 ] ( θ 2 ) ,
L [ 3,2 ] ( θ 2 ) = L [ 1,4 ] ( θ 2 ) ,
L [ 4,1 ] ( θ 2 ) = L [ 1,4 ] ( θ 2 ) ,
L [ 4,2 ] ( θ 2 ) = L [ 1,3 ] ( θ 2 ) ,
L ( Δ ) = L ( Δ 1 )   cos ( 2 φ - 2 φ ( Δ ) ) + L ( Δ 2 )   sin ( 2 φ - 2 φ ( Δ ) ) ,
L ( Δ 1 ) = 1 2 L [ 1,1 ] ( Δ 1 ) L 1,2 ( Δ 1 ) 0 0 L [ 2,1 ] ( Δ 1 ) L [ 2,2 ] ( Δ 1 ) 0 0 0 0 L [ 3,3 ] ( Δ 1 ) L [ 3,4 ] ( Δ 1 ) 0 0 L [ 4,3 ] ( Δ 1 ) L [ 4,4 ] ( Δ 1 )
L [ 1,1 ] ( Δ 1 ) = - k 0 d A / B   sin ( k 0 d AB ) ,
L [ 1,2 ] ( Δ 1 ) = - ik 0 ( A / B ) cos ( k 0 d AB ) + i A / B 3   sin ( k 0 d AB ) ,
L [ 2,1 ] ( Δ 1 ) = - ik 0 d   cos ( k 0 d AB ) - i 1 / AB   sin ( k 0 d AB ) ,
L [ 2,2 ] ( Δ 1 ) = L [ 1,1 ] ( Δ 1 )
L [ 3,3 ] ( Δ 1 ) = k 0 d 1 / C   sin ( k 0 d C ) ,
L [ 3,4 ] ( Δ 1 ) = ik 0 d ( 1 / C ) cos ( k 0 d C ) - i 1 / C 3   sin ( k 0 d C ) ,
L [ 4,3 ] ( Δ 1 ) = ik 0 d   cos ( k 0 d C ) + i 1 / C   sin ( k 0 d C ) ,
L [ 4,4 ] ( Δ 1 ) = L [ 3,3 ] ( Δ 1 )
L ( Δ 2 ) = 1 AB - C 0 0 L [ 1,3 ] ( Δ 2 ) L [ 1,4 ] ( Δ 2 ) 0 0 L [ 2,3 ] ( Δ 2 ) L [ 2,4 ] ( Δ 2 ) L [ 3,1 ] ( Δ 2 ) L [ 3,2 ] ( Δ 2 ) 0 0 L [ 4,1 ] ( Δ 2 ) L [ 4,2 ] ( Δ 2 ) 0 0
L [ 1,3 ] ( Δ 2 ) = - A   cos ( k 0 d AB ) + A   cos ( k 0 d C ) ,
L [ 1,4 ] ( Δ 2 ) = i A / B   sin ( k 0 d AB ) - iA 1 / C   sin ( k 0 d C ) ,
L [ 2,3 ] ( Δ 2 ) = i AB   sin ( k 0 d AB ) - i C   sin ( k 0 d C ) ,
L [ 2,4 ] ( Δ 2 ) = L [ 1,3 ] ( Δ 2 ) / A ,
L [ 3,1 ] ( Δ 2 ) = L [ 1,3 ] ( Δ 2 ) / A ,
L [ 3,2 ] ( Δ 2 ) = L [ 1,4 ] ( Δ 2 ) ,
L [ 4,1 ] ( Δ 2 ) = L [ 2,3 ] ( Δ 2 ) ,
L [ 4,2 ] ( Δ 2 ) = L [ 1,3 ] ( Δ 2 ) .
A x = 1 - sin ( ϕ i ) n x 2 ,
B x = n x 2 ,
C x = n x 2 - sin ( ϕ i ) 2
L x = cos ( ξ x k 0 d x ) - i   ξ x n x 2 sin ( ξ x k 0 d x ) 0 0 - i   n x 2 ξ x sin ( ξ x k 0 d x ) cos ( ξ x k 0 d x ) 0 0 0 0 cos ( ξ x k 0 d x ) - i   1 ξ x sin ( ξ x k 0 d x ) 0 0 - i ξ x   sin ( ξ x k 0 d x ) cos ( ξ x k 0 d x ) ,
L = L x L = L x L ( ipi ) + η ( θ ) L x L ( θ ) + η ( Δ ) L x L ( Δ ) .
R = r ˜ pp r ˜ ps r ˜ sp r ˜ ss
R = - [ KL K r ] - 1 [ KL K i ] ,
K = - n s cos ( ϕ t ) 0 0 0 0 n s   cos ( ϕ t ) - 1 ,
K r = - cos ( ϕ i ) 0 1 0 0 1 0 - cos ( ϕ i ) ,
K i = cos ( ϕ i ) 0 1 0 0 1 0 cos ( ϕ i ) ,
R = R ( ipi ) + η ( θ ) [ R ( θ 1 )   cos ( φ - φ ( θ ) ) + R ( θ 2 )   sin ( φ - φ ( θ ) ) ] + η ( Δ ) [ R ( Δ 1 )   cos ( 2 φ - 2 φ ( Δ ) ) + R ( Δ 2 )   sin ( 2 φ - 2 φ ( Δ ) ) ] ,
R ( ipi ) = - [ L r ( ipi ) ] - 1 [ L i ( ipi ) ] ,
R ( θ 1 ) = - [ L r ( ipi ) ] - 1 ( L i ( θ 1 ) + L r ( θ 1 ) R ( ipi ) ) ,
R ( θ 2 ) = - [ L r ( ipi ) ] - 1 ( L i ( θ 2 ) + L r ( θ 2 ) R ( ipi ) ) ,
R ( Δ 1 ) = - [ L r ( ipi ) ] - 1 ( L i ( Δ 1 ) + L r ( Δ 1 ) R ( ipi ) ) ,
R ( Δ 2 ) = - [ L r ( ipi ) ] - 1 ( L i ( Δ 2 ) + L r ( Δ 2 ) R ( ipi ) ) ,
L i ( ξ ) = KL x L ( ξ ) K i ,
L r ( ξ ) = KL x L ( ξ ) K r .
R ( ipi ) = r ˜ pp ( ipi ) 0 0 r ˜ ss ( ipi ) ,
R ( θ 1 ) = 0 0 0 0 ,
R ( Δ 1 ) = r ˜ pp ( Δ ) 0 0 r ˜ ss ( Δ ) .
R ( θ 2 ) = 0 r ps ( θ ) r sp ( θ ) 0 ,
R ( Δ 2 ) = 0 r ps ( Δ ) r sp ( Δ ) 0 .
r ˜ pp = r ˜ pp ( ipi ) + η Δ r ˜ pp ( Δ )   cos ( 2 φ - 2 φ ( Δ ) ) ,
r ˜ ss = r ˜ ss ( ipi ) + η Δ r ˜ ss ( Δ )   cos ( 2 φ - 2 φ ( Δ ) ) ,
r ˜ ps = η ( θ ) r ˜ ps ( θ )   sin ( φ - φ ( θ ) ) + η ( Δ ) r ˜ ps ( Δ )   sin ( 2 φ - 2 φ ( Δ ) ) ,
r ˜ sp = η ( θ ) r ˜ sp ( θ )   sin ( φ - φ ( θ ) ) + η ( Δ ) r ˜ sp ( Δ )   sin ( 2 φ - 2 φ ( Δ ) ) .
I d = I 0 ( 1 + α   cos   2 θ pol + β   sin   2 θ pol ) ,
α = α p = 1 ,
β = β p = 2   Re ( r ˜ ps / r ˜ pp ) ,
α = α s = - 1 ,
β = β s = 2   Re ( r ˜ sp / r ˜ ss ) .
β p = A p ( θ )   sin ( φ - φ ( θ ) ) + A p ( Δ )   sin ( 2 φ - 2 φ ( Δ ) ) ,
β s = A s ( θ )   sin ( φ - φ ( θ ) ) + A s ( Δ )   sin ( 2 φ - 2 φ ( Δ ) ) ,
A p ( θ ) = B p ( θ ) × θ ,
A p ( Δ ) = B p ( Δ ) × Δ n ,
A s ( θ ) = B s ( θ ) × θ ,
A s ( Δ ) = B s ( Δ ) × Δ n ,
B p ( θ ) = 2   Re ( r ˜ ps ( θ ) / r ˜ pp ( ipi ) ) sin ( ϕ i ) ( 1 - n o 2 / n e 2 ) ,
B p ( Δ ) = 2   Re ( r ˜ ps ( Δ ) / r ˜ pp ( ipi ) ) n o ,
B s ( θ ) = 2   Re ( r ˜ sp ( θ ) / r ˜ ss ( ipi ) ) sin ( ϕ i ) ( 1 - n o 2 / n e 2 ) ,
B s ( Δ ) = 2   Re ( r ˜ sp ( Δ ) / r ˜ ss ( ipi ) ) n o .
α ± = tan 2   Ψ - 1 tan 2   Ψ + 1 4   tan 2   Ψ ( tan 2   Ψ + 1 ) 2 Re r ˜ ps r ˜ ss ± 4   tan 2   Ψ ( tan 2   Ψ + 1 ) 2 Re r ˜ sp r ˜ pp ,
β ± = ± 2   tan   Ψ   cos   Δ tan 2   Ψ + 1 + 2   tan 2   Ψ tan 2   Ψ + 1 × Re r ˜ ps r ˜ pp + 2 tan 2   Ψ + 1 × Re r ˜ sp r ˜ ss - 4   tan   Ψ   cos   Δ ( tan 2   Ψ + 1 ) 2 × Re r ˜ ps r ˜ ss - 4   tan 3   Ψ   cos   Δ ( tan 2   Ψ + 1 ) 2 Re r ˜ sp r ˜ pp ,
tan   Ψ   exp ( i Δ ) = r ˜ pp r ˜ ss .
α ¯ = α + + α - 2 ,
β ¯ = β + - β - 2 ,
n e = ( 1.56 ± 0.02 ) + ( 6 ± 2 ) × 10 - 3 / λ 2 ,
n o = ( 1.49 ± 0.03 ) + ( 6 ± 2 ) × 10 - 3 / λ 2 ,
d = ( 55.6 ± 6 ) Å ( 29.2 Å for each monolayer )
d x = 24 Å .
U = cos   ϕ 1   cos   ϕ 2   cos   ϕ 3 - sin   ϕ 1   sin   ϕ 3 - cos   ϕ 1   cos   ϕ 2   sin   ϕ 3 - sin   ϕ 1   cos   ϕ 3 cos   ϕ 1   sin   ϕ 2 sin   ϕ 1   cos   ϕ 2   cos   ϕ 3 + cos   ϕ 1   sin   ϕ 3 - sin   ϕ 1   cos   ϕ 2   sin   ϕ 3 + cos   ϕ 1   cos   ϕ 3 sin   ϕ 1   sin   ϕ 2 - sin   ϕ 2   cos   ϕ 3 sin   ϕ 2   sin   ϕ 3 cos   ϕ 2
[ ] ( a ,   b ,   c ) = n a 2 0 0 0 n b 2 0 0 0 n c 2 ,
[ ] ( x ,   y ,   z ) = U [ ] ( a ,   b ,   c ) U - 1 ,
( x ,   y ,   z ) R z ( ϕ 1 ) ( x ,   y ,   z ) R y ( ϕ 2 ) ( x ,   y ,   c )
R c ( ϕ 3 ) ( a ,   b ,   c ) .
[ ] ( x ,   y ,   z )
= n a + n b 2 2 0 0 0 n a + n b 2 2 0 0 0 n c 2 + ϕ 2 n c 2 - n a + n b 2 2 0 0 cos   ϕ 1 0 0 sin   ϕ 1 cos   ϕ 1 sin   ϕ 1 0 + ( n a - n b )   n a + n b 2 × cos ( 2 ϕ 1 + 2 ϕ 3 ) sin ( 2 ϕ 1 + 2 ϕ 3 ) 0 sin ( 2 ϕ 1 + 2 ϕ 3 ) - cos ( 2 ϕ 1 + 2 ϕ 3 ) 0 0 0 0 .
n o = ( n a + n b ) / 2 , n e = n c ,
Δ n = n a - n b , θ = - ϕ 2 ,
φ ( θ ) - φ = ϕ 1 , φ ( Δ ) - φ = ϕ 1 + ϕ 3 ,
[ ] ( x ,   y ,   z ) = n o 2 0 0 0 n o 2 0 0 0 n e 2 + θ ( n e 2 - n o 2 ) 0 0 - cos ( φ - φ ( θ ) ) 0 0 sin ( φ - φ ( θ ) ) - cos ( φ - φ ( θ ) ) sin ( φ - φ ( θ ) ) 0 + Δ n × n o cos ( 2 φ - 2 φ ( Δ ) ) - sin ( 2 φ - 2 φ ( Δ ) ) 0 - sin ( 2 φ - 2 φ ( Δ ) ) - cos ( 2 φ - 2 φ ( Δ ) ) 0 0 0 0 ,
L = exp ( - ik 0 d Δ ) ,
L = I + x · Δ + x 2 2 ! · Δ 2 + x 3 3 ! · Δ 3 + ,
L [ 1,2 ] = Ax + 1 6 A 2 Bx 3 + 1 120 A 3 B 2 x 5 + + η ( θ ) Ax 2 + 1 6 A 2 Bx 4 + 1 120 A 3 B 2 x 6 + × cos ( φ - φ ( θ ) ) + η ( Δ ) × 1 6 A 2 x 3 + 1 60 A 3 Bx 5 + cos ( 2 φ - 2 φ ( Δ ) ) ,
L [ 1,2 ] ( x ) = A B   sinh AB x + η ( θ ) A B   x   sinh AB x cos ( φ - φ ( θ ) ) + η ( Δ ) A B 2 B 3 / 2 x   cosh AB x - A 2 B 3 / 2 sinh AB x cos ( 2 φ - 2 φ ( Δ ) ) .
det ( Δ ( ipi ) + η ( θ ) + Δ ( θ ) + η ( Δ ) Δ ( Δ ) - q l ) = 0 ,
q l = q l ( ipi ) + η θ q l ( θ ) + η Δ q l ( Δ ) .
( Δ ( ipi ) + η ( θ ) Δ ( θ ) + η ( Δ ) Δ ( Δ ) - q l I ) Q l = 0 ,
Q l = Q l ( ipi ) + η ( θ ) Q l ( θ ) + η ( Δ ) Q l ( Δ ) .
q p ± ( ipi ) = ± AB , Q p ± ( ipi ) = ± A / B 1 0 0 , q s ± ( ipi ) = ± C , Q s ± ( ipi ) = 0 0 1 ± C ,
q p ± ( θ ) = cos ( φ - φ ( θ ) ) , Q p ± ( θ ) = 1 AB - C AB + C B cos ( φ - φ ( θ ) ) ± AB + C AB cos ( φ - φ ( θ ) ) - sin ( φ - φ ( θ ) ) ( AB ) sin ( φ - φ ( θ ) ) ,
q p ± ( θ ) = cos ( φ - φ ( θ ) ) , Q p ± ( θ ) = 1 AB - C AB + C B cos ( φ - φ ( θ ) ) ± AB + C AB cos ( φ - φ ( θ ) ) - sin ( φ - φ ( θ ) ) ( AB ) sin ( φ - φ ( θ ) ) ,
q s ± ( θ ) = 0 , Q s ± ( θ ) = 1 AB - C ± ( C ) sin ( φ - φ ( θ ) ) b   sin ( φ - φ ( θ ) ) 0 0 ,
q p ± ( Δ ) = ± 1 2 A B cos ( 2 φ - 2 φ ( Δ ) ) , Q s ± ( Δ ) = 1 AB - C ± 2 B + AB + C 2 B A B cos ( 2 φ - 2 φ ( Δ ) ) ( A + 1 ) cos ( 2 φ - 2 φ ( Δ ) ) A B sin ( 2 φ - 2 φ ( Δ ) ) - A   sin ( 2 φ - 2 φ ( Δ ) ) ,
q s ± ( Δ ) = 1 2 C cos ( 2 φ - 2 φ ( Δ ) ) , Q s ± ( Δ ) = A AB - C sin ( 2 φ - 2 φ ( Δ ) ) ± C AB - C sin ( 2 φ - 2 φ ( Δ ) ) 1 2 C cos ( 2 φ - 2 φ ( Δ ) ) 0 ,
A = 1 - sin ( ϕ i ) n e 2 , B = n o 2 , C = n o 2 - sin ( ϕ i ) 2 ,
L { Q l } = exp ( - ik 0 dq p + ) 0 0 0 0 exp ( - ik 0 dq p - ) 0 0 0 0 exp ( - ik 0 dq s + ) 0 0 0 0 exp ( - ik 0 dq s - ) { Q l } .
L = [ Q l ] L { Q l } [ Q l ] - 1 ,
[ Q l ] = [ Q p + Q p - Q s + Q s - ] .
Ψ t = L · ( Ψ i + Ψ r ) .
Ψ i = E ip   cos   ϕ i E ip E is E is   cos   ϕ i ,
Ψ r = - E rp   cos   ϕ i E rp E rs - E rs   cos   ϕ i ,
Ψ t = E tp   cos   ϕ t n s E tp E ts n s E ts   cos   ϕ t ,
n s   sin ( ϕ t ) = sin ( ϕ i ) .
Ψ i = K i · E i , Ψ r = K r · E r ,
K i = cos   ϕ i 0 1 0 0 1 0 cos   ϕ i ,
K r = - cos   ϕ i 0 1 0 0 1 0 - cos   ϕ i .
E tp   cos   ϕ t = L 1 · ( Ψ i + Ψ r ) ,
n s E tp = L 2 · ( Ψ i + Ψ r ) ,
E ts = L 3 · ( Ψ i + Ψ r ) ,
n s E ts   cos   ϕ t = L 4 · ( Ψ i + Ψ r ) .
( cos   ϕ t L 2 - n s L 1 ) · ( Ψ i + Ψ r ) = 0 ,
( n s   cos   ϕ t L 3 - L 4 ) · ( Ψ i + Ψ r ) = 0 ,
K · L · ( Ψ i + Ψ r ) = 0 ,
K = - n s cos   ϕ t 0 0 0 0 n s   cos   ϕ t - 1 .
KL ( K i E i + K r E r ) = 0 .
( K · L · K i ) · E i = - ( K · L · K r ) · R · E i .
R = - [ K · L · K r ] - 1 [ K · L · K i ] .

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