Abstract

The reflection of linearly polarized electromagnetic plane waves from an N-layer system of inhomogeneous, dielectric films on a transparent, homogeneous substrate is investigated in the long-wavelength limit. Approximate formulas are obtained for changes in the reflectance of s- or p-polarized light and in the ellipsometric angles that are caused by a multilayer, thin-film system. An analysis of the influence of a multilayer, ultrathin surface film on the reflectance of p-polarized light at the Brewster angle is carried out. All approximate analytical results are correlated with the exact computer solution of the reflection problem for a multilayer system of inhomogeneous films. The possibilities are discussed of using the obtained approximate expressions for resolving the inverse problem of ultrathin dielectric films on transparent substrates. Novel options are developed for determining the parameters of ultrathin films by integrating differential reflectance and ellipsometry.

© 2004 Optical Society of America

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References

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  1. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, 1987).
  2. H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
    [CrossRef]
  3. A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
    [CrossRef] [PubMed]
  4. D. E. Aspnes, “Real-time optical diagnostics for epitaxial growth,” Surf. Sci. 307–309, 1017–1027 (1994).
    [CrossRef]
  5. P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
    [CrossRef]
  6. N. Kobayashi and Y. Kobayashi, “In situ monitoring and control of atomic layer epitaxy by surface photoabsorption,” Thin Solid Films 225, 32–39 (1993).
    [CrossRef]
  7. L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
    [CrossRef]
  8. P. Adamson, “Differential reflection spectroscopy of surface layers on thick transparent substrates with normally incident light,” Opt. Spectrosc. 80, 459–468 (1996).
  9. P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752–759 (2003).
    [CrossRef]
  10. P. M. Fauchet, “Enhanced sensitivity of time-resolved reflectivity measurements near Brewster’s angle,” IEEE J. Quantum Electron. 25, 1072–1078 (1989).
    [CrossRef]
  11. S. Henon and J. Meunier, “Ellipsometry and reflectivity at the Brewster angle: tools to study the bending elasticity and phase transitions in monolayers at liquid interfaces,” Thin Solid Films 234, 471–474 (1993).
    [CrossRef]
  12. E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
    [CrossRef]
  13. R. C. Maclaurin, “Theory of the reflection of light near the polarizing angle,” Proc. R. Soc. London, Ser. A 76, 49–65 (1905).
    [CrossRef]
  14. M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
    [CrossRef]
  15. M. T. Haarmans and D. Bedeaux, “Optical properties of thin films up to second order in the thickness,” Thin Solid Films 258, 213–223 (1995).
    [CrossRef]
  16. F. Abeles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).
  17. H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1986).
  18. M. Kildemo, O. Hunderi, and B. Drevillon, “Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films,” J. Opt. Soc. Am. A 14, 931–939 (1997).
    [CrossRef]
  19. 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]
  20. M. J. Minot, “The angular reflectance of single-layer gradient refractive index films,” J. Opt. Soc. Am. 67, 1046–1050 (1977).
    [CrossRef]
  21. S. A. Tretyakov and A. H. Sihvola, “On the homogenization of isotropic layers,” IEEE Trans. Antennas Propag. 48, 1858–1861 (2000).
    [CrossRef]

2003 (1)

2000 (1)

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

1997 (1)

1996 (3)

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
[CrossRef]

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

1995 (1)

M. T. Haarmans and D. Bedeaux, “Optical properties of thin films up to second order in the thickness,” Thin Solid Films 258, 213–223 (1995).
[CrossRef]

1994 (1)

D. E. Aspnes, “Real-time optical diagnostics for epitaxial growth,” Surf. Sci. 307–309, 1017–1027 (1994).
[CrossRef]

1993 (3)

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

N. Kobayashi and Y. Kobayashi, “In situ monitoring and control of atomic layer epitaxy by surface photoabsorption,” Thin Solid Films 225, 32–39 (1993).
[CrossRef]

S. Henon and J. Meunier, “Ellipsometry and reflectivity at the Brewster angle: tools to study the bending elasticity and phase transitions in monolayers at liquid interfaces,” Thin Solid Films 234, 471–474 (1993).
[CrossRef]

1989 (1)

P. M. Fauchet, “Enhanced sensitivity of time-resolved reflectivity measurements near Brewster’s angle,” IEEE J. Quantum Electron. 25, 1072–1078 (1989).
[CrossRef]

1988 (1)

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

1987 (1)

P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
[CrossRef]

1982 (1)

1977 (1)

1971 (1)

M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

1950 (1)

F. Abeles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1905 (1)

R. C. Maclaurin, “Theory of the reflection of light near the polarizing angle,” Proc. R. Soc. London, Ser. A 76, 49–65 (1905).
[CrossRef]

Abeles, F.

F. Abeles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Adamson, P.

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752–759 (2003).
[CrossRef]

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

Aspnes, D. E.

D. E. Aspnes, “Real-time optical diagnostics for epitaxial growth,” Surf. Sci. 307–309, 1017–1027 (1994).
[CrossRef]

Baumann, U.

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

Bedeaux, D.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

M. T. Haarmans and D. Bedeaux, “Optical properties of thin films up to second order in the thickness,” Thin Solid Films 258, 213–223 (1995).
[CrossRef]

Bleckmann, L.

L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
[CrossRef]

de Boeij, P. L.

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Dejardin, P.

P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
[CrossRef]

Dhathathreyan, A.

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

Dignam, M. J.

M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Drevillon, B.

Emslie, A. G.

Fauchet, P. M.

P. M. Fauchet, “Enhanced sensitivity of time-resolved reflectivity measurements near Brewster’s angle,” IEEE J. Quantum Electron. 25, 1072–1078 (1989).
[CrossRef]

Haarmans, M. T.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

M. T. Haarmans and D. Bedeaux, “Optical properties of thin films up to second order in the thickness,” Thin Solid Films 258, 213–223 (1995).
[CrossRef]

Haggerty, J. S.

Henon, S.

S. Henon and J. Meunier, “Ellipsometry and reflectivity at the Brewster angle: tools to study the bending elasticity and phase transitions in monolayers at liquid interfaces,” Thin Solid Films 234, 471–474 (1993).
[CrossRef]

Hunderi, O.

Kildemo, M.

Kobayashi, N.

N. Kobayashi and Y. Kobayashi, “In situ monitoring and control of atomic layer epitaxy by surface photoabsorption,” Thin Solid Films 225, 32–39 (1993).
[CrossRef]

Kobayashi, Y.

N. Kobayashi and Y. Kobayashi, “In situ monitoring and control of atomic layer epitaxy by surface photoabsorption,” Thin Solid Films 225, 32–39 (1993).
[CrossRef]

Koper, G. J. M.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Maclaurin, R. C.

R. C. Maclaurin, “Theory of the reflection of light near the polarizing angle,” Proc. R. Soc. London, Ser. A 76, 49–65 (1905).
[CrossRef]

Mann, E. K.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Meunier, J.

S. Henon and J. Meunier, “Ellipsometry and reflectivity at the Brewster angle: tools to study the bending elasticity and phase transitions in monolayers at liquid interfaces,” Thin Solid Films 234, 471–474 (1993).
[CrossRef]

Minot, M. J.

Möbius, D.

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

Moskovits, M.

M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Müller, A.

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

Richter, W.

L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
[CrossRef]

Sagis, L. M. C.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Schaaf, P.

P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
[CrossRef]

Schmitt, A.

P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
[CrossRef]

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]

Stobie, R. W.

M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
[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]

van der Zeeuw, E. A.

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

van Silfhout, A.

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Wentink, D. J.

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Wijers, C. M. J.

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Wold, E.

L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
[CrossRef]

Wormeester, H.

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abeles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Biochim. Biophys. Acta (1)

A. Dhathathreyan, U. Baumann, A. Müller, and D. Möbius, “Characterization of complex gramicidin monolayers by light reflection and Fourier transform infrared spectroscopy,” Biochim. Biophys. Acta 944, 265–272 (1988).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

P. M. Fauchet, “Enhanced sensitivity of time-resolved reflectivity measurements near Brewster’s angle,” IEEE J. Quantum Electron. 25, 1072–1078 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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

J. Chem. Phys. (1)

E. A. van der Zeeuw, L. M. C. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of scanning angle reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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

Langmuir (1)

P. Schaaf, P. Dejardin, and A. Schmitt, “Reflectometry as a technique to study the adsorption of human fibrinogen at the silica/solution interface,” Langmuir 3, 1131–1135 (1987).
[CrossRef]

Opt. Spectrosc. (1)

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

Phys. Rev. B (1)

H. Wormeester, D. J. Wentink, P. L. de Boeij, C. M. J. Wijers, and A. van Silfhout, “Surface states of the clean and oxidized Ge(001) surface studied with normal-incidence ellipsometry,” Phys. Rev. B 47, 12663–12671 (1993).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

R. C. Maclaurin, “Theory of the reflection of light near the polarizing angle,” Proc. R. Soc. London, Ser. A 76, 49–65 (1905).
[CrossRef]

Surf. Sci. (2)

D. E. Aspnes, “Real-time optical diagnostics for epitaxial growth,” Surf. Sci. 307–309, 1017–1027 (1994).
[CrossRef]

L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277–284 (1996).
[CrossRef]

Thin Solid Films (3)

N. Kobayashi and Y. Kobayashi, “In situ monitoring and control of atomic layer epitaxy by surface photoabsorption,” Thin Solid Films 225, 32–39 (1993).
[CrossRef]

S. Henon and J. Meunier, “Ellipsometry and reflectivity at the Brewster angle: tools to study the bending elasticity and phase transitions in monolayers at liquid interfaces,” Thin Solid Films 234, 471–474 (1993).
[CrossRef]

M. T. Haarmans and D. Bedeaux, “Optical properties of thin films up to second order in the thickness,” Thin Solid Films 258, 213–223 (1995).
[CrossRef]

Trans. Faraday Soc. (1)

M. J. Dignam, M. Moskovits, and R. W. Stobie, “Specular reflectance and ellipsometric spectroscopy of oriented molecular layers,” Trans. Faraday Soc. 67, 3306–3317 (1971).
[CrossRef]

Other (2)

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1986).

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

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Figures (9)

Fig. 1
Fig. 1

(a) Profiles according to Eqs. (17) and (19) (dashed curve) and (b) according to Eq. (18) of the refractive index ni(z) for different values of g (the numbers on the curves) when n0i=1.5 and ndi=2.

Fig. 2
Fig. 2

Relative errors of approximation formulas for (a) ΔR2(s)/R0(s) [relation (6)] and for (b) ΔR2(p)/R0(p) [relation (8)] as functions of λ for a two-film system with d1=d2=4 when ϕa=45°; na=1; ns=1.5 (4, 5), 4 (the other curves); n01=1.3 (4), 1.4 (2, 3), 4 (1, 5); nd1=n02=1.5 (1), 2 (2, 5), 2.5 (4), 4 (3); nd2=1.3 (5), 1.5 (4), 2 (3), 4 (2), 4.5 (1). Preceding numbers in parentheses are curve labels.

Fig. 3
Fig. 3

Relative errors of approximation formulas for (a) ΔR2(s)/R0(s) [relation (6)] and for (b) ΔR2(p)/R0(p) [relation (8)] as functions of ϕa for a two-film system with d1/λ=0.01 and d2/λ=0.02 when na=1; ns=2; n01=1.3 (1), 2.5 (2); nd1=n02=1.5 (2), 4.3 (1); nd2=1.3 (1), 2 (2). Preceding numbers in parentheses are curve labels.

Fig. 4
Fig. 4

Relative errors of approximation formulas for (a) δΔ [relation (14)] when ϕa=45° (solid curves) and 75° (dashed curves) and for (b) δΨ [relation (16)] as functions of λ for a three-film system with d1=d2=d3=1 at na=1; ns=1.5; n01=1 (6), 1.5 (1, 4), 2 (7), 3 (2, 5), 4 (3); nd1=n02=1.3 (2, 5), 1.5 (3), 2 (1, 4, 7), 4 (6); nd2=n03=1.5 (6), 2.5 (1, 3, 4), 3 (2, 5), 4 (7); nd3=1.2 (3), 1.5 (1, 4, 6), 4 (2, 5, 7). Preceding numbers in parentheses are curve labels.

Fig. 5
Fig. 5

Reflectance R3(p)ϕB as a function of ns for a three-film system when d1/λ=0.005, d2/λ=0.003, and d3/λ=0.002 at na=1; n01=3; nd1=n02=4; nd2=n03=1.5; nd3=4. Dashed curve corresponds to calculation by relation (20).

Fig. 6
Fig. 6

Relative errors of approximate relation (21) as functions of (a) d1/λ when k1=0.5 and of (b) k1 when d1/λ=0.01 at na=1; ns=1.5 (1, 6, 7), 4 (2–5); n1=1.3 (7), 1.5 (5), 2 (4, 6), 3.5 (3), 4 (1), 4.5 (2). Preceding numbers in parentheses are curve labels.

Fig. 7
Fig. 7

Relative errors of approximate relations (24) and (25) (solid curves) and (24) and (29) (dashed curves) as functions of (a) d1/λ if ns=1.5 and of (b) ns if d1/λ=0.01 at na=1; n1=2; ϕa(Δ)=ϕa(s)=65°; μ1=0 (1), 3% (2), -3% (3). The variable μ1 is the relative error of the quantities (δΔ)2/(ΔR1(s)/R0(s)) and ΔR1(p)(ϕB)/ΔR1(s)(ϕB). Preceding numbers in parentheses are curve labels.

Fig. 8
Fig. 8

Relative error of Eq. (32) as a function of λ when na=1; n1=3; n2=2; ns=1.5 (2, 3), 4 (1, 4–7); d1=1 (1–3, 5, 7), 5 (4), 10 (6); d2=0.1 (1, 2, 7), 0.2 (3, 5), 0.5 (4), 2 (6); μ5=0 (solid curves), 1% (2), 3% (7), -3% (1). The variable μ5 is the relative error of the quantity V. Preceding numbers in parentheses are curve labels.

Fig. 9
Fig. 9

Relative errors of approximate relations (36) (solid curves) and (37) (dashed curves) as functions of (a) λ when d1=3; d2=2; ns=1.5 (1, 2), 4 (3), and of (b) ns when d1/λ=0.0015; d2/λ=0.001 at ϕa(1)=0°; ϕa(2)=45°; na=1; n1=n02=1.5 (4), 2 (2, 3), 3 (1), 4 (5); nd2=1.5 (1), 2 (4), 2.5 (2, 3, 5). Profiles n2(z) are described by Eq. (18) with g=1. Preceding numbers in parentheses are curve labels.

Equations (61)

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MN(σ)=m11(σ)m12(σ)m21(σ)m22(σ),
m11(σ)=1-k2i=1Nai(σ)di2-k2i=1N-1bi(σ)dij=i+1Ncj(σ)dj,
m12(σ)=-iki=1Nbi(σ)di,m21(σ)=-iki=1Nci(σ)di,
m22(σ)=1-k2i=1Ngi(σ)di2-k2i=1N-1ci(σ)dij=i+1Nbj(σ)dj,
ai(s)=di-20didzzdi[i(ζ)-a sin2 ϕa]dζ,
bi(s)=1,ci(s)=di-10di[i(z)-a sin2 ϕa]dz,
gi(s)=di-10di[i(z)-a sin2 ϕa]dz-di-20di[i(z)-a sin2 ϕa]zdz,
ai(p)=di-20di[1-i-1(z)a sin2 ϕa]dzzdii(ζ)dζ,
bi(p)=1-di-1a sin2 ϕa0dii-1(z)dz,
ci(p)=di-10dii(z)dz,
gi(p)=di-10dii(z)dz-di-20dii(z)zdz-di-2a sin2 ϕa0dii(z)dzzdii-1(ζ)dζ,
rN(σ)={[m11(σ)+m12(σ)Ps(σ)]Pa(σ)-[m21(σ)+m22(σ)Ps(σ)]}×{[m11(σ)+m12(σ)Ps(σ)]Pa(σ)+[m21(σ)+m22(σ)Ps(σ)]}-1,
Pa(s)=na cos ϕa,Ps(s)=ns cos ϕs,
Pa(p)=na/cos ϕa,Ps(p)=ns/cos ϕs,
cos ϕs=(1-as-1 sin2 ϕa)1/2,
rN(σ)r0(σ)1+2k2Pa(σ)Ps(σ)[Pa(σ)]2-[Ps(σ)]2 i=1N[gi(σ)-ai(σ)]di2+i=1N-1j=i+1N[ci(σ)bj(σ)-cj(σ)bi(σ)]didj-{Ps(σ)[Pa(σ)+Ps(σ)]}-1j=1Ni=1N{ci(σ)-[Ps(σ)]2bi(σ)}{cj(σ)+Pa(σ)Ps(σ)bj(σ)}didj+i 2kPa(σ)[Pa(σ)]2-[Ps(σ)]2 i=1N{ci(σ)-[Ps(σ)]2bi(σ)}di.
RN(σ)R0(σ)1+4k2Pa(σ)Ps(σ)[Pa(σ)]2-[Ps(σ)]2 i=1N[gi(σ)-ai(σ)]di2+i=1N-1j=i+1N[ci(σ)bj(σ)-bi(σ)cj(σ)]didj+{[Pa(σ)]2-[Ps(σ)]2}-1i=1Nj=1N{ci(σ)-[Ps(σ)]2bi(σ)}{cj(σ)-[Pa(σ)]2bj(σ)}didj,
ΔrN(s)r0(s)ABλ2 i=1N(ti-2αi)di2+i=1N-1j=i+1N(ti-tj)didj-i=1Nj=1N(ti-s)(tj-a sin2 ϕa+nans cos ϕa cos ϕs)(s cos2 ϕs+nans cos ϕa cos ϕs)didj+i 4πna cos ϕaBλ i=1N(ti-s)di,
ΔRN(s)R0(s)2ABλ2 i=1N(ti-2αi)di2+i=1N-1j=i+1N(ti-tj)didj+B-1i=1Nj=1N(s-ti)(a-tj)didj,
ΔrN(p)r0(p)ACλ2 i=1N(ti-2αi+βia sin2 ϕa)di2+i=1N-1j=i+1N(tiηj-tjηi)didj-i=1Nj=1N (ti cos2 ϕs-sηi)(tjnj cos ϕa cos ϕs+nansnj-nansa sin2 ϕa)nj cos ϕs(s cos ϕa+nans cos ϕs)didj+i 4πna cos ϕaCλ i=1N(ti cos2 ϕs-sηi)di,
ΔRN(p)R0(p)2ACλ2 i=1N(ti-2αi+βia sin2 ϕa)di2+i=1N-1j=i+1N(tiηj-tjηi)didj+C-1i=1Nj=1N(sηi-ti cos2 ϕs)(aηj-tj cos2 ϕa)didj,
ti=di-10dii(z)dz,
ni-1=di-10dii-1(z)dz,
αi=di-20dii(z)zdz,
βi=di-20dii(z)0zi-1(ζ)dζ-i-1(z)0zi(ζ)dζdz.
ΔR1(0)/R0(0)16π2nans(a-s)-2[(a-s)(t1-2α1)+(a-t1)(s-t1)](d1/λ)2.
ΔIN(R)/I0(R)=[IN(R)-I0(R)]/I0(R)=[RN(σ)I(i)-R0(σ)I(i)]/R0(σ)I(i)=ΔRN(σ)/R0(σ),
δΔ4πna cos ϕa sin2 ϕa(s cos2 ϕa-a cos2 ϕs)-1i=1Nγi(di/λ),
γi=di-10di[i(z)-a][s-i(z)]i-1(z)dz=[a(ni-s)+ni(s-ti)]ni-1,
δΨπ(a+s)1/2(s-a)-1i=1Nγi(di/λ),
ni(z)=n0i+(ndi-n0i)(z/di)g,
ni(z)=n0indi[ndig-(ndig-n0ig)(z/di)]-1/g,
ni(z)=n0i(ndi/n0i)z/din0i exp[ln(ndi/n0i)(z/di)],
ΔRN(p)RN(p)(ϕB)π2(a+s)-1i=1Nγi(di/λ)2.
R1(p)(ϕB)R10(p)(ϕB)[1-2π(a+s)-1/2×(1+as|ˆ1|-2)ξ1(d1/λ)],
R10(p)(ϕB)π2|ˆ1|-2[(a+s)[(|ˆ1|2-2as1+as2)1/2+|ˆ1|2(a+s)-1]2-2(1|ˆ1|2+(ξ12-12)as+{[1|ˆ1|2+(ξ12-12)as]2+(|ˆ1|2-2as1)2ξ12}1/2)](d1/λ)2,
(δΔ)2/[ΔR1(s)/R0(s)]
na[ns cos ϕa(s) cos ϕs(s)]-1[cos2 ϕa(Δ) sin4 ϕa(Δ)](s-a)2×[s cos2 ϕa(Δ)-a cos2 ϕs(Δ)]-2×(1-a)(1-s)1-2.
1{s+a±[(s-a)2+4ast]1/2}[2(1-t)]-1,
tns cos ϕa(s) cos ϕs(s)[s cos2 ϕa(Δ)-a cos2 ϕs(Δ)]2na cos2 ϕa(Δ) sin4 ϕa(Δ)(s-a)2×(δΔ)2ΔR1(s)/R0(s).
ΔR1(0)16π2nans(na+ns)-4(1-a)×(1-s)(d1/λ)2,
ΔR1(p)(ϕB)π2(s+a)-1(1-a)2×(s-1)21-2(d1/λ)2.
t16nans(na+ns)-4(a+s)[ΔR1(p)(ϕB)/ΔR1(0)].
ΔI1(R)(ϕa=ϕB)/ΔI1(R)(ϕa=0)=ΔR1(p)(ϕa=ϕB)/ΔR1(p)(ϕa=0)ΔR1(p)(ϕB)/ΔR1(0),
t16as(a+s)-2[ΔR1(p)(ϕB)/ΔR1(s)(ϕB)].
ΔR21(σ)(ϕB)/R1(σ)(ϕB)=[R2(σ)(1, 2, d1, d2, ϕB)-R1(σ)×(1, d1, ϕB)]/R1(σ)×(1, d1, ϕB)
ΔR21(p)(ϕB)/R1(p)(ϕB)212-1(2-a)(2-s)(1-a)-1(1-s)-1d2d1-1,
ΔR21(s)(ϕB)/R1(s)(ϕB)32π2as(2-a)(1-s)×(s-a)-2(d1d2/λ2).
2=s[1-16π2as(s-a)-21-1(1-a)×(s-1)2(d1/λ)2V]-1,
ΔR21(p)(ϕa)/R1(p)(ϕa)2A(Cλ)-2[2(aη2-2 cos2 ϕa)×(sη1-t1 cos2 ϕs)d1d2+(aη2-2 cos2 ϕa)(sη2-2 cos2 ϕs)d22].
s-sa sin2 ϕan1-1-cos2 ϕst1T(ϕa)(λ/d1),
T(ϕa)
ΔR21(p)(ϕa)R1(p)(ϕa) (a cos2 ϕs-s cos2 ϕa)232π2nans cos ϕa cos ϕs(aη2-2 cos2 ϕa)×λd2+(2 cos2 ϕs-sη2)2 d2λ.
t1s+{T[ϕa(1)]sin2 ϕa(2)-T[ϕa(2)]sin2 ϕa(1)}A-1(λ/d1),
n1-1s-1+(sa)-1{T[ϕa(2)]cos2 ϕs(1)-T[ϕa(1)]cos2 ϕs(2)}A-1(λ/d1),
AT[ϕa(3)]=T[ϕa(1)][sin2 ϕa(3)-sin2 ϕa(2)]+T[ϕa(2)]×[sin2 ϕa(1)-sin2 ϕa(3)].
A[s cos2 ϕa(Δ)-a cos2 ϕs(Δ)][4πna cos ϕa(Δ) sin2 ϕa(Δ)]-1δΔ
=T[ϕa(1)][cos2 ϕs(2)-sin2 ϕa(2)]-T[ϕa(2)][cos2 ϕs(1)-sin2 ϕa(1)].
t1s-{T[ϕa(1)]-P sin2 ϕa(1)}×[cos2 ϕs(1)-sin2 ϕa(1)]-1(λ/d1),
n1-1s-1+(as)-1{T[ϕa(1)]-P cos2 ϕs(1)}×[cos2 ϕs(1)-sin2 ϕa(1)]-1(λ/d1),
P=δΔ(s cos2 ϕa(Δ)-a cos2 ϕs(Δ))×(4πna cos ϕa(Δ) sin2 ϕa(Δ))-1.

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