Abstract

We show that reflection of a monochromatic light from a semi-infinite medium covered with a stack of layered media is equivalent to that from an effective “semi-infinite medium” characterized by two distinct optical dielectric constants for the s- and p-polarized components, respectively. Such an effective-substrate approach simplifies the analysis of ellipsometry measurements of a wide range of surface-bound processes including thin-film growth and surface-bound reactions.

© 2008 Optical Society of America

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).
  2. B. Drevillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1-87 (1993).
    [CrossRef]
  3. J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
    [CrossRef]
  4. L. Bleckmann, O. Hunderi, W. Richter, and E. Wold, “Surface studies by means of 45° reflectometry,” Surf. Sci. 351, 277-284 (1996).
    [CrossRef]
  5. N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
    [CrossRef]
  6. P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
    [CrossRef]
  7. E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
    [CrossRef]
  8. K. Holst and H. Raether, “The influence of thin surface films on the plasma resonance emmision,” Opt. Commun. 2, 312-316 (1970).
    [CrossRef]
  9. W. H. Weber, “Modulated surface-plasmon resonances for in situ metal-film surface studies,” Phys. Rev. Lett. 39, 153-156 (1977).
    [CrossRef]
  10. F. Abeles and T. Lopez-Rios, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), p. 239.
  11. K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
    [CrossRef]
  12. P. A. Flournoy, R. W. McClure, and G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907-1915 (1972).
    [CrossRef] [PubMed]
  13. A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
    [CrossRef]
  14. J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
    [CrossRef] [PubMed]
  15. D. E. Aspnes, “Above-bandgap optical anisotropies in cubic semiconductors: a visible-near ultraviolet probe of surfaces,” J. Vac. Sci. Technol. B 3, 1498-1506 (1985).
    [CrossRef]
  16. J. F. McGilp, “Optical characterisation of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1-106 (1995).
    [CrossRef]
  17. N. Kobayashi and Y. Horikoshi, “Spectral dependence of optical reflectance during flow-rate modulation epitaxy of GaAs by the surface photo-absorption method,” Jpn. J. Appl. Phys., Part 2 29, L702-L705 (1990).
    [CrossRef]
  18. M. Born and W. Wolf, Principles of Optics (Pergamon, 1993), p. 51.
  19. J. D. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434 (1971).
    [CrossRef]
  20. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von. heterogenen Substanzen,” Ann. Phys. 24, 636-664 (1935).
    [CrossRef]
  21. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
    [CrossRef]
  22. R. H. Muller and J. C. Farmer, “Macroscopic optical models for the ellipsometry of an under-potential deposit: lead in copper and silver,” Surf. Sci. 135, 521-531 (1983).
    [CrossRef]
  23. X. D. Zhu, “Oblique-incidence optical reflectivity difference from a rough film of crystalline material,” Phys. Rev. B 69, 115407 (2004).
    [CrossRef]
  24. X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
    [CrossRef]
  25. G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
    [CrossRef]
  26. B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
    [CrossRef]
  27. B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
    [CrossRef]
  28. J. S. Shumaker-Parry and C. T. Campbell, “Quantitative methods for spatially-resolved adsorption/desorption measurements in real time by SPR microscopy,” Anal. Chem. 76, 907-917 (2004).
    [CrossRef] [PubMed]
  29. X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
    [CrossRef]
  30. Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
    [CrossRef]
  31. W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
    [CrossRef]
  32. J. P. Landry, X. D. Zhu, and J. Gregg, “Label-free detection of microarrays of biomolecules using oblique-incidence reflectivity difference microscopy,” Opt. Lett. 29, 581-583 (2004).
    [CrossRef] [PubMed]
  33. J. P. Landry, J. Gray, M. K. O'Toole, and X. D. Zhu, “Incidence-angle dependence of optical reflectivity difference from an ultrathin film on solid surface,” Opt. Lett. 31, 531-533 (2006).
    [CrossRef] [PubMed]
  34. A. V. Tikhonravov, M. K. Truetskov, and A. V. Krasilnikova, “Spectroscopic ellipsometry of slightly inhomogeneous nonabsorbing thin films with arbitrary reflective-index profiles: theoretical study,” Appl. Opt. 37, 5902-5911 (1998).
    [CrossRef]
  35. 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]
  36. 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. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
    [CrossRef]

2007 (1)

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

2006 (1)

2004 (6)

J. P. Landry, X. D. Zhu, and J. Gregg, “Label-free detection of microarrays of biomolecules using oblique-incidence reflectivity difference microscopy,” Opt. Lett. 29, 581-583 (2004).
[CrossRef] [PubMed]

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. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
[CrossRef]

X. D. Zhu, “Oblique-incidence optical reflectivity difference from a rough film of crystalline material,” Phys. Rev. B 69, 115407 (2004).
[CrossRef]

J. S. Shumaker-Parry and C. T. Campbell, “Quantitative methods for spatially-resolved adsorption/desorption measurements in real time by SPR microscopy,” Anal. Chem. 76, 907-917 (2004).
[CrossRef] [PubMed]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

2003 (2)

W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
[CrossRef]

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]

1999 (1)

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

1998 (2)

A. V. Tikhonravov, M. K. Truetskov, and A. V. Krasilnikova, “Spectroscopic ellipsometry of slightly inhomogeneous nonabsorbing thin films with arbitrary reflective-index profiles: theoretical study,” Appl. Opt. 37, 5902-5911 (1998).
[CrossRef]

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

1997 (1)

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

1996 (3)

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

J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
[CrossRef] [PubMed]

G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
[CrossRef]

1995 (1)

J. F. McGilp, “Optical characterisation of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1-106 (1995).
[CrossRef]

1993 (1)

B. Drevillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1-87 (1993).
[CrossRef]

1992 (1)

A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
[CrossRef]

1990 (1)

N. Kobayashi and Y. Horikoshi, “Spectral dependence of optical reflectance during flow-rate modulation epitaxy of GaAs by the surface photo-absorption method,” Jpn. J. Appl. Phys., Part 2 29, L702-L705 (1990).
[CrossRef]

1985 (1)

D. E. Aspnes, “Above-bandgap optical anisotropies in cubic semiconductors: a visible-near ultraviolet probe of surfaces,” J. Vac. Sci. Technol. B 3, 1498-1506 (1985).
[CrossRef]

1983 (2)

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
[CrossRef]

R. H. Muller and J. C. Farmer, “Macroscopic optical models for the ellipsometry of an under-potential deposit: lead in copper and silver,” Surf. Sci. 135, 521-531 (1983).
[CrossRef]

1979 (1)

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

1978 (1)

J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
[CrossRef]

1977 (1)

W. H. Weber, “Modulated surface-plasmon resonances for in situ metal-film surface studies,” Phys. Rev. Lett. 39, 153-156 (1977).
[CrossRef]

1976 (1)

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[CrossRef]

1974 (1)

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

1972 (1)

1971 (1)

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

1970 (1)

K. Holst and H. Raether, “The influence of thin surface films on the plasma resonance emmision,” Opt. Commun. 2, 312-316 (1970).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von. heterogenen Substanzen,” Ann. Phys. 24, 636-664 (1935).
[CrossRef]

Abeles, F.

F. Abeles and T. Lopez-Rios, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), p. 239.

Adamson, P.

Alexander, R. W.

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[CrossRef]

Arwin, H.

G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Above-bandgap optical anisotropies in cubic semiconductors: a visible-near ultraviolet probe of surfaces,” J. Vac. Sci. Technol. B 3, 1498-1506 (1985).
[CrossRef]

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

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

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

Bachmann, K. J.

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

Bartelt, M. C.

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

Bell, R. J.

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[CrossRef]

Bhasin, K.

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[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]

Born, M.

M. Born and W. Wolf, Principles of Optics (Pergamon, 1993), p. 51.

Brecht, A.

J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
[CrossRef] [PubMed]

A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
[CrossRef]

Brockman, J. M.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von. heterogenen Substanzen,” Ann. Phys. 24, 636-664 (1935).
[CrossRef]

Bryan, D.

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[CrossRef]

Burstein, E.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

Campbell, C. T.

J. S. Shumaker-Parry and C. T. Campbell, “Quantitative methods for spatially-resolved adsorption/desorption measurements in real time by SPR microscopy,” Anal. Chem. 76, 907-917 (2004).
[CrossRef] [PubMed]

Chen, W. P.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

Chen, Y. J.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

Chen, Z. H.

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

Corn, R. M.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

Dietz, N.

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

Drevillon, B.

B. Drevillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1-87 (1993).
[CrossRef]

Farmer, J. C.

R. H. Muller and J. C. Farmer, “Macroscopic optical models for the ellipsometry of an under-potential deposit: lead in copper and silver,” Surf. Sci. 135, 521-531 (1983).
[CrossRef]

Fei, Y. Y.

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

Flournoy, P. A.

Fong, C. Y.

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

Frutos, A. G.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

Gauglitz, G.

J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
[CrossRef] [PubMed]

A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
[CrossRef]

Gray, J.

J. P. Landry, J. Gray, M. K. O'Toole, and X. D. Zhu, “Incidence-angle dependence of optical reflectivity difference from an ultrathin film on solid surface,” Opt. Lett. 31, 531-533 (2006).
[CrossRef] [PubMed]

W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
[CrossRef]

Gregg, J.

Gu, B. Y.

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Hallais, J.

J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
[CrossRef]

Harris, C.

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

Hartstein, A.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

Holst, K.

K. Holst and H. Raether, “The influence of thin surface films on the plasma resonance emmision,” Opt. Commun. 2, 312-316 (1970).
[CrossRef]

Horikoshi, Y.

N. Kobayashi and Y. Horikoshi, “Spectral dependence of optical reflectance during flow-rate modulation epitaxy of GaAs by the surface photo-absorption method,” Jpn. J. Appl. Phys., Part 2 29, L702-L705 (1990).
[CrossRef]

Hottier, F.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
[CrossRef]

Hunderi, O.

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

Ingenhoff, J.

A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
[CrossRef]

Jansson, R.

G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
[CrossRef]

Jin, G.

G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
[CrossRef]

Kobayashi, N.

N. Kobayashi and Y. Horikoshi, “Spectral dependence of optical reflectance during flow-rate modulation epitaxy of GaAs by the surface photo-absorption method,” Jpn. J. Appl. Phys., Part 2 29, L702-L705 (1990).
[CrossRef]

Krasilnikova, A. V.

Landry, J. P.

Li, Z. Y.

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Liedberg, B.

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Liu, L. F.

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

Lopez-Rios, T.

F. Abeles and T. Lopez-Rios, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), p. 239.

Lu, H. B.

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Lundstrom, I.

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
[CrossRef]

McClure, R. W.

McGilp, J. F.

J. F. McGilp, “Optical characterisation of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1-106 (1995).
[CrossRef]

McIntyre, J. D. E.

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

Muller, R. H.

R. H. Muller and J. C. Farmer, “Macroscopic optical models for the ellipsometry of an under-potential deposit: lead in copper and silver,” Surf. Sci. 135, 521-531 (1983).
[CrossRef]

Nabighian, E.

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

Nelson, B. P.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

Nylander, C.

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
[CrossRef]

O'Toole, M. K.

Piehler, J.

J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
[CrossRef] [PubMed]

Raether, H.

K. Holst and H. Raether, “The influence of thin surface films on the plasma resonance emmision,” Opt. Commun. 2, 312-316 (1970).
[CrossRef]

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]

Schwarzacher, W.

W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
[CrossRef]

Shumaker-Parry, J. S.

J. S. Shumaker-Parry and C. T. Campbell, “Quantitative methods for spatially-resolved adsorption/desorption measurements in real time by SPR microscopy,” Anal. Chem. 76, 907-917 (2004).
[CrossRef] [PubMed]

Sukidi, N.

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
[CrossRef]

Thomas, P.

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

Tikhonravov, A. V.

Truetskov, M. K.

Wang, X.

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

Weber, W. H.

W. H. Weber, “Modulated surface-plasmon resonances for in situ metal-film surface studies,” Phys. Rev. Lett. 39, 153-156 (1977).
[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]

Wolf, W.

M. Born and W. Wolf, Principles of Optics (Pergamon, 1993), p. 51.

Wyntjes, G.

Yang, G. Z.

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Zhang, D. Z.

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Zhu, X. D.

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

J. P. Landry, J. Gray, M. K. O'Toole, and X. D. Zhu, “Incidence-angle dependence of optical reflectivity difference from an ultrathin film on solid surface,” Opt. Lett. 31, 531-533 (2006).
[CrossRef] [PubMed]

J. P. Landry, X. D. Zhu, and J. Gregg, “Label-free detection of microarrays of biomolecules using oblique-incidence reflectivity difference microscopy,” Opt. Lett. 29, 581-583 (2004).
[CrossRef] [PubMed]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

X. D. Zhu, “Oblique-incidence optical reflectivity difference from a rough film of crystalline material,” Phys. Rev. B 69, 115407 (2004).
[CrossRef]

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
[CrossRef]

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Anal. Chem. (3)

J. Piehler, A. Brecht, and G. Gauglitz, “Affinity detection of low molecular weight analytes,” Anal. Chem. 68, 139-143 (1996).
[CrossRef] [PubMed]

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928-3934 (1999).
[CrossRef]

J. S. Shumaker-Parry and C. T. Campbell, “Quantitative methods for spatially-resolved adsorption/desorption measurements in real time by SPR microscopy,” Anal. Chem. 76, 907-917 (2004).
[CrossRef] [PubMed]

Ann. Phys. (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von. heterogenen Substanzen,” Ann. Phys. 24, 636-664 (1935).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. A (1)

P. Thomas, E. Nabighian, M. C. Bartelt, C. Y. Fong, and X. D. Zhu, “An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate,” Appl. Phys. A 79, 131-137 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

J. B. Theeten, F. Hottier, and J. Hallais, “On-time determination of the composition of III-V ternary layers during VPE growth,” Appl. Phys. Lett. 32, 576-578 (1978).
[CrossRef]

Electrochem. Solid-State Lett. (1)

W. Schwarzacher, J. Gray, and X. D. Zhu, “Oblique-incidence reflectivity difference as an in situ probe of Co electrodeposition on polycrystalline Au,” Electrochem. Solid-State Lett. 6, C73-C76 (2003).
[CrossRef]

J. Chem. Phys. (1)

K. Bhasin, D. Bryan, R. W. Alexander, and R. J. Bell, “Absorption in the infrared of surface electromagnetic waves by adsorbed molecules on a copper surface,” J. Chem. Phys. 64, 5019-5025 (1976).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Vac. Sci. Technol. (1)

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons--propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004-1019 (1974).
[CrossRef]

J. Vac. Sci. Technol. A (1)

N. Dietz, N. Sukidi, C. Harris, and K. J. Bachmann, “Real-time monitoring of surface processes by p-polarized reflectance,” J. Vac. Sci. Technol. A 15, 807-815 (1997).
[CrossRef]

J. Vac. Sci. Technol. B (1)

D. E. Aspnes, “Above-bandgap optical anisotropies in cubic semiconductors: a visible-near ultraviolet probe of surfaces,” J. Vac. Sci. Technol. B 3, 1498-1506 (1985).
[CrossRef]

Jpn. J. Appl. Phys., Part 2 (1)

N. Kobayashi and Y. Horikoshi, “Spectral dependence of optical reflectance during flow-rate modulation epitaxy of GaAs by the surface photo-absorption method,” Jpn. J. Appl. Phys., Part 2 29, L702-L705 (1990).
[CrossRef]

Opt. Commun. (1)

K. Holst and H. Raether, “The influence of thin surface films on the plasma resonance emmision,” Opt. Commun. 2, 312-316 (1970).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (5)

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

X. D. Zhu, H. B. Lu, G. Z. Yang, Z. Y. Li, B. Y. Gu, and D. Z. Zhang, “Epitaxial growth of SrTiO3 on SrTiO3(001) using an oblique-incidence reflectance-difference technique,” Phys. Rev. B 57, 2514-2519 (1998).
[CrossRef]

Y. Y. Fei, X. D. Zhu, L. F. Liu, H. B. Lu, Z. H. Chen, and G. Z. Yang, “Oscillations in oblique-incidence optical reflection from a growth surface during layer-by-layer epitaxy,” Phys. Rev. B 69, 233405 (2004).
[CrossRef]

X. D. Zhu, “Oblique-incidence optical reflectivity difference from a rough film of crystalline material,” Phys. Rev. B 69, 115407 (2004).
[CrossRef]

X. D. Zhu, Y. Y. Fei, X. Wang, H. B. Lu, and G. Z. Yang, “General theory of optical reflection from a thin film on a solid and its application to heteroepitaxy,” Phys. Rev. B 75, 245434 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

W. H. Weber, “Modulated surface-plasmon resonances for in situ metal-film surface studies,” Phys. Rev. Lett. 39, 153-156 (1977).
[CrossRef]

Prog. Cryst. Growth Charact. Mater. (1)

B. Drevillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1-87 (1993).
[CrossRef]

Prog. Surf. Sci. (1)

J. F. McGilp, “Optical characterisation of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1-106 (1995).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67, 2930-2936 (1996).
[CrossRef]

Sens. Actuators (1)

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Sens. Actuators B (1)

A. Brecht, J. Ingenhoff, and G. Gauglitz, “Direct monitoring of antigen-antibody interactions by spectral interferometry,” Sens. Actuators B 6, 96-100 (1992).
[CrossRef]

Surf. Sci. (3)

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

R. H. Muller and J. C. Farmer, “Macroscopic optical models for the ellipsometry of an under-potential deposit: lead in copper and silver,” Surf. Sci. 135, 521-531 (1983).
[CrossRef]

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

Other (3)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

F. Abeles and T. Lopez-Rios, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), p. 239.

M. Born and W. Wolf, Principles of Optics (Pergamon, 1993), p. 51.

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Figures (4)

Fig. 1
Fig. 1

Reflection of a monochromatic light at vacuum wavelength λ from the surface of a semi-infinite substrate ( ε s ) covered with an ultrathin layer ( ε d ) with a thickness of d λ . The beam travels in a semi-infinite ambient ( ε 0 ) before reflection.

Fig. 2
Fig. 2

(a) Reflection of a monochromatic light at vacuum wavelength λ from the bare surface of a semi-infinite substrate ( ε s ) ; (b) reflection from the surface of a semi-infinite substrate ( ε s ) covered with a uniform dielectric layer ( ε 1 ) with arbitrary thickness d 1 ; (c) reflection from the surface of a semi-infinite substrate covered with two uniform dielectric layers ( ε 1 and ε 2 ) with arbitrary thicknesses d 1 and d 2 .

Fig. 3
Fig. 3

Reflection from the surface of an effective substrate covered with an ultrathin layer ( ε d ) with thickness d λ . The effective substrate is characterized by two optical constants, ε s , eff ( p ) and ε s , eff ( s ) -, for the s and p-polarized components of the incident light.

Fig. 4
Fig. 4

Reflection from the surface of a semi-infinite substrate covered with an ultrathin film ( ε d ) with a thickness of d λ and a uniform dielectric layer ( ε 1 ) with arbitrary thickness d 1 .

Equations (44)

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Δ p Δ s δ L n ( r ( p ) r ( s ) ) = r ( p ) r ( p 0 ) r ( p 0 ) ( r ( s ) r ( s 0 ) ) r ( s 0 ) + i [ ( Φ ( p ) Φ ( p 0 ) ) ( Φ ( s ) Φ ( s 0 ) ) ] .
Δ p Δ s [ ( i ) 4 π cos ϕ 0 sin 2 ϕ 0 ε 0 ε s λ ( ε s ε 0 ) ( ε s cos 2 ϕ 0 ε 0 sin 2 ϕ 0 ) ] ( ε d ε 0 ) ( ε d ε s ) ε d d .
Δ p Δ s α eff ( ε d ε 0 ) ( ε d ε s , eff ) ε d d .
r 0 s ( s ) = n 0 cos ϕ 0 n s cos ϕ s n 0 cos ϕ 0 + n s cos ϕ s ,
r 01 s ( s ) = r 01 ( s ) + r 1 s ( s ) e i Ψ 1 1 + r 01 ( s ) r 1 s ( s ) e i Ψ 1 ,
r a b ( s ) = n a cos ϕ a n b cos ϕ b n a cos ϕ a + n b cos ϕ b ,
Ψ 1 = 4 π n 1 d 1 cos ϕ 1 λ ,
n s , eff ( s ) ( 1 , s ) sin ϕ s , eff ( s ) ( 1 , s ) = n 0 sin ϕ 0 ,
r 01 s ( s ) r 0 s , eff ( s ) ( 1 , s ) n 0 cos ϕ 0 n s , eff ( s ) ( 1 , s ) cos ϕ s , eff ( s ) ( 1 , s ) n 0 cos ϕ 0 + n s , eff ( s ) ( 1 , s ) cos ϕ s , eff ( s ) ( 1 , s ) .
ε s , eff ( s ) ( 1 , s ) = n 0 2 sin 2 ϕ 0 + n 1 2 cos 2 ϕ 1 ( 1 r 1 s ( s ) e i Ψ 1 1 + r 1 s ( s ) e i Ψ 1 ) 2 .
n s , eff ( s ) ( 2 , 1 , s ) sin ϕ s , eff ( s ) ( 2 , 1 , s ) = n 0 sin ϕ 0 ,
r 0 s , eff ( s ) ( 2 , 1 , s ) = n 0 cos ϕ 0 n s , eff ( s ) ( 2 , 1 , s ) cos ϕ s , eff ( s ) ( 2 , 1 , s ) n 0 cos ϕ 0 + n s , eff ( s ) ( 2 , 1 , s ) cos ϕ s , eff ( s ) ( 2 , 1 , s ) .
ε s , eff ( s ) ( 2 , 1 , s ) = n 0 2 sin 2 ϕ 0 + n 2 2 cos 2 ϕ 2 ( 1 r 21 s ( s ) e i Ψ 2 1 + r 21 s ( s ) e i Ψ 2 ) 2 = n 0 2 sin 2 ϕ 0 + n 2 2 cos 2 ϕ 2 ( 1 r 2 s , eff ( s ) ( 1 , s ) e i Ψ 2 1 + r 2 s , eff ( s ) ( 1 , s ) e i Ψ 2 ) 2 ,
r 21 s ( s ) r 2 s , eff ( s ) ( 1 , s ) n 2 cos ϕ 2 n s , eff ( s ) ( 1 , s ) cos ϕ s , eff ( s ) ( 1 , s ) n 2 cos ϕ 2 + n s , eff ( s ) ( 1 , s ) cos ϕ s , eff ( s ) ( 1 , s ) ,
r 0 s , eff ( s ) ( m , m 1 , , 1 , s ) = n 0 cos ϕ 0 n s , eff ( s ) cos ϕ s , eff ( s ) n 0 cos ϕ 0 + n s , eff ( s ) cos ϕ s , eff ( s ) ,
n s , eff ( s ) sin ϕ s , eff ( s ) = n 0 sin ϕ 0 .
ε s , eff ( s ) ( m , m 1 , , 1 , s ) = n 0 2 sin 2 ϕ 0 + n m 2 cos 2 ϕ m ( 1 r m s , eff ( s ) ( m 1 , , 1 , s ) e i Ψ m 1 + r m s , eff ( s ) ( m 1 , , 1 , s ) e i Ψ m ) 2 ,
r m s , eff ( s ) ( m 1 , , 1 , s ) = n m cos ϕ m n s , eff ( s ) ( m 1 , , 1 , s ) cos ϕ s , eff ( s ) ( m 1 , , 1 , s ) n m cos ϕ m + n s , eff ( s ) ( m 1 , , 1 , s ) cos ϕ s , eff ( s ) ( m 1 , , 1 , s ) .
r 0 s ( p ) = n 0 cos ϕ s n s cos ϕ 0 n 0 cos ϕ s + n s cos ϕ 0 ,
r 01 s ( p ) = r 01 ( p ) + r 1 s ( p ) e i Ψ 1 1 + r 01 ( p ) r 1 s ( p ) e i Ψ 1 ,
r a b ( p ) = n a cos ϕ b n b cos ϕ a n a cos ϕ b + n b cos ϕ a ,
n s , eff ( p ) ( 1 , s ) sin ϕ s , eff ( p ) ( 1 , s ) = n 0 sin ϕ 0 ,
r 01 s ( p ) r 0 s , eff ( p ) n 0 cos ϕ s , eff ( p ) ( 1 , s ) n s , eff ( p ) ( 1 , s ) cos ϕ 0 n 0 cos ϕ s , eff ( p ) ( 1 , s ) + n s , eff ( p ) ( 1 , s ) cos ϕ 0 .
ε s , eff ( p ) ( 1 , s ) ( n 1 2 cos 2 ϕ 1 ) ( 1 r 1 s ( p ) e i Ψ 1 1 + r 1 s ( p ) e i Ψ 1 ) 2 + n 0 2 sin 2 ϕ 0 ε s , eff ( p ) ( 1 , s ) = 1 .
r 0 s , eff ( p ) ( m , m 1 , , 1 , s ) = n 0 cos ϕ s , eff ( s ) n s , eff ( s ) cos ϕ 0 n 0 cos ϕ s , eff ( s ) + n s , eff ( s ) cos ϕ 0 ,
n s , f f ( p ) sin ϕ s , f f ( p ) = n 0 sin ϕ 0 .
ε s , eff ( p ) ( m , m 1 , , 1 , s ) ( n m 2 cos 2 ϕ m ) ( 1 r m s , eff ( p ) ( m 1 , , 1 , s ) e i Ψ m 1 + r m s , eff ( p ) ( m 1 , , 1 , s ) e i Ψ m ) 2 + n 0 2 sin 2 ϕ 0 ε s , eff ( p ) ( m , m 1 , , 1 , s ) = 1 ,
r m s , eff ( p ) ( m 1 , , 1 , s ) n m cos ϕ s , eff ( p ) ( m 1 , , 1 , s ) n s , eff ( p ) ( m 1 , , 1 , s ) cos ϕ m n m cos ϕ s , eff ( p ) ( m 1 , , 1 , s ) + n s , eff ( p ) ( m 1 , , 1 , s ) cos ϕ m .
r 0 s , eff ( s ) = n 0 cos ϕ 0 n s , eff ( s ) cos ϕ s , eff ( s ) n 0 cos ϕ 0 + n s , eff ( s ) cos ϕ s , eff ( s ) ,
r 0 d s , eff ( s ) = r 0 d ( s ) + r d s , eff ( s ) e i Ψ d 1 + r 0 d ( s ) r d s , eff ( s ) e i Ψ d ,
r d s , eff ( s ) = n d cos ϕ d n s , eff ( s ) cos ϕ s , eff ( s ) n d cos ϕ d + n s , eff ( s ) cos ϕ s , eff ( s ) ,
r 0 d ( s ) = n 0 cos ϕ 0 n d cos ϕ d n 0 cos ϕ 0 + n d cos ϕ d ,
Ψ d = 4 π n d d cos ϕ d λ .
Δ s r 0 d s , eff ( s ) r 0 s , eff ( s ) r 0 s , eff ( s ) ( ε d ε s , eff ( s ) ) ( ε 0 ε s , eff ( s ) ) ( i 4 π n 0 cos ϕ 0 λ ) d ,
r 0 s , eff ( p ) = n 0 cos ϕ s , eff ( p ) n s , eff ( p ) cos ϕ 0 n 0 cos ϕ s , eff ( p ) + n s , eff ( p ) cos ϕ 0 ,
r 0 d s , eff ( p ) = r 0 d ( p ) + r d s , eff ( p ) e i Ψ d 1 + r 0 d ( p ) r d s , eff ( p ) e i Ψ d ,
r d s , eff ( p ) = n d cos ϕ s , eff ( p ) n s , eff ( p ) cos ϕ d n d cos ϕ s , eff ( p ) + n s , eff ( p ) cos ϕ d ,
r 0 d ( p ) = n 0 cos ϕ d n d cos ϕ 0 n 0 cos ϕ d + n d cos ϕ 0 .
Δ p r 0 d s , eff ( p ) r 0 s , eff ( p ) r 0 s , eff ( p ) [ ( ( ε d + ε s , eff ( p ) ) ε 0 sin 2 ϕ 0 ε d ε s , eff ( p ) ) ( ε d ε s , eff ( p ) ) ε d ( ε 0 sin 2 ϕ 0 ε s , eff ( p ) cos 2 ϕ 0 ) ( ε 0 ε s , eff ( p ) ) ] ( i 4 π n 0 cos ϕ 0 λ ) d .
Δ p Δ s α eff ( ε d ε 0 ) ( ε d ε s , eff ) ε d d ,
α eff ( i 4 π n 0 cos ϕ 0 λ ) ε 0 ε s , eff ( s ) sin 2 ϕ 0 + ( ε s , eff ( p ) ) 2 cos 2 ϕ 0 ε s , eff ( s ) ε s , eff ( p ) ( ε 0 ε s , eff ( p ) ) ( ε 0 ε s , eff ( s ) ) ( ε s , eff ( p ) cos 2 ϕ 0 ε 0 sin 2 ϕ 0 ) ,
ε s , eff ( ε s , eff ( p ) ) 2 ( ε 0 ε s , eff ( s ) ) sin 2 ϕ 0 ε 0 ε s , eff ( s ) sin 2 ϕ 0 ε s , eff ( p ) ( ε s , eff ( s ) ε s , eff ( p ) cos 2 ϕ 0 ) .
α eff ( i 4 π n 0 cos ϕ 0 λ ) ( 1 ε s , eff ( s ) ( 1 , s ) ε 0 ) ( cos 2 ϕ 0 ε s , eff ( s ) ( 1 , s ) ( cos ϕ 1 n 1 ) 2 ( 1 + r 1 s ( p ) e i Ψ 1 1 r 1 s ( p ) e i Ψ 1 ) 2 cos 2 ϕ 0 ε 0 ( cos ϕ 1 n 1 ) 2 ( 1 + r 1 s ( p ) e i Ψ 1 1 r 1 s ( p ) e i Ψ 1 ) 2 ) ,
ε s , eff ( ε 0 ε s , eff ( s ) ( 1 , s ) ) sin 2 ϕ 0 cos 2 ϕ 0 ε s , eff ( s ) ( 1 , s ) ( cos ϕ 1 n 1 ) 2 ( 1 + r 1 s ( p ) e i Ψ 1 1 r 1 s ( p ) e i Ψ 1 ) 2 .

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