Abstract

Ellipsometric measurements give information on two film properties with high precision, thickness and refractive index. In the simplest case, the substrate is covered with a single homogenous, transparent film. Yet, with ellipsometry, it is only possible to determine the two film properties thickness and refractive simultaneously if the layer thickness exceeds 15 nm – a restriction well known for a century. Here we present a technique to cross this limitation: A series expansion of the ellipsometric ratio ρ to the second order of the layer thickness relative to the wavelength reveals the first and second ellipsometric moment. These moments are properties of the thin film and independent of incident angle. Using both moments and one additional reference measurement enables to determine simultaneously both thickness and refractive index of ultra-thin films down to 5 nm thickness.

© 2017 Optical Society of America

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References

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  1. K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
    [Crossref] [PubMed]
  2. G. Sauerbrey, “Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung,” Z. Phys. 155(2), 206–222 (1959).
    [Crossref]
  3. M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
    [Crossref]
  4. B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature 332(6165), 615–617 (1988).
    [Crossref]
  5. E. Hecht, Optics (Addison-Wesley, 1990).
  6. G. B. Airy, “On the phænomena of Newton’s rings when formed between two transparent substances of different refractive powers,” Philosoph. Mag. Ser. 3 2(7), 20–30 (1833).
  7. P. Drude, The Theory of Optics (Longmains, Green and Co., 1902).
  8. P. Yeh, Optical Waves in Layered Media (John Wiley, 1988).
  9. M. M. Ibrahim and N. M. Bashara, “Parameter-correlation and computational considerations in multiple-angle ellipsometry,” J. Opt. Soc. Am. 61(12), 1622–1629 (1971).
    [Crossref]
  10. H. G. Tompkins, A Users Guide to Ellipsometry (Academic Press, 1993).
  11. D. Johannsmann, “Investigations of Soft Organic Films with Ellipsometry,” in Functional Polymer Films, W. Knoll and R.C. Advincula, ed.(Wiley-VCH, 2011).
  12. R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (Elsevier,1989).
  13. J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
    [Crossref] [PubMed]
  14. S. Hénon and J. Meunier, “Microscope at the brewster-angle - direct observation of 1st order phase-transitions in monolayers,” Rev. Sci. Instrum. 62(4), 936–939 (1991).
    [Crossref]
  15. G. T. Ayoub and N. M. Bashara, “Characterization of a very thin uniaxial film on a nonabsorbing substrate by multiple wavelength ellipsometry: Palmitic acid on water,” J. Opt. Soc. Am. 68(7), 978–983 (1978).
    [Crossref]
  16. A. N. Saxena, “Changes in the Phase and Amplitude of Polarized Light Reflected from a Film-Covered Surface and Their Relations with the Film Thickness,” J. Opt. Soc. Am. 55(9), 1061–1067 (1965).
    [Crossref]
  17. M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
    [Crossref]
  18. E. D. Palik, Handbook of Optical Constants of Solids (Elsevier, 1998).
  19. G. Decher and J.-D. Hong, “Buildup of ultrathin multilayer films by a self-assembly process, 1 consecutive adsorption of anionic and cationic bipolar amphiphiles on charged surfaces,” Macromol. Symp. 46(1), 321–327 (1991).
    [Crossref]
  20. K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
    [Crossref]
  21. M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
    [Crossref]
  22. M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
    [Crossref]

2017 (1)

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

2005 (1)

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

2002 (2)

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

1999 (1)

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

1998 (1)

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

1992 (1)

M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
[Crossref]

1991 (2)

G. Decher and J.-D. Hong, “Buildup of ultrathin multilayer films by a self-assembly process, 1 consecutive adsorption of anionic and cationic bipolar amphiphiles on charged surfaces,” Macromol. Symp. 46(1), 321–327 (1991).
[Crossref]

S. Hénon and J. Meunier, “Microscope at the brewster-angle - direct observation of 1st order phase-transitions in monolayers,” Rev. Sci. Instrum. 62(4), 936–939 (1991).
[Crossref]

1988 (1)

B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature 332(6165), 615–617 (1988).
[Crossref]

1978 (1)

1971 (1)

1965 (1)

1959 (1)

G. Sauerbrey, “Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung,” Z. Phys. 155(2), 206–222 (1959).
[Crossref]

1833 (1)

G. B. Airy, “On the phænomena of Newton’s rings when formed between two transparent substances of different refractive powers,” Philosoph. Mag. Ser. 3 2(7), 20–30 (1833).

Ahrens, H.

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

Airy, G. B.

G. B. Airy, “On the phænomena of Newton’s rings when formed between two transparent substances of different refractive powers,” Philosoph. Mag. Ser. 3 2(7), 20–30 (1833).

Ayoub, G. T.

Bashara, N. M.

Bouwman, W. G.

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

Büscher, K.

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

Cavalcanti, L. P.

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

Decher, G.

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

G. Decher and J.-D. Hong, “Buildup of ultrathin multilayer films by a self-assembly process, 1 consecutive adsorption of anionic and cationic bipolar amphiphiles on charged surfaces,” Macromol. Symp. 46(1), 321–327 (1991).
[Crossref]

Gopinadhan, M.

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

Graf, K.

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

Günther, J.-U.

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

Helm, C. A.

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

Hénon, S.

S. Hénon and J. Meunier, “Microscope at the brewster-angle - direct observation of 1st order phase-transitions in monolayers,” Rev. Sci. Instrum. 62(4), 936–939 (1991).
[Crossref]

Hong, J.-D.

G. Decher and J.-D. Hong, “Buildup of ultrathin multilayer films by a self-assembly process, 1 consecutive adsorption of anionic and cationic bipolar amphiphiles on charged surfaces,” Macromol. Symp. 46(1), 321–327 (1991).
[Crossref]

Ibrahim, M. M.

Jonson, M.

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

Kasemo, B.

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

Kjaer, K.

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

Knoll, W.

B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature 332(6165), 615–617 (1988).
[Crossref]

Lee, K.-B.

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Lösche, M.

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

Malfatti-Gasperini, A. A.

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

Meunier, J.

S. Hénon and J. Meunier, “Microscope at the brewster-angle - direct observation of 1st order phase-transitions in monolayers,” Rev. Sci. Instrum. 62(4), 936–939 (1991).
[Crossref]

Mirkin, C. A.

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Mrksich, M.

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Oliveira, R. G.

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

Park, S.-J.

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Paudler, M.

M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
[Crossref]

Puentes-Martinez, X. E.

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

Pusterla, J. M.

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

Riegler, H.

M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
[Crossref]

Rodahl, M.

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

Rothenhäusler, B.

B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature 332(6165), 615–617 (1988).
[Crossref]

Ruths, J.

M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
[Crossref]

Sauerbrey, G.

G. Sauerbrey, “Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung,” Z. Phys. 155(2), 206–222 (1959).
[Crossref]

Saxena, A. N.

Schmitt, J.

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

Smith, J. C.

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Steitz, R.

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

Voinova, M. V.

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

Biochim. Biophys. Acta (1)

J. M. Pusterla, A. A. Malfatti-Gasperini, X. E. Puentes-Martinez, L. P. Cavalcanti, and R. G. Oliveira, “Refractive index and thickness determination in Langmuir monolayers of myelin lipids,” Biochim. Biophys. Acta 1859(5), 924–930 (2017).
[Crossref] [PubMed]

J. Opt. Soc. Am. (3)

Langmuir (2)

M. Paudler, J. Ruths, and H. Riegler, “Analysis of Multiple-Angle Ellipsometry of Uniaxial Ultrathin Organic Films at the Air-Water Interface and Determination of the Refractive Indices in Behenic Acid Monolayers,” Langmuir 8(1), 184–189 (1992).
[Crossref]

K. Büscher, K. Graf, H. Ahrens, and C. A. Helm, “Influence of adsorption conditions on the structure of polyelectrolyte multilayers,” Langmuir 18(9), 3585–3591 (2002).
[Crossref]

Macromol. Symp. (1)

G. Decher and J.-D. Hong, “Buildup of ultrathin multilayer films by a self-assembly process, 1 consecutive adsorption of anionic and cationic bipolar amphiphiles on charged surfaces,” Macromol. Symp. 46(1), 321–327 (1991).
[Crossref]

Macromolecules (2)

M. Gopinadhan, H. Ahrens, J.-U. Günther, R. Steitz, and C. A. Helm, “Approaching the precipitation temperature of the deposition solution and the effects on the internal order of polyelectrolyte multilayers,” Macromolecules 38(12), 5228–5235 (2005).
[Crossref]

M. Lösche, J. Schmitt, G. Decher, W. G. Bouwman, and K. Kjaer, “Detailed Structure of Molecularly Thin Polyelectrolyte Multiliater Films on Solid Substrates as Revealed from Neutron Reflectometry,” Macromolecules 31(25), 8893–8906 (1998).
[Crossref]

Nature (1)

B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature 332(6165), 615–617 (1988).
[Crossref]

Philosoph. Mag. Ser. 3 (1)

G. B. Airy, “On the phænomena of Newton’s rings when formed between two transparent substances of different refractive powers,” Philosoph. Mag. Ser. 3 2(7), 20–30 (1833).

Phys. Scr. (1)

M. V. Voinova, M. Rodahl, M. Jonson, and B. Kasemo, “Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: continuum mechanics approach,” Phys. Scr. 59(5), 391–396 (1999).
[Crossref]

Rev. Sci. Instrum. (1)

S. Hénon and J. Meunier, “Microscope at the brewster-angle - direct observation of 1st order phase-transitions in monolayers,” Rev. Sci. Instrum. 62(4), 936–939 (1991).
[Crossref]

Science (1)

K.-B. Lee, S.-J. Park, C. A. Mirkin, J. C. Smith, and M. Mrksich, “Protein nanoarrays generated by dip-pen nanolithography,” Science 295(5560), 1702–1705 (2002).
[Crossref] [PubMed]

Z. Phys. (1)

G. Sauerbrey, “Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung,” Z. Phys. 155(2), 206–222 (1959).
[Crossref]

Other (7)

E. Hecht, Optics (Addison-Wesley, 1990).

P. Drude, The Theory of Optics (Longmains, Green and Co., 1902).

P. Yeh, Optical Waves in Layered Media (John Wiley, 1988).

H. G. Tompkins, A Users Guide to Ellipsometry (Academic Press, 1993).

D. Johannsmann, “Investigations of Soft Organic Films with Ellipsometry,” in Functional Polymer Films, W. Knoll and R.C. Advincula, ed.(Wiley-VCH, 2011).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (Elsevier,1989).

E. D. Palik, Handbook of Optical Constants of Solids (Elsevier, 1998).

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Figures (3)

Fig. 1
Fig. 1 Color-coded representation of the squared deviation χ2 between measured and calculated ellipsometric angles. The investigated sample is a silicon wafer with native oxide film. The actual film properties are dF = 1.6 nm and nF = 1.457. (a) Model parameter for the calculation of ellipsometric angles are dF and nF. The solid white line corresponds to all pairs (dF, nF), for which the first ellipsometric moment is constant E1 = 11 nm. (b) Model parameters are the first and second ellipsometric moment, E1 and E2 defined in Eq. (4). The position of the minimum is E1 = 11 nm and E2 = 575 nm, respectively (white point). The white ellipse corresponds to all pairs (E1, E2), which lead to the constant deviation χ2 = 0.008.
Fig. 2
Fig. 2 Influence of film thickness dF on ellipsometric angles in case of (a) SiO2 and (b) polymer films. Ellipsometric angles Δ and Ψ are converted to Reρ and (Imρ)2, respectively. Measured data (black points) are compared to calculated trajectories using Airy's formula, Drude approximation and Eq. (5). ρ0 is calculated (not measured) using Fresnel's reflection coefficients and literature values (red open circle). The nominal incidence angle is αinc = 60° and wavelength λ = 633 nm.(c) Apparent refractive index nF of a polymer film as function of the film thickness. The actual value is nF = 1.57 (solid line). A conventional data analysis (black squares) gives systematically wrong values for films thinner than 15 nm. By means of a reference measurement on the uncoated substrate and the use of Eq. (7), reliable values are obtained for films thicker than dF = 5 nm.
Fig. 3
Fig. 3 Position-resolved ellipsometric angles (a) Δ and (b) Ψ of a polymer film partially covering a Si wafer. The boundary line between uncoated substrate (region B) and polymer film (region A) is located at x = 500 µm. (c) Average ellipsometric angles Δ and Ψ of region A and B converted to Reρ and (Imρ)2, respectively. Thus, nF = 1.57 and dF = 5.2 nm is obtained.

Equations (8)

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

ρ=tanΨexp iΔ
r p = n 2 cos α inc n 1 cos α tra +i 2π d F λ A + n 2 cos α inc + n 1 cos α tra +i 2π d F λ A r s = n 1 cos α inc n 2 cos α tra +i 2π d F λ B + n 1 cos α inc + n 2 cos α tra +i 2π d F λ B
ρ ρ 0 +i ρ 2π d F λ +( ρ A + ρ B ) ( 2π d F λ ) 2
ρ =2 n 1 n 2 2 n 1 2 sin 2 α inc cos α inc cos 2 ( α inc α tra ) ( n F 2 n 1 2 )( n F 2 n 2 2 ) n F 2 ρ A =2 n 1 n 2 ( n 2 2 n 1 2 ) 2 sin 2 α inc cos 2 α inc cos 3 ( α inc α tra ) ( ( n F 2 n 1 2 )( n F 2 n 2 2 ) n F 2 ) 2 ρ B =2 n 1 n 2 cos α inc sin 2 α inc cos α tra ( n 2 2 n 1 2 ) 2 cos 2 ( α inc α tra ) ( n F 2 n 1 2 )( n F 2 n 2 2 ) n F 2 ( n F 2 n 1 2 ) n 2 4 +( n F 2 n 2 2 ) n F 2 n 1 2 n F 2
E 1 = d F ( n F 2 n 1 2 )( n F 2 n 2 2 ) n F 2 E 2 = d F ( n F 2 n 1 2 ) n 2 4 +( n F 2 n 2 2 ) n F 2 n 1 2 n F 2
Reρ= ρ 0 1 2 n 2 n 1 cos( α inc α tra ) sin 2 α inc ( 1 cos α tra cos( α inc α tra ) n 2 2 cos α inc E 2 E 1 ) (Imρ) 2 = ρ 0 S (Imρ) 2
E 2 E 1 = n 2 2 cos α inc cos α tra cos( α inc α tra ) ( 2S n 1 n 2 sin 2 α inc cos( α inc α tra ) +1 )
n F = ( n 1 2 + n 2 2 ) E 2 E 1 n 2 2 ( n 2 2 n 1 2 )+ ( E 2 E 1 n 2 2 ) 2 ( n 2 2 n 1 2 ) 2 +4 ( n 1 n 2 ) 4 2( E 2 E 1 + n 1 2 )

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