Abstract

We present an improved theory of image formation by reflection interference contrast microscopy (RICM) for structural studies of stratified films on planar substrates and propose a new theoretical approach to analyzing the surface profile of nonplanar films. We demonstrate the validity of the new approach by analyzing the fringe patterns of RICM images from wedge-shaped liquid films and spherical probes. By simulation of various scenarios, we study the effect of finite-aperture illumination and the shape of the nonplanar interface on the interference fringe pattern of RICM images. We show how the reconstruction of the microscopic topography of the sample from the fringe spacing is corrected by angular and curvature correction terms. We discuss the variation of the mean intensity of the fringe patterns and the decay in the fringe amplitude with increasing fringe order that is caused by nonplanar interfaces of different slope.

© 1998 Optical Society of America

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References

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  1. A. S. G. Curtis, “The mechanism of adhesion of cells to glass. A study by interference reflection microscopy,” J. Cell Biol. 20, 199–215 (1964).
    [CrossRef] [PubMed]
  2. J. S. Ploem, “Reflection contrast microscopy as a tool for investigation of the attachment of living cells to a glass surface,” Mononuclear Phagocytes in Immunity, Infection and Pathology, R. van Furth, ed. (Blackwell, Oxford, 1975), pp. 405–421.
  3. D. Gingell, I. Todd, “Interference reflection microscopy: a quantitative theory for image interpretation and its application to cell-substratum separation measurement,” Biophys. J. 26, 507–526 (1979).
    [CrossRef] [PubMed]
  4. H. Verschueren, “Interference reflection microscopy in cell biology: methodology and applications,” J. Cell. Sci. 75, 279–301 (1985).
    [PubMed]
  5. R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
    [CrossRef] [PubMed]
  6. A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
    [CrossRef] [PubMed]
  7. J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
    [CrossRef]
  8. J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
    [CrossRef]
  9. J. Rädler, E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. France II 3, 727–748 (1993).
    [CrossRef]
  10. M. Kühner, E. Sackmann, “Ultrathin hydrated dextran films grafted on glass: preparation and characterization of structural, viscous, and elastic properties by quantitative microinterferometry,” Langmuir 12, 4866–4876 (1996).
    [CrossRef]
  11. G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993).
  13. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).
  14. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1975).
  15. H. Beyer, Theorie und Praxis der Interferenzmikroskopie, 1st ed. (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1974).
  16. H. Mykura, “Interference microscopy at high wedge angles,” Proc. Phys. Soc. B67, 281–289 (1954).
  17. H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopes,” J. Sci. Instrum. 40, 313–315 (1963).
    [CrossRef]
  18. F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopy,” J. Sci. Instrum. 33, 236–238 (1956).
    [CrossRef]
  19. J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
    [CrossRef]
  20. M. Pluta, Advanced Light Microscopy Vol. II: Specialized Methods (PWN-Polish Scientific Publishers, Warszawa, Poland, 1988).
  21. nih image, developed by W. Rasband, National Institutes of Health, Bethesda, Md., 20815.
  22. M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

1998 (1)

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

1997 (3)

A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
[CrossRef] [PubMed]

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

1996 (1)

M. Kühner, E. Sackmann, “Ultrathin hydrated dextran films grafted on glass: preparation and characterization of structural, viscous, and elastic properties by quantitative microinterferometry,” Langmuir 12, 4866–4876 (1996).
[CrossRef]

1995 (1)

J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
[CrossRef]

1993 (1)

J. Rädler, E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. France II 3, 727–748 (1993).
[CrossRef]

1985 (1)

H. Verschueren, “Interference reflection microscopy in cell biology: methodology and applications,” J. Cell. Sci. 75, 279–301 (1985).
[PubMed]

1979 (1)

D. Gingell, I. Todd, “Interference reflection microscopy: a quantitative theory for image interpretation and its application to cell-substratum separation measurement,” Biophys. J. 26, 507–526 (1979).
[CrossRef] [PubMed]

1964 (1)

A. S. G. Curtis, “The mechanism of adhesion of cells to glass. A study by interference reflection microscopy,” J. Cell Biol. 20, 199–215 (1964).
[CrossRef] [PubMed]

1963 (1)

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopes,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

1956 (2)

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopy,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

1954 (1)

H. Mykura, “Interference microscopy at high wedge angles,” Proc. Phys. Soc. B67, 281–289 (1954).

Albersdörfer, A.

A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
[CrossRef] [PubMed]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1975).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1975).

Beyer, H.

H. Beyer, Theorie und Praxis der Interferenzmikroskopie, 1st ed. (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1974).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993).

Bruinsma, R.

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

Curtis, A. S. G.

A. S. G. Curtis, “The mechanism of adhesion of cells to glass. A study by interference reflection microscopy,” J. Cell Biol. 20, 199–215 (1964).
[CrossRef] [PubMed]

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).

Faix, J.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

Feder, T.

A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
[CrossRef] [PubMed]

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

Gates, J. W.

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Gerisch, G.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

Gingell, D.

D. Gingell, I. Todd, “Interference reflection microscopy: a quantitative theory for image interpretation and its application to cell-substratum separation measurement,” Biophys. J. 26, 507–526 (1979).
[CrossRef] [PubMed]

Jaworek, T.

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

Kühner, M.

M. Kühner, E. Sackmann, “Ultrathin hydrated dextran films grafted on glass: preparation and characterization of structural, viscous, and elastic properties by quantitative microinterferometry,” Langmuir 12, 4866–4876 (1996).
[CrossRef]

Mykura, H.

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopes,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

H. Mykura, “Interference microscopy at high wedge angles,” Proc. Phys. Soc. B67, 281–289 (1954).

Nardi, J.

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

Niewöhner, J.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).

Ploem, J. S.

J. S. Ploem, “Reflection contrast microscopy as a tool for investigation of the attachment of living cells to a glass surface,” Mononuclear Phagocytes in Immunity, Infection and Pathology, R. van Furth, ed. (Blackwell, Oxford, 1975), pp. 405–421.

Pluta, M.

M. Pluta, Advanced Light Microscopy Vol. II: Specialized Methods (PWN-Polish Scientific Publishers, Warszawa, Poland, 1988).

Rädler, J.

J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
[CrossRef]

J. Rädler, E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. France II 3, 727–748 (1993).
[CrossRef]

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).

Rhead, G. E.

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopes,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

Sackmann, E.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
[CrossRef] [PubMed]

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

M. Kühner, E. Sackmann, “Ultrathin hydrated dextran films grafted on glass: preparation and characterization of structural, viscous, and elastic properties by quantitative microinterferometry,” Langmuir 12, 4866–4876 (1996).
[CrossRef]

J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
[CrossRef]

J. Rädler, E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. France II 3, 727–748 (1993).
[CrossRef]

Simson, R.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

Strey, H.

J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
[CrossRef]

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).

Todd, I.

D. Gingell, I. Todd, “Interference reflection microscopy: a quantitative theory for image interpretation and its application to cell-substratum separation measurement,” Biophys. J. 26, 507–526 (1979).
[CrossRef] [PubMed]

Tolmon, F. R.

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopy,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

Verschueren, H.

H. Verschueren, “Interference reflection microscopy in cell biology: methodology and applications,” J. Cell. Sci. 75, 279–301 (1985).
[PubMed]

Wallraff, E.

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

Wegner, G.

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

Wiegand, G.

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993).

Wood, J. G.

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopy,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

Biophys. J. (3)

D. Gingell, I. Todd, “Interference reflection microscopy: a quantitative theory for image interpretation and its application to cell-substratum separation measurement,” Biophys. J. 26, 507–526 (1979).
[CrossRef] [PubMed]

R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, E. Sackmann, “Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin of cortexillins,” Biophys. J. 74, 514–522 (1998).
[CrossRef] [PubMed]

A. Albersdörfer, T. Feder, E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repulsion and short-range attraction force: a model membrane study,” Biophys. J. 73, 245–257 (1997).
[CrossRef] [PubMed]

Europhys. Lett. (1)

J. Nardi, T. Feder, R. Bruinsma, E. Sackmann, “Electrostatic adhesion between fluid membranes: phase separation and blistering,” Europhys. Lett. 37, 371–376 (1997).
[CrossRef]

J. Cell Biol. (1)

A. S. G. Curtis, “The mechanism of adhesion of cells to glass. A study by interference reflection microscopy,” J. Cell Biol. 20, 199–215 (1964).
[CrossRef] [PubMed]

J. Cell. Sci. (1)

H. Verschueren, “Interference reflection microscopy in cell biology: methodology and applications,” J. Cell. Sci. 75, 279–301 (1985).
[PubMed]

J. Colloid Interface Sci. (1)

G. Wiegand, T. Jaworek, G. Wegner, E. Sackmann, “Studies of structure and local wetting properties on heterogeneous, micropatterned solid surfaces by microinterferometry,” J. Colloid Interface Sci. 196, 299–312 (1997).
[CrossRef]

J. Phys. France II (1)

J. Rädler, E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. France II 3, 727–748 (1993).
[CrossRef]

J. Sci. Instrum. (3)

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopes,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopy,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Langmuir (2)

M. Kühner, E. Sackmann, “Ultrathin hydrated dextran films grafted on glass: preparation and characterization of structural, viscous, and elastic properties by quantitative microinterferometry,” Langmuir 12, 4866–4876 (1996).
[CrossRef]

J. Rädler, H. Strey, E. Sackmann, “Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces,” Langmuir 11, 4539–4548 (1995).
[CrossRef]

Proc. Phys. Soc. (1)

H. Mykura, “Interference microscopy at high wedge angles,” Proc. Phys. Soc. B67, 281–289 (1954).

Other (8)

J. S. Ploem, “Reflection contrast microscopy as a tool for investigation of the attachment of living cells to a glass surface,” Mononuclear Phagocytes in Immunity, Infection and Pathology, R. van Furth, ed. (Blackwell, Oxford, 1975), pp. 405–421.

M. Pluta, Advanced Light Microscopy Vol. II: Specialized Methods (PWN-Polish Scientific Publishers, Warszawa, Poland, 1988).

nih image, developed by W. Rasband, National Institutes of Health, Bethesda, Md., 20815.

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1990).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1975).

H. Beyer, Theorie und Praxis der Interferenzmikroskopie, 1st ed. (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1974).

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Figures (7)

Fig. 1
Fig. 1

Schematic views of the traces of light beams in RICM: (a) Normal-incidence illumination of objects composed of stratified planar interfaces. (b) Illumination with a finite aperture in a multilayer system composed of stratified planar interfaces (multireflections are indicated). (c) Representation of the geometric conditions at an interface inclined by the angle β with respect to the Z axis. Top: Light incident through area a and reflected in area s interferes at location B. Note that a and s are deformed from a circular to an elliptical shape at the inclined interfaces. Bottom: Interference of two waves at B(x, y) with an optical path difference of AS + SB - CB.

Fig. 2
Fig. 2

Schematic view of the RICM setup with the antiflex technique (see Section 3).

Fig. 3
Fig. 3

(a) Plot of the intensity as a function of the thickness of an adsorbed layer in model systems with (solid curves) and without (dashed curves) a contrast-enhancing coating (95-nm MgF2/25-nm SiO2). For ambient media, air and water are considered. (b) Vertical distribution of the refractive indices n of the model systems with the same contrast-enhancing coating as in (a): For the solid curve, segment 1 represents the glass substrate, segment 2 represents MgF2, segment 3 represents SiO2, and segment 4 represents the adsorbed layer. In segment 5, air is represented by the solid portion and water by the dashed portion. (c) RICM image of a polymer pattern adsorbed on a contrast-enhancement coated (95-nm MgF2/25-nm SiO2) substrate that is exposed to air. The refractive-index distribution coincides with that of the model system shown in (b). (d) RICM image of a single lipid bilayer observed during spreading on a contrast-enhancement coated (45-nm MgF2/5-nm SiO2) substrate. The membrane and the substrate are separated by an ultrathin water film of approximately 2 nm in thickness.

Fig. 4
Fig. 4

(a) RICM interferogram of the contact zone of a water droplet on an indium tin oxide substrate exposed to air. The droplet forms a wedge-shaped film with a specific contact angle. (b) RICM interferogram of a water droplet on a glass substrate exposed to air. (c) Plot of the intensity along the line shown in (a). The measured data were fitted by use of the theory of nonplanar interfaces (NI) and the finite-aperture (FA) theory. (d) Plot of the intensity along the line shown in (b). Again, the measured data were fitted by the theories of nonplanar interfaces (NI) and of a finite aperture (FA). The intensity curve simulated with the finite-aperture theory does not reproduce the measured data in a satisfactory way. For the calculation we assumed the same intensity from the ambient media–substrate interface (left-hand side) and the same fringe spacing (right-hand side).

Fig. 5
Fig. 5

(a) RICM image of a latex sphere, in water, that is close to the substrate surface. In addition to the fringe pattern, a brighter region can be observed below the sphere. (b) Plot of the intensity along the line shown in (a). The solid curve corresponds to fitting the data on the basis of the theory of nonplanar interfaces (NI).

Fig. 6
Fig. 6

(a) Plot of the angular correction term δ c as a function of the uncorrected angle of inclination βUC for the systems listed in Table 2. The data for each system were fitted by use of Eq. (32). (b) Plot of the correction term δ c,sphere of the radius of curvature of a sphere as functions of the uncorrected radius ρUC for the system of the glass–water–latex bead with a value of INA = 0.48. The data were fitted by use of Eqs. (34) and (35).

Fig. 7
Fig. 7

(a) Plots of the mean intensity of the fringe pattern of wedge-shaped films as a function of the inclination angle βc. Shown are examples for a glass–air–water system with INA = 0.48 (crosses), a glass–air–water system with INA = 0.75 (inverted triangles), and a glass–hexadecane–air system with INA = 0.48 (circles with a dot). (b) Plots of the envelopes of the fringe patterns of the glass–water–air system (INA = 0.48) as functions of the fringe order. The envelopes are plotted for different angles of inclination. The intensity values of the envelopes of the fringe patterns are normalized by the mean intensity of the fringe patterns.

Tables (2)

Tables Icon

Table 1 Refractive Indices of the Materials Used in the Experiments

Tables Icon

Table 2 Fitting Parameters c1, c2, and c3 of the Angular Correction Term δ c for the Simulated System with Various Compositions and INA’sa

Equations (52)

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

I x ,   y = E 2 t ,
E r ,   t = E r exp i ϕ r exp i ω t ,
I x ,   y =   E   d Ω   d Ω *   E   d Ω   d Ω t ,
I x ,   y =   E * E   d Ω   d Ω t ,
Ω = ϑ ,   φ
E = E refl = E 1 + E 2 + + .
E refl = r 01 E 0 + 1 - r 01 2 r 12 E 0 exp - ik Δ 1 + + = RE 0 ,
r ij s ϑ = n i cos   ϑ i - n j cos   ϑ j n i cos   ϑ i + n j cos   ϑ j ,
r ij p ϑ = n j cos   ϑ i - n i cos   ϑ j n j cos   ϑ i + n i cos   ϑ j ,
R = r 01 + i = 1 m s = 1 i 1 - r s - 1 , s 2 exp - ik Δ s r i , i + 1 ,
I x ,   y = R * RE 0 2 = R * RI 0 ,
Δ i = 2 n i d i x ,   y ,
S = I 01 L 1 I 12 L 2 I m - 1 m L m I m m + 1 ,
I ij = 1 t ij 1 r ij r ij 1 ,
L i = exp ik Δ i 0 0 exp - ik Δ i ,
Δ i = 2 n i cos   ϑ i d i x ,   y = 2 n i 2 - n 0 2 sin 2   ϑ 0 1 / 2 d i x ,   y ,
E 0 E refl = S 11 S 12 S 21 S 22 E trans 0 ,
R s , p ϑ = S 21 s , p ϑ S 11 s , p ϑ .
I x ,   y = I refl s + I refl p ,
I refl s , p = 0 2 π 0 θ R s , p ϑ * R s , p ϑ E 0 s , p ϑ 2 sin   ϑ   d ϑ d φ 0 2 π 0 θ sin   ϑ   d ϑ d φ .
E 0 s , p ϑ = I 0 / 2 ϑ α IA 0 ϑ > α IA ,
α IA = arcsin INA n 0 ,
ζ θ = 0 2 π 0 θ sin   ϑ   d ϑ d φ = 2 π 1 - cos   θ .
I refl s , p = 2 π ζ α IA I 0 2 0 α IA R s , p ϑ * R s , p ϑ sin   ϑ   d ϑ .
Δ 1 = n 1 AS ¯ + SB ¯ - n 0 CB ¯ .
R s , p ϑ ,   φ = Θ α IA - ϑ inc , E 1 r 01 s , p ϑ inc , E 1 + Θ α IA - ϑ inc , E 2 t 01 s , p ϑ inc , E 2 r 12 s , p ϑ refl t 10 s , p ϑ × exp - ik Δ 1 .
I refl s , p = 1 ζ α IA I 0 2 0 2 π 0 α DA R s , p ϑ ,   φ * R s , p ϑ ,   φ ×   sin   ϑ d ϑ d φ ,
v = I max - I min I max + I min .
tan   β x = d h x d x = Δ h Δ x ,
β UC = tan - 1 λ / 4 n i s x s ,
β C = β UC δ c β UC .
δ c β UC = 1 + 1 c 1 c 2 - β UC c 3 ,
λ 4 n i   s = h x s - h x 0 = ρ UC 2 - x 0 2 1 / 2 - ρ UC 2 - x s 2 1 / 2 ,
ρ C = ρ UC δ c , sphere ρ UC ,
δ c , sphere ρ UC = 1 - 1 c 1 c 2 + ρ UC .
IF 01   :   n 01 X - 0 = 0 ,
n 01 = 0 0 1 ,     0 = 0 0 0 .
IF 12   :   n 12 X - A = 0 ,
n 12 = - sin   β 0 cos   β ,     A = A x A y A z ,
IF 12   :   X - M 2 = R 2 ,
M = M x M y R + h .
g refl   :   B + λ r refl = X ,
r refl = - sin   ϑ cos   φ sin   ϑ sin   φ cos   ϑ ,     B = B x B y 0 .
n 12 = M - S | M - S | = 1 R M - S .
cos   ϑ refl = - r refl · n 12 | r refl | · | n 12 | = - r refl · n 12 ,
r inc   =   r refl + 2   cos   ϑ refl   n 12 .
g inc   :   S + η r inc = X .
cos   ϑ inc = r inc · n 01 | r inc | · | n 01 | = r inc · n 01 ,
CB ¯ = n 0 n 1 sin   ϑ inc AB ¯ .
ϑ inc , E 1 = ϑ inc if   E 2   exists   λ S < 0   or   λ S ,   η A ϑ otherwise .
ϑ inc , E 2 = ϑ inc if   | A | < r FieldStop ,   Δ 1 < 30   μ m > α IA otherwise .
n i + 1 r trans = n i r inc + n i + 1 cos   ϑ i + 1 - n i cos   ϑ i n i , i + 1 ,

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