Abstract

We describe a high-speed interferometric method, using multiple angles of incidence and multiple wavelengths, to measure the absolute thickness, tilt, the local angle between the surfaces, and the refractive index of a fluctuating transparent wedge. The method is well suited for biological, fluid and industrial applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. H. Kiessig, “Untersuchungen zur totalreflexion von röntgenstrahlen,” Ann. Phys. 402(6), 715–768 (1931).
    [Crossref]
  2. X.-L. Zhou and S.-H. Chen, “Theoretical foundation of x-ray and neutron reflectometry,” Phys. Rep. 257(4-5), 223–348 (1995).
    [Crossref]
  3. L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95(2), 359–369 (1954).
    [Crossref]
  4. T. Russell, “X-ray and neutron reflectivity for the investigation of polymers,” Mater. Sci. Rep. 5(4), 171–271 (1990).
    [Crossref]
  5. K. Stoev and K. Sakurai, “Recent theoretical models in grazing incidence x-ray reflectometry,” The Rigaku Journal 14, 22–37 (1997).
  6. C. Majkrzak, “Polarized neutron reflectometry,” Phys. B 173(1-2), 75–88 (1991).
    [Crossref]
  7. J. Ankner and G. Felcher, “Polarized-neutron reflectometry,” J. Magn. Magn. Mater. 200(1-3), 741–754 (1999).
    [Crossref]
  8. J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
    [Crossref]
  9. W. A. Pliskin and E. E. Conrad, “Nondestructive determination of thickness and refractive index of transparent films,” IBM J. Res. Dev. 8(1), 43–51 (1964).
    [Crossref]
  10. H. Takabayashi and T. Nakamura, “Apparatus for measuring thickness of object transparent to light utilizing interferometric method,” (1987). US Patent 4,660,980.
  11. T. Kihara and K. Yokomori, “Simultaneous measurement of refractive index and thickness of thin film by polarized reflectances,” Appl. Opt. 29(34), 5069–5073 (1990).
    [Crossref]
  12. K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
    [Crossref]
  13. H. J. Choi, H. H. Lim, H. S. Moon, T. B. Eom, J. J. Ju, and M. Cha, “Measurement of refractive index and thickness of transparent plate by dual-wavelength interference,” Opt. Express 18(9), 9429–9434 (2010).
    [Crossref]
  14. N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” (1991). US Patent 4,999,014.
  15. N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “High resolution ellipsometric apparatus,” (1991). US Patent 5,042,951.
  16. A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
    [Crossref]
  17. J. T. Fanton, J. Opsal, and A. Rosencwaig, “Method and apparatus for evaluating the thickness of thin films,” (1993). US Patent 5,181,080.
  18. W.-D. Joo, J. You, Y.-S. Ghim, and S.-W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” in Interferometry XIV: Techniques and Analysis, vol. 7063J. Schmit, K. Creath, and C. E. Towers, eds., International Society for Optics and Photonics (SPIE, 2008), pp. 245–252.
  19. J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
    [Crossref]
  20. J.-A. Kim, J.-W. Kim, J.-Y. Lee, and J.-H. Woo, “Thickness measuring apparatus and thickness measuring method,” (2018). US Patent 9,927,224.
  21. R. Kingslake, Lenses in photography: the practical guide to optics for photographers (Barnes, 1963).
  22. J. W. Strutt, “Liv. on the interference-rings, described by haidinger, observable by means of plates whose surfaces are absolutely parallel,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 12(71), 489–493 (1906).
    [Crossref]
  23. C. Raman and V. Rajagopalan, “L. haidinger’s rings in non-uniform plates,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29(196), 508–514 (1940).
    [Crossref]
  24. J. E. Park, J. Kim, and M. Cha, “Measurement of thickness profiles of glass plates by analyzing haidinger fringes,” Appl. Opt. 56(7), 1855–1860 (2017).
    [Crossref]
  25. M. He, “Long-time evolution of interfacial structure of partial wetting,” (2020).
  26. L. Hoyt, “New table of the refractive index of pure glycerol at 20 c,” Ind. Eng. Chem. 26(3), 329–332 (1934).
    [Crossref]
  27. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
  28. G. B. Thomas, Calculus and analytic geometry (1974).
  29. M. He and S. R. Nagel, “Characteristic interfacial structure behind a rapidly moving contact line,” Phys. Rev. Lett. 122(1), 018001 (2019).
    [Crossref]
  30. J. D. Jackson, Classical electrodynamics (John Wiley & Sons, 2007).
  31. H. Tompkins and E. A. Irene, Handbook of ellipsometry (William Andrew, 2005).
  32. H. Fujiwara, Spectroscopic ellipsometry: principles and applications (John Wiley & Sons, 2007).
  33. I. Bischofberger, R. Ramachandran, and S. R. Nagel, “Fingering versus stability in the limit of zero interfacial tension,” Nat. Commun. 5(1), 5265 (2014).
    [Crossref]
  34. T. E. Videbæk and S. R. Nagel, “Diffusion-driven transition between two regimes of viscous fingering,” Phys. Rev. Fluids 4(3), 033902 (2019).
    [Crossref]

2019 (2)

M. He and S. R. Nagel, “Characteristic interfacial structure behind a rapidly moving contact line,” Phys. Rev. Lett. 122(1), 018001 (2019).
[Crossref]

T. E. Videbæk and S. R. Nagel, “Diffusion-driven transition between two regimes of viscous fingering,” Phys. Rev. Fluids 4(3), 033902 (2019).
[Crossref]

2017 (2)

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

J. E. Park, J. Kim, and M. Cha, “Measurement of thickness profiles of glass plates by analyzing haidinger fringes,” Appl. Opt. 56(7), 1855–1860 (2017).
[Crossref]

2014 (1)

I. Bischofberger, R. Ramachandran, and S. R. Nagel, “Fingering versus stability in the limit of zero interfacial tension,” Nat. Commun. 5(1), 5265 (2014).
[Crossref]

2010 (1)

2004 (1)

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

1999 (1)

J. Ankner and G. Felcher, “Polarized-neutron reflectometry,” J. Magn. Magn. Mater. 200(1-3), 741–754 (1999).
[Crossref]

1998 (1)

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

1997 (1)

K. Stoev and K. Sakurai, “Recent theoretical models in grazing incidence x-ray reflectometry,” The Rigaku Journal 14, 22–37 (1997).

1995 (1)

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of x-ray and neutron reflectometry,” Phys. Rep. 257(4-5), 223–348 (1995).
[Crossref]

1992 (1)

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

1991 (1)

C. Majkrzak, “Polarized neutron reflectometry,” Phys. B 173(1-2), 75–88 (1991).
[Crossref]

1990 (2)

1964 (1)

W. A. Pliskin and E. E. Conrad, “Nondestructive determination of thickness and refractive index of transparent films,” IBM J. Res. Dev. 8(1), 43–51 (1964).
[Crossref]

1954 (1)

L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

1940 (1)

C. Raman and V. Rajagopalan, “L. haidinger’s rings in non-uniform plates,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29(196), 508–514 (1940).
[Crossref]

1934 (1)

L. Hoyt, “New table of the refractive index of pure glycerol at 20 c,” Ind. Eng. Chem. 26(3), 329–332 (1934).
[Crossref]

1931 (1)

H. Kiessig, “Untersuchungen zur totalreflexion von röntgenstrahlen,” Ann. Phys. 402(6), 715–768 (1931).
[Crossref]

1906 (1)

J. W. Strutt, “Liv. on the interference-rings, described by haidinger, observable by means of plates whose surfaces are absolutely parallel,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 12(71), 489–493 (1906).
[Crossref]

Ankner, J.

J. Ankner and G. Felcher, “Polarized-neutron reflectometry,” J. Magn. Magn. Mater. 200(1-3), 741–754 (1999).
[Crossref]

Asada, K.

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Bischofberger, I.

I. Bischofberger, R. Ramachandran, and S. R. Nagel, “Fingering versus stability in the limit of zero interfacial tension,” Nat. Commun. 5(1), 5265 (2014).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Cha, M.

Chen, S.-H.

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of x-ray and neutron reflectometry,” Phys. Rep. 257(4-5), 223–348 (1995).
[Crossref]

Choi, H. J.

Conrad, E. E.

W. A. Pliskin and E. E. Conrad, “Nondestructive determination of thickness and refractive index of transparent films,” IBM J. Res. Dev. 8(1), 43–51 (1964).
[Crossref]

Dura, J. A.

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

Eom, T. B.

Fanton, J.

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

Fanton, J. T.

J. T. Fanton, J. Opsal, and A. Rosencwaig, “Method and apparatus for evaluating the thickness of thin films,” (1993). US Patent 5,181,080.

Felcher, G.

J. Ankner and G. Felcher, “Polarized-neutron reflectometry,” J. Magn. Magn. Mater. 200(1-3), 741–754 (1999).
[Crossref]

Fujiwara, H.

H. Fujiwara, Spectroscopic ellipsometry: principles and applications (John Wiley & Sons, 2007).

Ghim, Y.-S.

W.-D. Joo, J. You, Y.-S. Ghim, and S.-W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” in Interferometry XIV: Techniques and Analysis, vol. 7063J. Schmit, K. Creath, and C. E. Towers, eds., International Society for Optics and Photonics (SPIE, 2008), pp. 245–252.

Gold, N.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “High resolution ellipsometric apparatus,” (1991). US Patent 5,042,951.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” (1991). US Patent 4,999,014.

He, M.

M. He and S. R. Nagel, “Characteristic interfacial structure behind a rapidly moving contact line,” Phys. Rev. Lett. 122(1), 018001 (2019).
[Crossref]

M. He, “Long-time evolution of interfacial structure of partial wetting,” (2020).

Hoyt, L.

L. Hoyt, “New table of the refractive index of pure glycerol at 20 c,” Ind. Eng. Chem. 26(3), 329–332 (1934).
[Crossref]

Irene, E. A.

H. Tompkins and E. A. Irene, Handbook of ellipsometry (William Andrew, 2005).

Ishikawa, K.

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Iwata, K.

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical electrodynamics (John Wiley & Sons, 2007).

Jin, J.

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

Joo, W.-D.

W.-D. Joo, J. You, Y.-S. Ghim, and S.-W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” in Interferometry XIV: Techniques and Analysis, vol. 7063J. Schmit, K. Creath, and C. E. Towers, eds., International Society for Optics and Photonics (SPIE, 2008), pp. 245–252.

Ju, J. J.

Kagawa, K.-i

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Kang, C.-S.

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

Kelso, S.

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

Kiessig, H.

H. Kiessig, “Untersuchungen zur totalreflexion von röntgenstrahlen,” Ann. Phys. 402(6), 715–768 (1931).
[Crossref]

Kihara, T.

Kim, J.

Kim, J. W.

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

Kim, J.-A.

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

J.-A. Kim, J.-W. Kim, J.-Y. Lee, and J.-H. Woo, “Thickness measuring apparatus and thickness measuring method,” (2018). US Patent 9,927,224.

Kim, J.-W.

J.-A. Kim, J.-W. Kim, J.-Y. Lee, and J.-H. Woo, “Thickness measuring apparatus and thickness measuring method,” (2018). US Patent 9,927,224.

Kim, S.-W.

W.-D. Joo, J. You, Y.-S. Ghim, and S.-W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” in Interferometry XIV: Techniques and Analysis, vol. 7063J. Schmit, K. Creath, and C. E. Towers, eds., International Society for Optics and Photonics (SPIE, 2008), pp. 245–252.

Kingslake, R.

R. Kingslake, Lenses in photography: the practical guide to optics for photographers (Barnes, 1963).

Lee, J. Y.

J.-A. Kim, J. W. Kim, C.-S. Kang, J. Jin, and J. Y. Lee, “An interferometric system for measuring thickness of parallel glass plates without 2π ambiguity using phase analysis of quadrature haidinger fringes,” Rev. Sci. Instrum. 88(5), 055108 (2017).
[Crossref]

Lee, J.-Y.

J.-A. Kim, J.-W. Kim, J.-Y. Lee, and J.-H. Woo, “Thickness measuring apparatus and thickness measuring method,” (2018). US Patent 9,927,224.

Lim, H. H.

Majkrzak, C.

C. Majkrzak, “Polarized neutron reflectometry,” Phys. B 173(1-2), 75–88 (1991).
[Crossref]

Majkrzak, C. F.

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

Moon, H. S.

Nagel, S. R.

M. He and S. R. Nagel, “Characteristic interfacial structure behind a rapidly moving contact line,” Phys. Rev. Lett. 122(1), 018001 (2019).
[Crossref]

T. E. Videbæk and S. R. Nagel, “Diffusion-driven transition between two regimes of viscous fingering,” Phys. Rev. Fluids 4(3), 033902 (2019).
[Crossref]

I. Bischofberger, R. Ramachandran, and S. R. Nagel, “Fingering versus stability in the limit of zero interfacial tension,” Nat. Commun. 5(1), 5265 (2014).
[Crossref]

Nakamura, T.

H. Takabayashi and T. Nakamura, “Apparatus for measuring thickness of object transparent to light utilizing interferometric method,” (1987). US Patent 4,660,980.

Nguyen, N. V.

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

Opsal, J.

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “High resolution ellipsometric apparatus,” (1991). US Patent 5,042,951.

J. T. Fanton, J. Opsal, and A. Rosencwaig, “Method and apparatus for evaluating the thickness of thin films,” (1993). US Patent 5,181,080.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” (1991). US Patent 4,999,014.

Park, J. E.

Parratt, L. G.

L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

Pliskin, W. A.

W. A. Pliskin and E. E. Conrad, “Nondestructive determination of thickness and refractive index of transparent films,” IBM J. Res. Dev. 8(1), 43–51 (1964).
[Crossref]

Rajagopalan, V.

C. Raman and V. Rajagopalan, “L. haidinger’s rings in non-uniform plates,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29(196), 508–514 (1940).
[Crossref]

Ramachandran, R.

I. Bischofberger, R. Ramachandran, and S. R. Nagel, “Fingering versus stability in the limit of zero interfacial tension,” Nat. Commun. 5(1), 5265 (2014).
[Crossref]

Raman, C.

C. Raman and V. Rajagopalan, “L. haidinger’s rings in non-uniform plates,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29(196), 508–514 (1940).
[Crossref]

Richter, C. A.

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

Rosencwaig, A.

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “High resolution ellipsometric apparatus,” (1991). US Patent 5,042,951.

J. T. Fanton, J. Opsal, and A. Rosencwaig, “Method and apparatus for evaluating the thickness of thin films,” (1993). US Patent 5,181,080.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” (1991). US Patent 4,999,014.

Russell, T.

T. Russell, “X-ray and neutron reflectivity for the investigation of polymers,” Mater. Sci. Rep. 5(4), 171–271 (1990).
[Crossref]

Sakurai, K.

K. Stoev and K. Sakurai, “Recent theoretical models in grazing incidence x-ray reflectometry,” The Rigaku Journal 14, 22–37 (1997).

Stoev, K.

K. Stoev and K. Sakurai, “Recent theoretical models in grazing incidence x-ray reflectometry,” The Rigaku Journal 14, 22–37 (1997).

Strutt, J. W.

J. W. Strutt, “Liv. on the interference-rings, described by haidinger, observable by means of plates whose surfaces are absolutely parallel,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 12(71), 489–493 (1906).
[Crossref]

Takabayashi, H.

H. Takabayashi and T. Nakamura, “Apparatus for measuring thickness of object transparent to light utilizing interferometric method,” (1987). US Patent 4,660,980.

Thomas, G. B.

G. B. Thomas, Calculus and analytic geometry (1974).

Tompkins, H.

H. Tompkins and E. A. Irene, Handbook of ellipsometry (William Andrew, 2005).

Ueda, M.

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Videbæk, T. E.

T. E. Videbæk and S. R. Nagel, “Diffusion-driven transition between two regimes of viscous fingering,” Phys. Rev. Fluids 4(3), 033902 (2019).
[Crossref]

Willenborg, D.

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

Willenborg, D. L.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “High resolution ellipsometric apparatus,” (1991). US Patent 5,042,951.

N. Gold, D. L. Willenborg, J. Opsal, and A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” (1991). US Patent 4,999,014.

Wolf, E.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Woo, J.-H.

J.-A. Kim, J.-W. Kim, J.-Y. Lee, and J.-H. Woo, “Thickness measuring apparatus and thickness measuring method,” (2018). US Patent 9,927,224.

Yamano, H.

K. Ishikawa, H. Yamano, K.-i Kagawa, K. Asada, K. Iwata, and M. Ueda, “Measurement of thickness of a thin film by means of laser interference at many incident angles,” Opt. Laser Eng. 41(1), 19–29 (2004).
[Crossref]

Yokomori, K.

You, J.

W.-D. Joo, J. You, Y.-S. Ghim, and S.-W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” in Interferometry XIV: Techniques and Analysis, vol. 7063J. Schmit, K. Creath, and C. E. Towers, eds., International Society for Optics and Photonics (SPIE, 2008), pp. 245–252.

Zhou, X.-L.

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of x-ray and neutron reflectometry,” Phys. Rep. 257(4-5), 223–348 (1995).
[Crossref]

Ann. Phys. (1)

H. Kiessig, “Untersuchungen zur totalreflexion von röntgenstrahlen,” Ann. Phys. 402(6), 715–768 (1931).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

J. A. Dura, C. A. Richter, C. F. Majkrzak, and N. V. Nguyen, “Neutron reflectometry, x-ray reflectometry, and spectroscopic ellipsometry characterization of thin sio 2 on si,” Appl. Phys. Lett. 73(15), 2131–2133 (1998).
[Crossref]

A. Rosencwaig, J. Opsal, D. Willenborg, S. Kelso, and J. Fanton, “Beam profile reflectometry: A new technique for dielectric film measurements,” Appl. Phys. Lett. 60(11), 1301–1303 (1992).
[Crossref]

IBM J. Res. Dev. (1)

W. A. Pliskin and E. E. Conrad, “Nondestructive determination of thickness and refractive index of transparent films,” IBM J. Res. Dev. 8(1), 43–51 (1964).
[Crossref]

Ind. Eng. Chem. (1)

L. Hoyt, “New table of the refractive index of pure glycerol at 20 c,” Ind. Eng. Chem. 26(3), 329–332 (1934).
[Crossref]

J. Magn. Magn. Mater. (1)

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Figures (11)

Fig. 1.
Fig. 1. Setup for measuring thickness and refractive index of a parallel slab with no tilt ($\alpha =\gamma =0$). Rays marked by blue and red meet at the upper focal plane (dotted line) of the lens. This leads to the interference pattern of concentric circles in the center $S$ of the focal plane (as shown schematically above that plane).
Fig. 2.
Fig. 2. Experimental setup for measuring a fluctuating liquid layer. ①: high-speed camera; ②: laser beam ($\lambda =$ 532 nm); ③: beam splitter; ④: focal plane + reference scale; ⑤: convex lens (Nikon f=50mm, 1:1.8); ⑥: liquid layer; ⑦: substrate; ⑧: interference pattern.
Fig. 3.
Fig. 3. (a) Interference pattern produced by a film of liquid (a water/glycerol mixture: glycerol mass fraction $65.2\%$). Background subtracted for clarity. Scale bar: 2mm. (b) $h/n$ versus accumulated fringe number $p$, for water (blue circles) and a water/glycerol mixture (red squares; glycerol mass fraction $65.2\%$). Inset: refractive indices deduced from the slopes of the linear fits.
Fig. 4.
Fig. 4. Ray tracing diagram when $\alpha =0,\gamma \ne 0$. The pattern center $S'$ at the upper focal plane is shifted by $\phi_0=-\gamma$, while the shape and spacing of the fringes remain approximately unchanged.
Fig. 5.
Fig. 5. Conical surface formed by the segment of $\overline {OQ}$ when $P$ cycles along a closed fringe. The shape of the cone does not change with the orientation of the sample. The conic section at $Q$ (shape of the fringe) remains approximately unchanged from a) $\gamma =0$ to b) $\gamma > 0$.
Fig. 6.
Fig. 6. Schematics for $\alpha \ne 0,\gamma =0$. (a) The transverse plane intersects the sample to form a rectangular cross section (shaded area). Light rays interfering at point $P$ lie approximately in the transverse plane. Analyzing the ray tracing diagram projected onto this plane gives $h$ and $n$. (b) The longitudinal plane intersects the sample to form a wedge cross section (shaded area) of slope $\alpha$. Analyzing the ray tracing diagram in this plane gives $h$ and $\alpha$.
Fig. 7.
Fig. 7. Ray tracing diagram when $\alpha \ne 0,\gamma = 0$. The pattern center $S'$ at the upper focal plane is shifted by $\phi_{0,\alpha }$ (Eq. (18)), while the shape of the fringes are also skewed. $\overline {S'B}$: the longitudinal direction; $\overline {S'A}$: the transverse direction.
Fig. 8.
Fig. 8. (a) A typical simulated curve of $\sigma (\vec {h})$ as a function of $\alpha$, for a sample of $h=95$ $\mu$m, $\alpha =5^{\circ }$ (solid curve). The location of the minimum (solid vertical line) indicates the best estimate of $\alpha_{\textrm{{fit}}}=4.8^{\circ }$. Dashed line: ideal curve without uncertainties. (b) and (c) Histograms of $h_{\textrm{{fit}}}$ and $\alpha_{\textrm{{fit}}}$, showing mean values (red vertical lines) and widths of the distributions for $N=5000$ simulations.
Fig. 9.
Fig. 9. Projected ray tracing diagram when $\alpha \ne 0,\gamma \ne 0$. Tilded symbols: corresponding projected values. $\gamma \ne 0$ shifts the pattern by $\vec {\phi }_{0,\gamma }$ (green). $\alpha \ne 0$ shifts the pattern by $\vec {\phi }_{0,\alpha }$ (orange) and skews the shape of the fringes. The net shift of the pattern $\vec {\phi }_0$ (purple) is the vector sum of $\vec {\phi }_{0,\alpha }+\vec {\phi }_{0,\gamma }$.
Fig. 10.
Fig. 10. An observed fringe configuration, due to curvature of the sample surface, that cannot be modeled by the simple analysis of this work. Background subtracted for clarity.
Fig. 11.
Fig. 11. A ray tracing diagram in plane $ABCD$ (red solid arrows) is projected onto a nearby plane $AB'C'D$ (blue dashed arrows). The projection obeys Snell’s law to the first order of the angle $\Omega$ between the planes.

Tables (1)

Tables Icon

Table 1. Measured thicknesses for different pieces of glass ( h g0 , h g1 , h g2 ), and refractive indices for air ( n a ), water ( n w ), and a water/glyceral mixture ( n wg , glycerol mass fraction 53.5 %). “Micrometer” values of h ’s were measured mechanically with a micrometer or caliper. “Literature” values of n ’s were taken from literature [26,27]. The values obtained by the interference measurement are shown in the right column.

Equations (32)

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ϕ = arctan S P ¯ S T ¯ = arctan R f ,
Δ L = 2 n h cos θ ,
destructive: 2 n h cos θ m = m λ ,
constructive: 2 n h cos θ m + 1 2 = ( m + 1 2 ) λ ,
sin ϕ = n sin θ .
destructive: 2 n h 1 sin 2 ϕ m n 2 = m λ ,
constructive: 2 n h 1 sin 2 ϕ m + 1 2 n 2 = ( m + 1 2 ) λ .
destructive: 2 n h ( 1 sin 2 ϕ m 2 n 2 ) = m λ ,
constructive: 2 n h ( 1 sin 2 ϕ m + 1 2 2 n 2 ) = ( m + 1 2 ) λ .
1 λ [ cos ( 2 ϕ m + 1 2 ) cos ( 2 ϕ m ) ] = n h .
Δ L max = 2 n h .
2 n h ( t + Δ t ) 2 n h ( t ) = Δ p λ ,
h ( t + Δ t ) n h ( t ) n = Δ p λ 2 n 2 ,
Δ h n Δ p = λ 2 n 2 .
1 λ 2 [ cos 2 ϕ m + 1 2 + cos 2 ϕ m 1 2 2 cos ϕ m ] = 1 4 h 2 .
sin ( ϕ + γ ) = n sin θ .
ϕ 0 = γ .
2 a = Q Q ¯ = Q T ¯ + T Q ¯ = f [ tan ϕ + tan ( ϕ + 2 γ ) ] = 2 f tan [ arcsin ( n sin θ ) ] + O ( γ 2 ) .
2 b = 2 a 1 sin 2 γ cos 2 ( ϕ + γ ) = 2 a + O ( γ 2 ) .
sin ( ϕ + α ) = n sin ( θ + α ) .
ϕ 0 , α = arcsin ( n sin α ) α .
Φ m ϕ m ϕ 0 , α ,
λ 4 n h = cos ( arcsin sin [ Φ m + arcsin ( n sin α ) ] n α ) cos ( arcsin sin [ Φ m + 1 2 + arcsin ( n sin α ) ] n α ) .
( h 0 , h 1 2 , h 1 , ) = f λ ( Φ 0 , Φ 1 2 , Φ 1 , Φ 3 2 , ; α ) ,
α f i t = argmin α σ λ ( f λ ( Φ 0 , Φ 1 2 , Φ 1 , Φ 3 2 , ; α ) ) .
h f i t = f λ ( Φ 0 , Φ 1 2 , Φ 1 , Φ 3 2 , ; α f i t ) .
ϕ 0 , α = arcsin ( n sin α ) α ,
ϕ 0 , γ = ϕ 0 ϕ 0 , α ,
A O ¯ O C ¯ = n 1 n 2 ,
A O ¯ O C ¯ = n 1 n 2 cos ω 1 cos ω 2 ,
1 Ω 2 2 + = cos Ω < cos ω 1 < cos ω 1 cos ω 2 < 1.
A O ¯ O C ¯ = n 1 n 2 ,