Abstract

A technique for measuring dielectric tensors in anisotropic layered structures, such as thin films of biaxial materials, is demonstrated. The ellipsometric data are collected in a quasi-monochromatic Mueller matrix image acquired over a large range of incident and azimuthal angles by illuminating a very small area on the sample with a focused beam from a modulating polarization state generator. After the beam interacts with the sample, the reflected and/or transmitted light is collected using an imaging polarization state analyzer. An image of the exit pupil of a collection objective lens is formed across a CCD such that each pixel collects light from a different angle incident on the sample, thus acquiring ellipsometric data at numerous incident angles simultaneously. The large range of angles and orientations is necessary to accurately determine dielectric tensors. The small but significant polarization aberrations of the low-polarization objective lenses used to create and collect the focused beams provide a significant challenge to accurate measurement. Measurements are presented of a thin-film E-type polarizer and a stretched plastic biaxial film.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (Elsevier, 1987).
  2. J. Lekner, "Ellipsometry of anisotropic media," J. Opt. Soc. Am. A 10, 1579-1581 (1993).
    [CrossRef]
  3. J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
    [CrossRef]
  4. G. E. J. Jellison, F. A. Modine, and L. A. Boatner, "Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett. 22, 1808-1810 (1997).
    [CrossRef]
  5. A. En-Naciri, L. Johann, R. Kleim, M. Sieskind, and M. Amann, "Spectroscopic ellipsometry of anisotropic materials: application to the optical constants of HgI2," Appl. Opt. 38, 647-654 (1998).
    [CrossRef]
  6. M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
    [CrossRef]
  7. M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, "Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2," J. Opt. Soc. Am. A 13, 875-883 (1996).
    [CrossRef]
  8. G. E. Jellison and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
    [CrossRef]
  9. G. E. Jellison and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
    [CrossRef]
  10. M. Schubert, B. Rheinlander, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, and B. Johs, "Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals," J. Opt. Soc. Am. A 13, 1930-1940 (1996).
    [CrossRef]
  11. M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
    [CrossRef]
  12. M. Schubert, T. E. Tiwald, and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
    [CrossRef]
  13. A. En-Naciri, L. Johann, and R. Kleim, "Spectroscopic generalized ellipsometry based on Fourier analysis," Appl. Opt. 38, 4802-4811 (1999).
    [CrossRef]
  14. J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).
  16. G. E. Jellison, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
    [CrossRef]
  17. R. M. A. Azzam, "Photopolarimetric measurements of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1978).
    [CrossRef] [PubMed]
  18. R. M. A. Azzam, "A simple Fourier photo-polarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices," Opt. Commun. 25, 137-140 (1978).
    [CrossRef]
  19. R. A. Chipman, "Polarimetry," in OSA Handbook of Optics (McGraw-Hill, 1995), pp. 22.21-22.35.
  20. D. H. Goldstein, "Mueller matrix dual-rotating retarder polarimeter," Appl. Opt. 31, 6676-6683 (1992).
    [CrossRef] [PubMed]
  21. R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).
  22. M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).
  23. K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).
  24. P. Varga, "Use of confocal microscopes in conoscopy and ellipsometry. 1. Electromagnetic theory," Appl. Opt. 39, 6360-6365 (2000).
    [CrossRef]
  25. D. W. Berreman, "Optics in stratified and anisotropic media: 4×4-matrix formulation," J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  26. P. Yeh, "Optics of anisotropic layered media: a new 4×4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
    [CrossRef]
  27. M. Schubert, "Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems," Phys. Rev. B 53, 4265-4274 (1996).
    [CrossRef]
  28. M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
    [CrossRef]
  29. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
    [CrossRef]
  30. R. A. Chipman, "Depolarization index and the average degree of polarization," Appl. Opt. 44, 2490-2495 (2005).
    [CrossRef] [PubMed]
  31. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).
  32. R. A. Chipman, "Polarization aberrations," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1987).
  33. P. Yeh and M. Paukshto, "Molecular crystaline thin film E-polarizer," Mol. Mater. 14, 1-19 (2001).
  34. P. L. Lazarev and M. V. Paukshto, "Low-leakage off-angle in E-polarizers," J. Soc. Inf. Disp. 9, 101-105 (2001).
    [CrossRef]

2005 (1)

2002 (1)

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

2001 (2)

P. Yeh and M. Paukshto, "Molecular crystaline thin film E-polarizer," Mol. Mater. 14, 1-19 (2001).

P. L. Lazarev and M. V. Paukshto, "Low-leakage off-angle in E-polarizers," J. Soc. Inf. Disp. 9, 101-105 (2001).
[CrossRef]

2000 (1)

1999 (2)

1998 (5)

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

G. E. Jellison, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).

J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

A. En-Naciri, L. Johann, R. Kleim, M. Sieskind, and M. Amann, "Spectroscopic ellipsometry of anisotropic materials: application to the optical constants of HgI2," Appl. Opt. 38, 647-654 (1998).
[CrossRef]

1997 (4)

1996 (3)

1993 (1)

1992 (1)

1990 (1)

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

1980 (1)

P. Yeh, "Optics of anisotropic layered media: a new 4×4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
[CrossRef]

1978 (2)

R. M. A. Azzam, "Photopolarimetric measurements of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, "A simple Fourier photo-polarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices," Opt. Commun. 25, 137-140 (1978).
[CrossRef]

1973 (1)

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

1972 (1)

Amann, M.

Arnold, M.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, "A simple Fourier photo-polarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices," Opt. Commun. 25, 137-140 (1978).
[CrossRef]

R. M. A. Azzam, "Photopolarimetric measurements of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (Elsevier, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (Elsevier, 1987).

Berreman, D. W.

Boatner, L. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

Chipman, R. A.

R. A. Chipman, "Depolarization index and the average degree of polarization," Appl. Opt. 44, 2490-2495 (2005).
[CrossRef] [PubMed]

R. A. Chipman, "Polarization aberrations," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1987).

R. A. Chipman, "Polarimetry," in OSA Handbook of Optics (McGraw-Hill, 1995), pp. 22.21-22.35.

Collins, G.

M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).

Cramer, C.

Dollase, W.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Eifler, A.

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Elman, J. F.

J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

En-Naciri, A.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).

Franke, E.

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

Goldstein, D. H.

Gottschalch, V.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Greener, U. J.

J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Hahn, J.

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

Haller, M. E.

M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).

R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).

Hecht, J.-D.

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Herzinger, C. M.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, "Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2," J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

M. Schubert, B. Rheinlander, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, and B. Johs, "Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals," J. Opt. Soc. Am. A 13, 1930-1940 (1996).
[CrossRef]

Hodgkinson, I.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Hofmann, T.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Jellison, G. E.

Jellison, G. E. J.

Johann, L.

Johs, B.

Kasic, A.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Kleim, R.

Kramer, V.

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Kraub, G.

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Lazarev, P. L.

P. L. Lazarev and M. V. Paukshto, "Low-leakage off-angle in E-polarizers," J. Soc. Inf. Disp. 9, 101-105 (2001).
[CrossRef]

Lekner, J.

Loiterman, R. M.

R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).

Mansuripur, M.

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

Modine, F. A.

Neumann, H.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

Off, J.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Paukshto, M.

P. Yeh and M. Paukshto, "Molecular crystaline thin film E-polarizer," Mol. Mater. 14, 1-19 (2001).

Paukshto, M. V.

P. L. Lazarev and M. V. Paukshto, "Low-leakage off-angle in E-polarizers," J. Soc. Inf. Disp. 9, 101-105 (2001).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).

Rheinlander, B.

Richter, F.

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

Riede, V.

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Roder, M.

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

Schmiedel, H.

Scholz, F.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Schubert, E.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Schubert, M.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

M. Schubert, T. E. Tiwald, and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
[CrossRef]

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

M. Schubert, B. Rheinlander, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, and B. Johs, "Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals," J. Opt. Soc. Am. A 13, 1930-1940 (1996).
[CrossRef]

M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, "Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2," J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

M. Schubert, "Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems," Phys. Rev. B 53, 4265-4274 (1996).
[CrossRef]

Semenenko, A. I.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

Semenenko, L. V.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

Sieskind, M.

Sokolov, V. K.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

Spanier, R. F.

R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).

Sullivan, J.

M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).

Svitashev, K. K.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).

Tiwald, T. E.

Varga, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

Wolf, R. G.

R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).

Woollam, J. A.

Yeh, P.

P. Yeh and M. Paukshto, "Molecular crystaline thin film E-polarizer," Mol. Mater. 14, 1-19 (2001).

P. Yeh, "Optics of anisotropic layered media: a new 4×4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

M. Schubert, B. Rheinlander, E. Franke, H. Neumann, J. Hahn, M. Roder, and F. Richter, "Anisotropy of boron nitride thin-film reflectivity spectra by generalized ellipsometry," Appl. Phys. Lett. 70, 1819-1821 (1997).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Soc. Inf. Disp. (1)

P. L. Lazarev and M. V. Paukshto, "Low-leakage off-angle in E-polarizers," J. Soc. Inf. Disp. 9, 101-105 (2001).
[CrossRef]

Mol. Mater. (1)

P. Yeh and M. Paukshto, "Molecular crystaline thin film E-polarizer," Mol. Mater. 14, 1-19 (2001).

Opt. Commun. (1)

R. M. A. Azzam, "A simple Fourier photo-polarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices," Opt. Commun. 25, 137-140 (1978).
[CrossRef]

Opt. Lett. (2)

Opt. Spektrosk. (1)

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, "Ellipsometery based on a convergent light beam," Opt. Spektrosk. 34, 542-544 (1973).

Phys. Rev. B (2)

M. Schubert, "Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems," Phys. Rev. B 53, 4265-4274 (1996).
[CrossRef]

J.-D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Kraub, and V. Kramer, "Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry," Phys. Rev. B 57, 7037-7042 (1998).
[CrossRef]

Proc. SPIE (1)

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, W. Dollase, and C. M. Herzinger, "Generalized ellipsometry of complex mediums in layered systems," Proc. SPIE 4806, 264-276 (2002).
[CrossRef]

Semicond. Int. (1)

M. E. Haller, J. Sullivan, and G. Collins, "Multidomain ellipsometry for thin film process control," Semicond. Int. 21, 269-272 (1998).

Surf. Sci. (1)

P. Yeh, "Optics of anisotropic layered media: a new 4×4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
[CrossRef]

Thin Solid Films (2)

J. F. Elman, U. J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

G. E. Jellison, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

Other (7)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (Elsevier, 1987).

R. A. Chipman, "Polarimetry," in OSA Handbook of Optics (McGraw-Hill, 1995), pp. 22.21-22.35.

R. F. Spanier, R. G. Wolf, R. M. Loiterman, and M. E. Haller, "Simultaneous multiple angle/multiple wavelength ellipsometer and method," U.S. patent 5,166,752 (24 November 1992).

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988).

R. A. Chipman, "Polarization aberrations," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Diagram of the Mueller matrix imaging ellipsometer. The configuration for a specular reflection measurement is shown in (A). The polarization state generator (PSG) of the ellipsometer is mounted on a swing arm to allow for a rotation of the system optical axis about an incident angle θ 0 . By setting θ 0 to 90 ° and aligning the PSG and polarization state analyzer (PSA), transmission measurements can also be performed, as shown in (B). The sample is mounted in a stage (not shown) that allows for rotation about the normal as well as tilt adjustment. The exit pupil of the collection objective is focused onto the CCD array (dashed lines).

Fig. 2
Fig. 2

Geometry of angle of incidence measurement in a reflection configuration. Each position in the incident beam focusing on the sample represents a ray illuminating the sample with an incident angle θ and azimuthal angle ϕ. The Jones matrices are calculated in s and p coordinates for each ray, which are then rotated through angle ψ into the x and y coordinate system of the microscope objective, which is then measured by the polarimeter.

Fig. 3
Fig. 3

(A) Polarization aberration measurements of our microscope objective pair. On the left is a quiver plot of the diattenuation introduced by the objective pair. The length of the lines indicates the magnitude of the diattenuation, which varies from 0 to 9.8%. The orientation of the lines indicates the orientation of the transmission axis. On the right is a quiver plot of the linear retardance introduced by the microscope objectives. The length of the lines indicates the magnitude of the retardance, which varies from 0 to 5.4 ° ; the orientation of lines indicates the orientation of the fast axis. These polarization aberrations are about 50 times larger than the accuracy of the ellipsometer, and must be calibrated. (B) Diattenuation and the retardance that results after the calibration is applied. After calibration, the residual diattenuation varies between 0 and 2.1%, and the residual retardance varies between 0 and 2 ° .

Fig. 4
Fig. 4

Mueller matrix images for the E-type polarizer sample measured in reflection. This Mueller matrix image is a 4 × 4 array of circular images due to the circular aperture stop of the collection objective. (A) Measured Mueller matrix image. (B) Fitted Mueller matrix image calculated using dielectric tensor values of ε x = ( 1.95 + 0.09 i ) 2 , ε y = ( 1.48 + 0.005 i ) 2 , and ε z = ( 1.83 + 0.03 i ) 2 and Euler angles of Θ = 0.1 ° , Φ = 2.2 ° , and Λ = 2.2 ° .

Fig. 5
Fig. 5

Direct comparison of 60 data points in the Mueller matrix image for the E-type polarizer sample measured in reflection. (A) Grid of points in the Mueller matrix image used for comparison. (B) Comparison of the measured data and fitted values in each element of the Mueller matrix image. Fitted values are the same as those given in Fig. 4.

Fig. 6
Fig. 6

Maps of the ellipsometric parameters (A) ( Ψ x x , Ψ x y , Ψ y x ) and (B) ( Δ x x , Δ x y , Δ y x ) expressed in degrees, for the E-type polarizer sample measured in reflection. The top rows of (A) and (B) show the data derived from measured Mueller matrix images, and the bottom rows show the maps calculated using the best fit dielectric tensor values of Fig. 4. The crescent-shaped artifact in the measured Ψ x y and Δ x y results from the mathematical removal of depolarization.

Fig. 7
Fig. 7

Mueller matrix images for the E-type polarizer sample measured in reflection where the sample has been rotated about its normal by 60 ° . (A) Measured Mueller matrix image. (B) Simulated Mueller matrix image calculated using the dielectric tensor values of Fig. 4, except that the new Euler angles are Θ = 0.1 ° , Φ = 2.2 ° , and Λ = 62.2 ° .

Fig. 8
Fig. 8

Plots of the ellipsometric parameters Ψ x x and Δ x x along a horizontal cross section for the E-type polarizer measured in transmission. The circles are taken from the measurement and the dashed curves show the values generated using the fitted dielectric tensor values of ε x = ( 1.95 + 0.07 i ) 2 , ε y = ( 1.51 + 0.007 i ) 2 , and ε z = ( 1.75 + 0.03 i ) 2 , and Euler angles of Θ = 0.07 ° , Φ = 2.2 ° , and Λ = 2.2 ° .

Fig. 9
Fig. 9

(A) Biaxial plastic film Mueller matrix image measured in transmission (B) Fitted Mueller matrix image calculated using the diagonal dielectric tensor ε x = ( 1.715 ) 2 , ε y = ( 1.701 ) 2 , and ε z = ( 1.512 ) 2 and Euler angles of Θ = 0 ° , Φ = 6.19 ° , and Λ = 0.19 ° .

Fig. 10
Fig. 10

Direct comparison of 47 data points in the normalized Mueller matrix image for the stretched plastic film measured in transmission. (A) Grid of points in the Mueller matrix image used for comparison. (B) Comparison of the measured data and fitted values in each element of the Mueller matrix image. Fitted values are the same as those given in Fig. 9.

Fig. 11
Fig. 11

Comparison of the fit of the M 12 Mueller matrix element from the stretched plastic sample with the microscope objective included and removed.

Fig. 12
Fig. 12

Maps of the ellipsometric parameters (A) ( Ψ x x , Ψ x y , Ψ y x ) and (B) ( Δ x x , Δ x y , Δ y x ) expressed in degrees, for the stretched biaxial plastic film measured in transmission. The top rows of (A) and (B) show the measurements, and the bottom rows show the maps generated using the fitted dielectric tensor values from Fig. 9.

Equations (19)

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

ε ̃ = R T ( Θ , Φ , Δ ) [ ( n x + i κ x ) 2 0 0 0 ( n y + i κ y ) 2 0 0 0 ( n z + i κ z ) 2 ] R ( Θ , Φ , Δ ) ,
E r = [ r p p ( θ , ϕ ) r s p ( θ , ϕ ) r p s ( θ , ϕ ) r s s ( θ , ϕ ) ] E 0 = r ( θ , ϕ ) E 0 ,
E t = [ t p p ( θ , ϕ ) t s p ( θ , ϕ ) t p s ( θ , ϕ ) t s s ( θ , ϕ ) ] E 0 = t ( θ , ϕ ) E 0 .
( k ̃ i + ε ̃ i ) E i = 0 ,
k ̃ i E i = H i .
k ̃ = 1 k 0 2 [ k y 2 k z 2 k x k y k x k z k x k y k x 2 k z 2 k y k z k x k z k y k z k x 2 k y 2 ] .
R ( ψ ) = [ 1 0 0 0 0 cos ψ sin ψ 0 0 sin ψ cos ψ 0 0 0 0 1 ]
M x y ( θ i , ϕ i ) = R ( ψ i ) M S P ( θ i , ϕ i ) R ( ψ i ) .
M = M depol M ret M diat .
tan Ψ x x = J x x J y y , tan Ψ x y = J x y J y y , tan Ψ y x = J y x J y y ,
Δ x x = arg [ J x x J y y ] , Δ x y = arg [ J x y J y y ] , Δ y x = arg [ J y x J y y ] .
μ i ( w ) = 1 16 k = 1 16 1 σ k 2 [ M ̂ k , i ( w ) M ̂ 1 , i ( w ) M k , i M 1 , i ] 2 ,
Δ w = ( A t A + I p ) 1 μ .
A i , j = k = 1 16 2 σ k 2 { w j [ M i , k ( w ) M i , 1 ( w ) ] } .
M Ill ( x , y ) = M ret [ δ ( x , y ) , ϕ ( x , y ) + π 2 ] M diat [ D ( x , y ) , ϕ ( x , y ) ] ,
M col ( x , y ) = M diat [ D ( x , y ) , ϕ ( x , y ) ] M ret [ δ ( x , y ) , ϕ ( x , y ) + π 2 ] ,
δ ( x , y ) = a δ + b δ θ ( x , y ) 2 + c δ θ ( x , y ) 4 ,
D ( x , y ) = a D + b D θ ( x , y ) 2 + c D θ ( x , y ) 4 .
M i = λ Δ λ 2 λ + Δ λ 2 S ( λ ) M i ( λ ) d λ ,

Metrics