Abstract

We describe the use of an ellipsometer having the configuration PCSC’A to determine the Mueller matrix of a linear optical system S, using imperfect compensators C and C’ and linear polarizer P and analyzer A. Two manual (16 intensity), one semiautomatic (rotating-compensator) and one fully automatic (dual rotating-compensator) methods are described. Real-time Fourier analysis (of both raw and reduced data) is used in the automatic methods to reduce random and systematic measurement errors. Calibration procedures given for each method permit Mueller matrix <i>spectroscopy</i> using (quasi-) achromatic retarders whose principal axes, as well as retardation and transmission ratios change slightly with wavelength.

© 1978 Optical Society of America

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  1. E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Stand. Misc. Publ. 256, U.S. G.P.O., Washington, 1964.
  2. Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall, Surf. Sci., 16, (1969).
  3. Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam, Surf. Sci., 56, (1976).
  4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1976).
  5. D. E. Aspnes, "Spectroscopic Ellipsometry of Solids," in Optical Properties of Solids: New Developments, edited by B. O. Seraphin, (North-Holland, Amsterdam, 1976).
  6. The Jones and Muller Calculi are discussed at length in Refs. 4, 7, and 8.
  7. W. A. Shurcliff, Polarized Light, (Harvard Univ., Cambridge, Mass., 1962).
  8. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, (Pergamon, New York, 1971).
  9. R. M. A. Azzam and N. M. Bashara, "Ellipsometric Measurement of the Polarization Transfer Function of an Optical System," J. Opt. Soc. Am., 62, 336–340, (1972).
  10. D. E. Aspnes and P. S. Hauge, "Rotating-Compensator/Analyzer Fixed-Analyzer Ellipsometer: Analysis and Comparison to Other Automatic Ellipsometers," J. Opt. Soc. Am., 66, 949–954, (1976).
  11. Planets, Stars and Nebulae Studies with Photopolarimetry, edited by T. Gehrels, (University of Arizona, 1974).
  12. Optical Polarimetry: Instrumentation and Applications, edited by R. M. A. Azzam and D. L. Coffeen, Proc. Soc. Photo-Opt. Instrum. Eng., 112, (1977).
  13. P. S. Hauge, "Survey of Methods for the Complete Determination of a State of Polarization," Proc. Soc. Photo-Opt. Instrum. Eng., 88, 3–10, (1976).
  14. P. S. Hauge, "Techniques of Measurement of the Polarization- Altering Properties of Linear Optical Systems," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 2–11, (1977).
  15. F. Perrin, "Polarization of Light Scattered by Isotropic Opalescent Media," J. Chem. Phys., 10, 415–427, (1942).
  16. H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).
  17. D. L. Coffeen and J. E. Hansen, "Polarization Studies of Planetary Atmospheres," see Ref. 11, 518–581.
  18. S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum., 40, 761–767, (1969); S. N. Jasperson, D. K. Burge, and R. C. O'Handley, "A Modulated Ellipsometer for Studying Thin Film Optical Properties and Surface Dynamics," Surf. Sci., 37, 548–558, (1973); J. I. Treu, A. B. Callender, and S. E. Schnatterly, "Use of a Stable Polarization Modulator in a Scanning Spectrophotometer and Ellipsometer," Rev. Sci. Instrum., 44, 793–797, (1973).
  19. A. J. Hunt and D. R. Huffman, "A New Polarization-Modulation Light Scattering Instrument," Rev. Sci. Instrum., 44, 1753–1762, (1973); R. J. Perry, A. J. Hunt, and D. R. Huffman, "Experimental determinations of Mueller scattering matrices for nonspherical particles," Appl. Opt., 17, 2700–2710, (1978).
  20. R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).
  21. R. H. Muller, "Definitions and Conventions in Ellipsometry," Surf. Sci., 16, 14–33, (1969).
  22. The compensator Jones and Mueller matrices in Eqs. (1), (3), (5), etc., have been normalized so that the first element is unity. The effect of this normalization is accounted for in the determination of the detector sensitivity g.
  23. In Ref. 33, the defect parameters were defined as q, r, and s. In this paper, p, r, and s are used, where p = q - 1.
  24. If absolute values of Stokes parameters are desired, at least one absolute intensity measurement must be made at each wavelength used. Ip and g may then be determined separately. This is unnecessary, however, both for MME and for determination of polarization state, which is completely determined by the normalized Stokes parameters sk = Sk/S0, (k = 1,…, 3).
  25. G. F. Beardsley, "Mueller Scattering Matrix of Sea Water," J. Opt. Soc. Am., 58, 52–57, (1968).
  26. A. C. Holland and G. Gagne, "The Scattering of Polarized Light by Polydisperse Systems of Irregular Particles," Appl. Opt., 9, 1113–1121, (1970).
  27. D. E. Aspnes, "Fourier Transform Detection System for Rotating- Analyzer Ellipsometers," Opt. Commun., 8, 222–225, (1973); D. E. Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Appl. Opt., 14, 220–228, (1975).
  28. P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev., 17, 472–489, (1973).
  29. Alternatively, the compensator can be driven by a stepping motor, and the reference and data pulses taken from the driving circuitry.
  30. Z. Sekera, "Recent Developments in the Study of the Polarization of Sky Light," Adv. Geophys., 3, 43–104, (1956); "Light Scattering in the Atmosphere and the Polarization of Sky Light," J. Opt. Soc., Am., 47, 484–490, (1957); "Polarization of Skylight," Encyclopedia of Phys., 48, Springer, Berlin, 1957, pp. 288–328.
  31. E. B. Hodgdon, "Theory, Design and Calibration of a uv Spectrophotopolarimeter," Appl. Opt., 4, 1479–1483, (1965).
  32. F. Q. Orrall, "A Complete Stokes Vector Polarimeter," Solar Magnetic Fields, edited by R. Howard, (Reidel, Dordrecht, 1971), pp. 30–36.
  33. P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun., 14, 431–437, (1975).
  34. P. S. Hauge, "Generalized Rotating-Compensator Ellipsometry," Surf. Sci., 56, 148–160, (1976).
  35. P. S. Hauge, "Automated Mueller Matrix Ellipsometry," Opt. Commun., 17, 74–76, (1976).
  36. D. E. Aspnes, "Photoelectric Ellipsometer for Measuring Partially Polarized Light," J. Opt. Soc. Am., 65, 1274–1278, (1975); "A Photometric Ellipsometer for Measuring Flux in a General State of Polarization," Surf. Sci., 56, 161–169, (1976).
  37. J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).
  38. H. G. Berry, G. Gabrielse, and E. E. Livingston, "Measurement of the Stokes Parameters of Light," Appl. Opt., 16, 3200–3205, (1977).
  39. To illustrate, reasonable values of l = 8, m = 360, and n = 8 give 23 C40 intensity readings which are condensed into 20 and then 16 matrix elements.
  40. R. M. A. Azzam, "Photopolarimetric Measurement of the Mueller Matrix by Fourier Analysis of a Single Detected Signal," Opt. Lett., 2, 148–150, (1978).
  41. Other rotation ratios are possible, including rotation in opposing directions. However, the ratio of 5:1 gives separation of all 12 frequencies at their lowest possible values.

1978 (1)

1977 (4)

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

H. G. Berry, G. Gabrielse, and E. E. Livingston, "Measurement of the Stokes Parameters of Light," Appl. Opt., 16, 3200–3205, (1977).

P. S. Hauge, "Techniques of Measurement of the Polarization- Altering Properties of Linear Optical Systems," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 2–11, (1977).

R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).

1976 (4)

D. E. Aspnes and P. S. Hauge, "Rotating-Compensator/Analyzer Fixed-Analyzer Ellipsometer: Analysis and Comparison to Other Automatic Ellipsometers," J. Opt. Soc. Am., 66, 949–954, (1976).

P. S. Hauge, "Survey of Methods for the Complete Determination of a State of Polarization," Proc. Soc. Photo-Opt. Instrum. Eng., 88, 3–10, (1976).

P. S. Hauge, "Generalized Rotating-Compensator Ellipsometry," Surf. Sci., 56, 148–160, (1976).

P. S. Hauge, "Automated Mueller Matrix Ellipsometry," Opt. Commun., 17, 74–76, (1976).

1975 (2)

1973 (3)

A. J. Hunt and D. R. Huffman, "A New Polarization-Modulation Light Scattering Instrument," Rev. Sci. Instrum., 44, 1753–1762, (1973); R. J. Perry, A. J. Hunt, and D. R. Huffman, "Experimental determinations of Mueller scattering matrices for nonspherical particles," Appl. Opt., 17, 2700–2710, (1978).

D. E. Aspnes, "Fourier Transform Detection System for Rotating- Analyzer Ellipsometers," Opt. Commun., 8, 222–225, (1973); D. E. Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Appl. Opt., 14, 220–228, (1975).

P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev., 17, 472–489, (1973).

1972 (1)

1970 (1)

1969 (2)

S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum., 40, 761–767, (1969); S. N. Jasperson, D. K. Burge, and R. C. O'Handley, "A Modulated Ellipsometer for Studying Thin Film Optical Properties and Surface Dynamics," Surf. Sci., 37, 548–558, (1973); J. I. Treu, A. B. Callender, and S. E. Schnatterly, "Use of a Stable Polarization Modulator in a Scanning Spectrophotometer and Ellipsometer," Rev. Sci. Instrum., 44, 793–797, (1973).

R. H. Muller, "Definitions and Conventions in Ellipsometry," Surf. Sci., 16, 14–33, (1969).

1968 (1)

1965 (1)

1956 (1)

Z. Sekera, "Recent Developments in the Study of the Polarization of Sky Light," Adv. Geophys., 3, 43–104, (1956); "Light Scattering in the Atmosphere and the Polarization of Sky Light," J. Opt. Soc., Am., 47, 484–490, (1957); "Polarization of Skylight," Encyclopedia of Phys., 48, Springer, Berlin, 1957, pp. 288–328.

1942 (1)

F. Perrin, "Polarization of Light Scattered by Isotropic Opalescent Media," J. Chem. Phys., 10, 415–427, (1942).

Aspnes, D. E.

D. E. Aspnes and P. S. Hauge, "Rotating-Compensator/Analyzer Fixed-Analyzer Ellipsometer: Analysis and Comparison to Other Automatic Ellipsometers," J. Opt. Soc. Am., 66, 949–954, (1976).

D. E. Aspnes, "Photoelectric Ellipsometer for Measuring Partially Polarized Light," J. Opt. Soc. Am., 65, 1274–1278, (1975); "A Photometric Ellipsometer for Measuring Flux in a General State of Polarization," Surf. Sci., 56, 161–169, (1976).

D. E. Aspnes, "Fourier Transform Detection System for Rotating- Analyzer Ellipsometers," Opt. Commun., 8, 222–225, (1973); D. E. Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Appl. Opt., 14, 220–228, (1975).

D. E. Aspnes, "Spectroscopic Ellipsometry of Solids," in Optical Properties of Solids: New Developments, edited by B. O. Seraphin, (North-Holland, Amsterdam, 1976).

Azzam, R. M. A.

Bashara, N. M.

Beardsley, G. F.

Berry, H. G.

Bottiger, J.

R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).

Calculi, Muller

The Jones and Muller Calculi are discussed at length in Refs. 4, 7, and 8.

Champion, J. V.

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

Clarke, D.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, (Pergamon, New York, 1971).

Coffeen, D. L.

D. L. Coffeen and J. E. Hansen, "Polarization Studies of Planetary Atmospheres," see Ref. 11, 518–581.

Dill, F. H.

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun., 14, 431–437, (1975).

P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev., 17, 472–489, (1973).

Downer, D.

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

Fry, E.

R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).

Gabrielse, G.

Gagne, G.

Gate, L. F.

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

Grainger, J. F.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, (Pergamon, New York, 1971).

Hansen, J. E.

D. L. Coffeen and J. E. Hansen, "Polarization Studies of Planetary Atmospheres," see Ref. 11, 518–581.

Hauge, P. S.

P. S. Hauge, "Techniques of Measurement of the Polarization- Altering Properties of Linear Optical Systems," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 2–11, (1977).

D. E. Aspnes and P. S. Hauge, "Rotating-Compensator/Analyzer Fixed-Analyzer Ellipsometer: Analysis and Comparison to Other Automatic Ellipsometers," J. Opt. Soc. Am., 66, 949–954, (1976).

P. S. Hauge, "Survey of Methods for the Complete Determination of a State of Polarization," Proc. Soc. Photo-Opt. Instrum. Eng., 88, 3–10, (1976).

P. S. Hauge, "Generalized Rotating-Compensator Ellipsometry," Surf. Sci., 56, 148–160, (1976).

P. S. Hauge, "Automated Mueller Matrix Ellipsometry," Opt. Commun., 17, 74–76, (1976).

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun., 14, 431–437, (1975).

P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev., 17, 472–489, (1973).

Hodgdon, E. B.

Holland, A. C.

Huffman, D. R.

A. J. Hunt and D. R. Huffman, "A New Polarization-Modulation Light Scattering Instrument," Rev. Sci. Instrum., 44, 1753–1762, (1973); R. J. Perry, A. J. Hunt, and D. R. Huffman, "Experimental determinations of Mueller scattering matrices for nonspherical particles," Appl. Opt., 17, 2700–2710, (1978).

Hunt, A. J.

A. J. Hunt and D. R. Huffman, "A New Polarization-Modulation Light Scattering Instrument," Rev. Sci. Instrum., 44, 1753–1762, (1973); R. J. Perry, A. J. Hunt, and D. R. Huffman, "Experimental determinations of Mueller scattering matrices for nonspherical particles," Appl. Opt., 17, 2700–2710, (1978).

Jasperson, S. N.

S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum., 40, 761–767, (1969); S. N. Jasperson, D. K. Burge, and R. C. O'Handley, "A Modulated Ellipsometer for Studying Thin Film Optical Properties and Surface Dynamics," Surf. Sci., 37, 548–558, (1973); J. I. Treu, A. B. Callender, and S. E. Schnatterly, "Use of a Stable Polarization Modulator in a Scanning Spectrophotometer and Ellipsometer," Rev. Sci. Instrum., 44, 793–797, (1973).

Kruger, J.

E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Stand. Misc. Publ. 256, U.S. G.P.O., Washington, 1964.

Livingston, E. E.

Meeten, G. H.

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

Muller, R. H.

R. H. Muller, "Definitions and Conventions in Ellipsometry," Surf. Sci., 16, 14–33, (1969).

Orrall, F. Q.

F. Q. Orrall, "A Complete Stokes Vector Polarimeter," Solar Magnetic Fields, edited by R. Howard, (Reidel, Dordrecht, 1971), pp. 30–36.

Passaglia, E.

E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Stand. Misc. Publ. 256, U.S. G.P.O., Washington, 1964.

Perrin, F.

F. Perrin, "Polarization of Light Scattered by Isotropic Opalescent Media," J. Chem. Phys., 10, 415–427, (1942).

Schnatterly, S. E.

S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum., 40, 761–767, (1969); S. N. Jasperson, D. K. Burge, and R. C. O'Handley, "A Modulated Ellipsometer for Studying Thin Film Optical Properties and Surface Dynamics," Surf. Sci., 37, 548–558, (1973); J. I. Treu, A. B. Callender, and S. E. Schnatterly, "Use of a Stable Polarization Modulator in a Scanning Spectrophotometer and Ellipsometer," Rev. Sci. Instrum., 44, 793–797, (1973).

Sekera, Z.

Z. Sekera, "Recent Developments in the Study of the Polarization of Sky Light," Adv. Geophys., 3, 43–104, (1956); "Light Scattering in the Atmosphere and the Polarization of Sky Light," J. Opt. Soc., Am., 47, 484–490, (1957); "Polarization of Skylight," Encyclopedia of Phys., 48, Springer, Berlin, 1957, pp. 288–328.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light, (Harvard Univ., Cambridge, Mass., 1962).

Stromberg, R. R.

E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Stand. Misc. Publ. 256, U.S. G.P.O., Washington, 1964.

Thompson, R.

R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).

Adv. Geophys. (1)

Z. Sekera, "Recent Developments in the Study of the Polarization of Sky Light," Adv. Geophys., 3, 43–104, (1956); "Light Scattering in the Atmosphere and the Polarization of Sky Light," J. Opt. Soc., Am., 47, 484–490, (1957); "Polarization of Skylight," Encyclopedia of Phys., 48, Springer, Berlin, 1957, pp. 288–328.

Appl. Opt. (3)

IBM J. Res. Dev. (1)

P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev., 17, 472–489, (1973).

J. Chem. Phys. (1)

F. Perrin, "Polarization of Light Scattered by Isotropic Opalescent Media," J. Chem. Phys., 10, 415–427, (1942).

J. Opt. Soc. Am. (4)

J. Phys. E: Sci. Instrum. (1)

J. V. Champion, D. Downer, G. H. Meeten, and L. F. Gate, "Measurement of Magnetically Induced Linear Optical Birefringence and Dichroism in Colloidal Dispersions," J. Phys. E: Sci. Instrum., 10, 1137–1141, (1977).

Opt. Commun. (3)

P. S. Hauge, "Automated Mueller Matrix Ellipsometry," Opt. Commun., 17, 74–76, (1976).

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun., 14, 431–437, (1975).

D. E. Aspnes, "Fourier Transform Detection System for Rotating- Analyzer Ellipsometers," Opt. Commun., 8, 222–225, (1973); D. E. Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Appl. Opt., 14, 220–228, (1975).

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. (1)

P. S. Hauge, "Survey of Methods for the Complete Determination of a State of Polarization," Proc. Soc. Photo-Opt. Instrum. Eng., 88, 3–10, (1976).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

P. S. Hauge, "Techniques of Measurement of the Polarization- Altering Properties of Linear Optical Systems," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 2–11, (1977).

R. Thompson, E. Fry, and J. Bottiger, "Polarization-Dependent Light-Scattering Measurements Using an Electro-Optical Modulator- Based Instrument," Proc. Soc. Photo-Opt. Instrum. Eng., 112, 152–157, (1977).

Rev. Sci. Instrum. (2)

S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum., 40, 761–767, (1969); S. N. Jasperson, D. K. Burge, and R. C. O'Handley, "A Modulated Ellipsometer for Studying Thin Film Optical Properties and Surface Dynamics," Surf. Sci., 37, 548–558, (1973); J. I. Treu, A. B. Callender, and S. E. Schnatterly, "Use of a Stable Polarization Modulator in a Scanning Spectrophotometer and Ellipsometer," Rev. Sci. Instrum., 44, 793–797, (1973).

A. J. Hunt and D. R. Huffman, "A New Polarization-Modulation Light Scattering Instrument," Rev. Sci. Instrum., 44, 1753–1762, (1973); R. J. Perry, A. J. Hunt, and D. R. Huffman, "Experimental determinations of Mueller scattering matrices for nonspherical particles," Appl. Opt., 17, 2700–2710, (1978).

Surf. Sci. (2)

R. H. Muller, "Definitions and Conventions in Ellipsometry," Surf. Sci., 16, 14–33, (1969).

P. S. Hauge, "Generalized Rotating-Compensator Ellipsometry," Surf. Sci., 56, 148–160, (1976).

Other (19)

Alternatively, the compensator can be driven by a stepping motor, and the reference and data pulses taken from the driving circuitry.

F. Q. Orrall, "A Complete Stokes Vector Polarimeter," Solar Magnetic Fields, edited by R. Howard, (Reidel, Dordrecht, 1971), pp. 30–36.

The compensator Jones and Mueller matrices in Eqs. (1), (3), (5), etc., have been normalized so that the first element is unity. The effect of this normalization is accounted for in the determination of the detector sensitivity g.

In Ref. 33, the defect parameters were defined as q, r, and s. In this paper, p, r, and s are used, where p = q - 1.

If absolute values of Stokes parameters are desired, at least one absolute intensity measurement must be made at each wavelength used. Ip and g may then be determined separately. This is unnecessary, however, both for MME and for determination of polarization state, which is completely determined by the normalized Stokes parameters sk = Sk/S0, (k = 1,…, 3).

H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).

D. L. Coffeen and J. E. Hansen, "Polarization Studies of Planetary Atmospheres," see Ref. 11, 518–581.

Planets, Stars and Nebulae Studies with Photopolarimetry, edited by T. Gehrels, (University of Arizona, 1974).

Optical Polarimetry: Instrumentation and Applications, edited by R. M. A. Azzam and D. L. Coffeen, Proc. Soc. Photo-Opt. Instrum. Eng., 112, (1977).

E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Stand. Misc. Publ. 256, U.S. G.P.O., Washington, 1964.

Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall, Surf. Sci., 16, (1969).

Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam, Surf. Sci., 56, (1976).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1976).

D. E. Aspnes, "Spectroscopic Ellipsometry of Solids," in Optical Properties of Solids: New Developments, edited by B. O. Seraphin, (North-Holland, Amsterdam, 1976).

The Jones and Muller Calculi are discussed at length in Refs. 4, 7, and 8.

W. A. Shurcliff, Polarized Light, (Harvard Univ., Cambridge, Mass., 1962).

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, (Pergamon, New York, 1971).

Other rotation ratios are possible, including rotation in opposing directions. However, the ratio of 5:1 gives separation of all 12 frequencies at their lowest possible values.

To illustrate, reasonable values of l = 8, m = 360, and n = 8 give 23 C40 intensity readings which are condensed into 20 and then 16 matrix elements.

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