Abstract

When a light beam whose polarization and intensity are weakly modulated at a frequency ω<sub>m</sub> passes through a periodic analyzer of frequency ω<sub>a</sub>(<ω<sub>m</sub>) and the transmitted flux is linearly detected, the resulting total signal S<sub>t</sub> consists of two components: (i) a periodic baseband signal S<sub>bb</sub> with harmonics of frequencies nω<sub>a</sub> (n = 0,1,2,…) and (ii) an amplitude-modulated-carrier signal δS<sub>mc</sub> with center (carrier) frequency ω<sub>m</sub> and sideband frequencies at ω<sub>m</sub> ± nω<sub>a</sub>(n = 1,2,…). In this paper we show that the average polarization of the beam is determined by a limited spectral analysis of S<sub>bb</sub>, whereas the polarization and intensity modulation are determined by a limited spectral analysis of δS<sub>mc</sub>, or the associated envelope signal δS<sub>e</sub>, where δS<sub>mc</sub> = δS<sub>e</sub>cosω<sub>m</sub>t. The theory of this frequency-mixing detection (FMD) of polarization modulation is developed for an arbitrary periodic analyzer. The specific case of a rotating analyzer is considered as an example. Applications of FMD include the retrieval of information impressed on light beams as polarization modulation in optical communication systems, and the automation of modulated ellipsometry, AIDER (angle-of- incidence-derivative ellipsometry and reflectometry), and modulated generalized ellipsometry.

© 1976 Optical Society of America

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  1. To prevent overlapping between the spectra of Sbb and δSmc, we select the frequency of the periodic analyzer ωa to be much smaller than the beam-modulation frequency ωm (e. g., ωm > 10ωa), and restrict ωma not to equal the ratio of two integers.
  2. Alternatively, we may measure the amplitudes of the cosine and sine components of one nonzero harmonic of Sbb. This is the case of the example considered in Sec. III.
  3. Dependent on the type of periodic analyzer that we choose, Eqs. (16) may or may not have an explicit, or a unique, solution for ¯ψ and ¯Δ.
  4. This requires, of course, that the three equations be linearly independent. This is satisfied in general, unless the periodic analyzer, the chosen harmonics (p, q), and/or the quiescent polarization (¯ψ,¯Δ) happen to be such that two (or all three) equations become linearly dependent.
  5. Such a constant can be absorbed in the multipler c that appears in Eq. (7).
  6. This is in agreement with results to be found in Refs. 8–10.
  7. See, for example, W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).
  8. P. S. Hauge and F. H. Dill, "Design and Operation of ETA, an Automated Ellipsometer," IBM J. Res. Devel. 17, 472–489 (1973).
  9. D. E. Aspnes, "Effects of Component Optical Activity in Data Reduction and Calibration of Rotating-Analyzer Ellipsometers," J. Opt. Soc. Am. 65, 812–819 (1975).
  10. R. M. A. Azzam and N. M. Bashara, "Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers," 64, 1459–1469 (1974).
  11. R. M. A. Azzam, "Oscillating-Analyzer Ellipsometer," Rev. Sci. Instrum. 47, (1976) (in press).
  12. R. W. Stobie, B. Rao, and M. J. Dignam, "Analysis of a Novel Ellipsometer Technique for Infrared Spectroscopy," J. Opt. Soc. Am. 65, 25–28 (1975).
  13. P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun. 14, 431–435 (1975).
  14. D. E. Aspnes, "Photometric Ellipsometer for Measuring Partially Polarized Light," J. Opt. Soc. Am. 65, 1274–1278, 1975.
  15. R. M. A. Azzam, "Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder," Optik. 45, (1976) (in press).
  16. A. B. Buckman and N. M. Bashara "Ellipsometry for Modulated Reflection Studies," J. Opt. Soc. Am. 58, 700–701 (1968).
  17. R. M. A. Azzam, "Modulated Generalized Ellipsometry," J. Opt. Soc. Am. 66, 520–524 (1976).
  18. R. M. A. Azzam, "AIDER: Angle-of-Incidence-Derivative Ellipsometry and Reflectometry," Opt. Commun. 16, 153–156 (1976).

1976 (4)

R. M. A. Azzam, "Oscillating-Analyzer Ellipsometer," Rev. Sci. Instrum. 47, (1976) (in press).

R. M. A. Azzam, "Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder," Optik. 45, (1976) (in press).

R. M. A. Azzam, "AIDER: Angle-of-Incidence-Derivative Ellipsometry and Reflectometry," Opt. Commun. 16, 153–156 (1976).

R. M. A. Azzam, "Modulated Generalized Ellipsometry," J. Opt. Soc. Am. 66, 520–524 (1976).

1975 (4)

R. W. Stobie, B. Rao, and M. J. Dignam, "Analysis of a Novel Ellipsometer Technique for Infrared Spectroscopy," J. Opt. Soc. Am. 65, 25–28 (1975).

D. E. Aspnes, "Photometric Ellipsometer for Measuring Partially Polarized Light," J. Opt. Soc. Am. 65, 1274–1278, 1975.

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun. 14, 431–435 (1975).

D. E. Aspnes, "Effects of Component Optical Activity in Data Reduction and Calibration of Rotating-Analyzer Ellipsometers," J. Opt. Soc. Am. 65, 812–819 (1975).

1973 (1)

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

1968 (1)

Aspnes, D. E.

D. E. Aspnes, "Photometric Ellipsometer for Measuring Partially Polarized Light," J. Opt. Soc. Am. 65, 1274–1278, 1975.

D. E. Aspnes, "Effects of Component Optical Activity in Data Reduction and Calibration of Rotating-Analyzer Ellipsometers," J. Opt. Soc. Am. 65, 812–819 (1975).

Azzam, R. M. A.

R. M. A. Azzam, "Modulated Generalized Ellipsometry," J. Opt. Soc. Am. 66, 520–524 (1976).

R. M. A. Azzam, "Oscillating-Analyzer Ellipsometer," Rev. Sci. Instrum. 47, (1976) (in press).

R. M. A. Azzam, "Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder," Optik. 45, (1976) (in press).

R. M. A. Azzam, "AIDER: Angle-of-Incidence-Derivative Ellipsometry and Reflectometry," Opt. Commun. 16, 153–156 (1976).

R. M. A. Azzam and N. M. Bashara, "Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers," 64, 1459–1469 (1974).

Bashara, N. M.

A. B. Buckman and N. M. Bashara "Ellipsometry for Modulated Reflection Studies," J. Opt. Soc. Am. 58, 700–701 (1968).

R. M. A. Azzam and N. M. Bashara, "Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers," 64, 1459–1469 (1974).

Buckman, A. B.

Dignam, M. J.

Dill, F. H.

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun. 14, 431–435 (1975).

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

Hauge, P. S.

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun. 14, 431–435 (1975).

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

Pratt, W. K.

See, for example, W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

Rao, B.

Stobie, R. W.

IBM J. Res. Devel. (1)

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

J. Opt. Soc. Am. (5)

Opt. Commun. (2)

R. M. A. Azzam, "AIDER: Angle-of-Incidence-Derivative Ellipsometry and Reflectometry," Opt. Commun. 16, 153–156 (1976).

P. S. Hauge and F. H. Dill, "A Rotating-Compensator Fourier Ellipsometer," Opt. Commun. 14, 431–435 (1975).

Optik. (1)

R. M. A. Azzam, "Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder," Optik. 45, (1976) (in press).

Rev. Sci. Instrum. (1)

R. M. A. Azzam, "Oscillating-Analyzer Ellipsometer," Rev. Sci. Instrum. 47, (1976) (in press).

Other (8)

R. M. A. Azzam and N. M. Bashara, "Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers," 64, 1459–1469 (1974).

To prevent overlapping between the spectra of Sbb and δSmc, we select the frequency of the periodic analyzer ωa to be much smaller than the beam-modulation frequency ωm (e. g., ωm > 10ωa), and restrict ωma not to equal the ratio of two integers.

Alternatively, we may measure the amplitudes of the cosine and sine components of one nonzero harmonic of Sbb. This is the case of the example considered in Sec. III.

Dependent on the type of periodic analyzer that we choose, Eqs. (16) may or may not have an explicit, or a unique, solution for ¯ψ and ¯Δ.

This requires, of course, that the three equations be linearly independent. This is satisfied in general, unless the periodic analyzer, the chosen harmonics (p, q), and/or the quiescent polarization (¯ψ,¯Δ) happen to be such that two (or all three) equations become linearly dependent.

Such a constant can be absorbed in the multipler c that appears in Eq. (7).

This is in agreement with results to be found in Refs. 8–10.

See, for example, W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

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