Abstract

When a light beam whose polarization and intensity are weakly modulated at a frequency ωm passes through a periodic analyzer of frequency ωa(≪ωm) and the transmitted flux is linearly detected, the resulting total signal St consists of two components: (i) a periodic baseband signal Sbb with harmonics of frequencies a(n = 0,1,2,…) and (ii) an amplitude-modulated-carrier signal δSmc with center (carrier) frequency ωm and sideband frequencies at ωm ± a(n = 1,2,…). In this paper we show that the average polarization of the beam is determined by a limited spectral analysis of Sbb, whereas the polarization and intensity modulation are determined by a limited spectral analysis of δSmc, or the associated envelope signal δSe, where δSmc = δSe cosωmt. 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

Full Article  |  PDF Article

Corrections

R. M. A. Azzam, "Errata: Frequency-mixing detection (FMD) of polarization-modulated light," J. Opt. Soc. Am. 67, 420-420 (1977)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-67-3-420

References

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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 ωm/ωa 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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  13. P. S. Hauge and F. H. Dill, “A Rotating-Compensator Fourier Ellipsometer,” Opt. Commun. 14, 431–435 (1975).
    [CrossRef]
  14. D. E. Aspnes, “Photometric Ellipsometer for Measuring Partially Polarized Light,” J. Opt. Soc. Am. 65, 1274–1278, 1975.
    [CrossRef]
  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).
    [CrossRef]
  17. R. M. A. Azzam, “Modulated Generalized Ellipsometry,” J. Opt. Soc. Am. 66, 520–524 (1976).
    [CrossRef]
  18. R. M. A. Azzam, “AIDER: Angle-of-Incidence-Derivative Ellipsometry and Reflectometry,” Opt. Commun. 16, 153–156 (1976).
    [CrossRef]

1976 (2)

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

R. M. A. Azzam, “Modulated Generalized Ellipsometry,” J. Opt. Soc. Am. 66, 520–524 (1976).
[CrossRef]

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).
[CrossRef]

D. E. Aspnes, “Photometric Ellipsometer for Measuring Partially Polarized Light,” J. Opt. Soc. Am. 65, 1274–1278, 1975.
[CrossRef]

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).

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

1974 (1)

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

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).
[CrossRef]

1968 (1)

Aspnes, D. E.

D. E. Aspnes, “Photometric Ellipsometer for Measuring Partially Polarized Light,” J. Opt. Soc. Am. 65, 1274–1278, 1975.
[CrossRef]

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).
[CrossRef]

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

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

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, “Oscillating-Analyzer Ellipsometer,” Rev. Sci. Instrum.47,(1976) (in press).
[CrossRef]

Bashara, N. M.

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

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

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).
[CrossRef]

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

Hauge, P. S.

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

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

Pratt, W. K.

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

Rao, B.

Stobie, R. W.

Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers (1)

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

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).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Commun. (2)

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

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

Other (9)

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, “Oscillating-Analyzer Ellipsometer,” Rev. Sci. Instrum.47,(1976) (in press).
[CrossRef]

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 ωm/ωa 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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Figures (3)

FIG. 1
FIG. 1

A light beam of polarization (ψ, Δ) and intensity (I) that are weakly modulated at a frequency ωm is transmitted through an analyzer A one or more of whose parameters αj is periodically swept at a frequency ωa. The interaction between the modulated beam and the periodic analyzer results in frequency mixing that is borne in the intensity variations of the light leaving the analyzer. The latter intensity variations are detected by a linear polarization-independent photodetector D, giving an electrical signal D(or St). By spectral analysis of St (Fig. 3), the average polarization as well as the polarization and intensity modulation of the beam can be determined. xyz represents a reference Cartesian coordinate system with the z axis along the direction of propagation of the light beam.

FIG. 2
FIG. 2

Frequency spectrum of the baseband (Sbb) and modulated-carrier (δSmc) components of the total detected signal St. ωm is the light-modulation frequency and ωa is the frequency of the periodic analyzer.

FIG. 3
FIG. 3

A block diagram of the electronic signal-processing units needed for frequency-mixing detection of polarization-modulated light. The output St of the photodetector (Fig. 1) is divided by the signal divider SD into two equal signals. One signal passes through channel I that consists of the low-pass filter LPF (of cutoff frequency ωc ≥ 10ωa, where ωa is the periodic-analyzer frequency) and the spectral analyzer SA1. The output of channel I determines the average polarization of the beam ψ ¯ , Δ ¯ . The second signal passes through channel II that consists of the band-pass filter BPF (of center frequency ωc = ωm, the light-modulation frequency, and bandwidth Δω ≥ 20ωa), the amplitude-modulation (or envelope) detector AMD, and the signal analyzer SA2. The output of channel II determines the intensity- ( δ I ^ / Ī ) and polarization-modulation ( δ ψ ^ , δ Δ ^ ) parameters of the beam. Sbb, δSmc and δSe are the baseband, modulated-carrier, and envelope signals, respectively.

Equations (43)

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I = E x E x * + E y E y * ,
χ = E y / E x = tan ψ e j Δ ,
I = Ī + δ I ,
ψ = ψ ¯ + δ ψ ,             Δ = Δ ¯ + δ Δ ,
δ I / Ī 1 ,             δ ψ , δ Δ < ~ π / 12.
δ X = δ X ^ cos ω m t ,             X = I , ψ ,             and Δ
I t = I f ( ψ , Δ , α j ) ,
D = c I t = c I f ( ψ , Δ , α j ) ,
δ D = ( D / I ) δ I + ( D / ψ ) δ ψ + ( D / Δ ) δ Δ .
δ D = c I [ f ( δ I / I ) + f ψ δ ψ + f Δ δ Δ ] ,
f ψ = f / ψ ,             f Δ = f / Δ .
f = f 0 + f 1 sin ( ω a l + θ 1 ) + f 2 sin ( 2 ω a l + θ 2 ) + ,
f ψ = f ψ 0 + f ψ 1 sin ( ω a t + θ ψ 1 ) + f ψ 2 sin ( 2 ω a t + θ ψ 2 ) + ,
f Δ = f Δ 0 + f Δ 1 sin ( ω a t + θ Δ 1 ) + f Δ 2 sin ( 2 ω a t + θ Δ 2 ) + .
S b b = D ω a on ω m off = c Ī [ f 0 + f 1 sin ( ω a t + θ 1 ) + f 2 sin ( 2 ω a t + θ 2 ) + ] ,
δ S m c = δ D ω a on ω m on = δ S e cos ω m t ,
δ S e = c Ī { [ f 0 + f 1 sin ( ω a t + θ 1 ) + f 2 sin ( 2 ω a t + θ 2 ) + ] ( δ I ^ / Ī ) + [ f ψ 0 + f ψ 1 sin ( ω a t + θ ψ 1 ) + f ψ 2 sin ( 2 ω a t + θ ψ 2 ) + ] δ ψ ^ + [ f Δ 0 + f Δ 1 sin ( ω a t + θ 1 ) + f Δ 2 sin ( 2 ω a t + θ Δ 2 ) + ] δ Δ ^ } .
S t = S b b + δ S m c .
η ω = s ω / s 0 .
η p ω a = f p / f 0 ,             η q ω a = f q / f 0 .
θ ψ n = θ Δ n = θ n             for all n ,
δ S m c = c Ī [ f 0 ( δ I ^ / Ī ) + f ψ 0 δ ψ ^ + f Δ 0 δ Δ ^ ] cos ω m t + c Ī { n = 1 [ f n ( δ I ^ / Ī ) + f ψ n δ ψ ^ + f Δ n δ Δ ^ ] sin ( n ω a t + θ n ) } × cos ω m t .
s ω m m c = c Ī [ f 0 ( δ I ^ / Ī ) + f ψ 0 δ ψ ^ + f Δ 0 δ Δ ^ ] , s ω m ± n ω a m c = 1 2 c Ī [ f n ( δ I ^ / Ī ) + f ψ n δ ψ ^ + f Δ n δ Δ ^ ] .
η ω m = ( δ I ^ / Ī ) + ( f ψ 0 / f 0 ) δ ψ ^ + ( f Δ 0 / f 0 ) δ Δ ^ , 2 η ω m ± p ω a = ( f p / f 0 ) ( δ I ^ / Ī ) + ( f ψ p / f 0 ) δ ψ ^ + ( f Δ p / f 0 ) δ Δ ^ , 2 η ω m ± q ω a = ( f q / f 0 ) ( δ I ^ / Ī ) + ( f ψ q / f 0 ) δ ψ ^ + ( f Δ q / f 0 ) δ Δ ^ .
s 0 e = c Ī [ f 0 ( δ I ^ / Ī ) + f ψ 0 δ ψ ^ + f Δ 0 δ Δ ^ ] , s p ω a e = c Ī [ f p ( δ I ^ / Ī ) sin ( p ω a t + θ p ) + f ψ p sin ( p ω a t + θ ψ a ) δ ψ ^ + f Δ p sin ( p ω a t + θ Δ p ) δ Δ ^ ] ,
η 0 e = ( δ I ^ / Ī ) + ( f ψ 0 / f 0 ) δ ψ ^ + ( f Δ 0 / f 0 ) δ Δ ^ , η p ω a e c = ( f p sin θ p / f 0 ) ( δ I ^ / Ī ) + ( f ψ p sin θ ψ p / f 0 ) δ ψ ^ + ( f Δ p sin θ Δ p / f 0 ) δ Δ ^ , η p ω a e s = ( f p cos θ p / f 0 ) ( δ I ^ / Ī ) + ( f ψ p cos θ ψ p / f 0 ) δ ψ ^ + ( f Δ p cos θ Δ p / f 0 ) δ Δ ^ .
f = ( 1 + cos 2 A ) + tan 2 ψ ( 1 - cos 2 A ) + 2 tan ψ cos Δ sin 2 A ,
f ψ = 2 tan ψ sec 2 ψ ( 1 - cos 2 A ) + 2 sec 2 ψ cos Δ sin 2 A , f Δ = - 2 tan ψ sin Δ sin 2 A .
A = 1 2 ω a t ,
f = sec 2 ψ + ( 1 - tan 2 ψ ) cos ω a t + 2 tan ψ cos Δ sin ω a t ,
f ψ = 2 tan ψ sec 2 ψ - 2 tan ψ sec 2 ψ cos ω a t + 2 sec 2 ψ cos Δ sin ω a t ,
f Δ = - 2 tan ψ sin Δ sin ω a t .
η ω a c = f 1 sin θ 1 / f 0 ,             η ω a s = f 1 cos θ 1 / f 0 .
f 0 = sec 2 ψ ¯ ,             f 1 sin θ 1 = ( 1 - tan 2 ψ ¯ ) ,             f 1 cos θ 1 = 2 tan ψ ¯ cos Δ ¯ ,
η ω a c = ( 1 - tan 2 ψ ¯ ) / sec 2 ψ ¯ = cos 2 ψ ¯ ,
η ω a s = 2 tan ψ ¯ cos Δ ¯ / sec 2 ψ ¯ = sin 2 ψ ¯ cos Δ ¯ .
f ψ 0 = 2 tan ψ ¯ sec 2 ψ ¯ ,             f ψ 1 sin θ ψ 1 = - 2 tan ψ ¯ sec 2 ψ ¯ , f ψ 1 cos θ ψ 1 = 2 sec 2 ψ ¯ cos Δ ¯ , f Δ 0 = 0 ,             f Δ 1 sin θ Δ 1 = 0 ,             f Δ 1 cos θ Δ 1 = - 2 tan ψ ¯ sin Δ ¯ ,
η 0 e = ( δ I ^ / Ī ) + ( 2 tan ψ ¯ ) δ ψ ^ ,
η ω a e c = ( cos 2 ψ ¯ ) ( δ I ^ / Ī ) + ( - 2 tan ψ ¯ ) δ ψ ^ ,
η ω a e s = ( sin 2 ψ ¯ cos Δ ¯ ) ( δ I ^ / Ī ) + ( 2 cos Δ ¯ ) δ ψ ^ + ( - 2 tan ψ ¯ sin Δ ¯ ) δ Δ ^ .
δ I ^ Ī = η 0 e + η ω a e c 1 + cos 2 ψ ¯ ,
δ ψ ^ = η 0 e - ( δ I ^ / Ī ) 2 tan 2 ψ ¯ ,
δ Δ ^ = - η ω a e s + ( sin 2 ψ ¯ cos Δ ¯ ) ( δ I ^ / Ī ) + ( 2 cos Δ ¯ ) δ ψ ^ 2 tan ψ ¯ sin Δ ¯ ,