Abstract

The first three columns of the instrument matrix A of the four-detector photopolarimeter (FDP) are determined by Fourier analysis of the output current vector I(P) as a function of the azimuth angle P of the incident linearly polarized light. Therefore 12 of the 16 elements of A are measured free of the imperfections of the (absent) quarter-wave retarder (QWR). The effect of angular beam deviation by the polarizer is compensated for by taking the average, (1/2) [I(P) + I(P + 180°)], of the FDP output at 180°-apart, optically equivalent, angular positions of the polarizer. The remaining fourth column of A is determined by the FDP’s response to the right- and left-handed circular polarization states. Because these states are impossible to generate with an imperfect QWR, a novel procedure is developed. In particular, the response of the FDP to the unattainable right- or left-handed circular polarization state is found by taking the average of the responses of the FDP to an elliptical near-circular state and that state rotated in azimuth by 90°. This calibration scheme is applied to measure A of our prototype FDP of four Si detectors at λ = 632.8 nm. A is determined, in external and internal reference frames, free of imperfections in the polarizing optical elements. The FDP, with its uncontaminated A matrix, is used subsequently to evaluate the imperfections of the QWR with the help of an appropriate model.

© 1989 Optical Society of America

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References

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  1. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985); U.S. Patent4,681,450 (July21, 1987).
    [CrossRef] [PubMed]
  2. R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
    [CrossRef]
  3. R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
    [CrossRef]
  4. See, for example, M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 554.
  5. The FDP has also been calibrated and operated at several other wavelengths in the visible spectrum by using a filtered continuum light source. See A. G. Lopez, R. M. A. Azzam, “Four-detector photopolarimeter (FDP): precision analysis and low-light-level measurements,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 181.
  6. R. M. A. Azzam, “Stationary property of normal-incidence reflection from isotropic surfaces,”J. Opt. Soc. Am. 72, 1187–1189 (1982).
    [CrossRef]
  7. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  8. The effect of angular beam derivation of the QWR and by the polarizer is also compensated for by taking the average of four readings of the output current vector I of the FDP corresponding to four optically equivalent settings of the two elements. These settings are obtained by rotating each element by 180° from a given position, so that I= (1/4)[I(P, C) + I(P+ 180°, C) + I(P, C+ 180°) + I(P+ 180°, C+ 180°)]. This average is a generalization of that indicated by Eq. (9) for the case in which the polarizer alone is used.
  9. R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,”J. Opt. Soc. Am. 61, 1380–1391 (1971).
    [CrossRef]
  10. P. S. Hauge, “Mueller-matrix ellipsometry with imperfect compensators,”J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  11. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982); “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
    [CrossRef]

1988 (2)

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

1985 (1)

1982 (2)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982); “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[CrossRef]

R. M. A. Azzam, “Stationary property of normal-incidence reflection from isotropic surfaces,”J. Opt. Soc. Am. 72, 1187–1189 (1982).
[CrossRef]

1978 (1)

1971 (1)

Azzam, R. M. A.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985); U.S. Patent4,681,450 (July21, 1987).
[CrossRef] [PubMed]

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982); “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[CrossRef]

R. M. A. Azzam, “Stationary property of normal-incidence reflection from isotropic surfaces,”J. Opt. Soc. Am. 72, 1187–1189 (1982).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,”J. Opt. Soc. Am. 61, 1380–1391 (1971).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

The FDP has also been calibrated and operated at several other wavelengths in the visible spectrum by using a filtered continuum light source. See A. G. Lopez, R. M. A. Azzam, “Four-detector photopolarimeter (FDP): precision analysis and low-light-level measurements,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 181.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,”J. Opt. Soc. Am. 61, 1380–1391 (1971).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 554.

Elminyawi, I. M.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

El-Saba, A. M.

Grosz, F. G.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

Hauge, P. S.

Lopez, A. G.

The FDP has also been calibrated and operated at several other wavelengths in the visible spectrum by using a filtered continuum light source. See A. G. Lopez, R. M. A. Azzam, “Four-detector photopolarimeter (FDP): precision analysis and low-light-level measurements,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 181.

Masetti, E.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 554.

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982); “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

Other (4)

See, for example, M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 554.

The FDP has also been calibrated and operated at several other wavelengths in the visible spectrum by using a filtered continuum light source. See A. G. Lopez, R. M. A. Azzam, “Four-detector photopolarimeter (FDP): precision analysis and low-light-level measurements,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 181.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

The effect of angular beam derivation of the QWR and by the polarizer is also compensated for by taking the average of four readings of the output current vector I of the FDP corresponding to four optically equivalent settings of the two elements. These settings are obtained by rotating each element by 180° from a given position, so that I= (1/4)[I(P, C) + I(P+ 180°, C) + I(P, C+ 180°) + I(P+ 180°, C+ 180°)]. This average is a generalization of that indicated by Eq. (9) for the case in which the polarizer alone is used.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the FDP and the PSG used for its calibration. L indicates the source of a collimated monochromatic unpolarized or circular polarized light beam. P and QWR indicate a linear polarizer and a quarter-wave retarder, respectively. BS indicates a (slightly tilted) beam splitter, and Dr indicates a reference detector.

Fig. 2
Fig. 2

Calibration of the FDP with LP light of variable azimuth P. Fourier analysis of the output current vector I(P) as a function of P determines the first three columns of the instrument matrix A.

Fig. 3
Fig. 3

The normalized response of each of the four detectors of the FDP as a function of the linear polarizer azimuth P when the calibration scheme of Fig. 2 is used: (a) i0,(b) i1, (c) i2, and (d) i3. The diamonds represent the measured response recorded at 10°-apart, equispaced, angular positions of the polarizer over the full range 0 ≤ P ≤ 180°. The continuous curves are obtained by a least-squares fit of the experimental data points with a function of the form in = an0 + an1 cos 2P + an2 sin 2P, where n = 0, 1, 2, 3 and an0, an1, and an2 specify the first three columns of the instrument matrix A.

Fig. 4
Fig. 4

An ENCS is specified by its major-axis azimuth θ (unrestricted) and the small deviation β of the ellipticity angle from 45° (β = 45° − ). The right-handed polarization is indicated by the clockwise arrow, for a beam that travels normal and out of the page toward the reader.

Fig. 5
Fig. 5

(a) An ENCS of azimuth θ, ENCS(θ), and a state produced from it by a 90° rotation, ENCS(θ + 90°). (b) Cross section of the Poincaré sphere by a plane that contains the polar axis LR and passes through the points that represent the pair of ENCS’s in (a). The small angle β has the meaning given in Fig. 4.

Fig. 6
Fig. 6

External and internal reference frames, te and p0s0, respectively, with respect to which the Stokes parameters of light and the instrument matrix A of the FDP are defined. t and e represent the directions of the transmission and extinction axes of the polarizer in its reference, P = 0, orientation. p0 and s0 are directions parallel and perpendicular to the plane of incidence for light reflection at the first detector inside the FDP. The relative orientation angle α0 is determined by a procedure given in the text.

Fig. 7
Fig. 7

Normalized Stokes parameters, (a) s1, (b) s2, and (c) s3, as measured by the well-calibrated FDP (diamonds) and as produced by an ideal PSG (curves). In this test the polarizer is set at zero azimuth (P = 0), and the QWR is rotated to vary its fast-axis azimuth C (in 10° steps). Deviation of the experimental points from the idealized theoretical curves is indicative of imperfections in the QWR.

Fig. 8
Fig. 8

Deviations of the FDP-determined normalized Stokes parameters from their ideal values: (a) Δs1, (b) Δs2, and (c) Δs3 are plotted versus the QWR fast-axis azimuth C with the polarizer set at P = 0. The discrete points (diamonds) are the measured deviations, and the curves are obtained by a least-squares fit, using Fourier series of the forms given by Eqs. (40)(42).

Fig. 9
Fig. 9

Deviations of the FDP-determined normalized Stokes parameters from their ideal values: (a) Δs1, (b) Δs2, and (c) Δs3 are plotted versus the polarizer azimuth P with the QWR set at C = 0. The Fourier series that fit the experimental data points are given by Eqs. (53)(55). The residual rms errors in fitting Δs1, Δs2, and Δs3 are 0.0011, 0.0042, and 0.0016, respectively.

Fig. 10
Fig. 10

DOAP, to which much of the analysis of this paper also applies. (See Ref. 11 for details.)

Tables (2)

Tables Icon

Table 1 Fourier Coefficients for the Stokes-Parameter Perturbations [Eqs. (40)(42)] in the P = 0, Variable-C Testa

Tables Icon

Table 2 Fourier Coefficients for the Stokes-Parameter Perturbations [Eqs. (53)(55)] in the C = 0, Variable-P Testa

Equations (59)

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I = AS
I n = I / i r
S = S n = [ 1 s 1 s 2 s 3 ] t
I n = A S n .
A = [ A 0 A 1 A 2 A 3 ] .
S LP ( P ) = [ 1 cos 2 P sin 2 P 0 ] t .
I LP ( P ) = A 0 + A 1 cos 2 P + A 2 sin 2 P .
A 0 = [ 0.746 2.433 0.694 2.597 ] t , A 1 = [ 0.157 - 1.180 - 0.244 - 1.934 ] t , A 2 = [ - 0.071 0.516 0.137 - 0.296 ] t .
I av ( P ) = ( 1 / 2 ) [ I ( P ) + I ( P + 180 ° ) ] ,
S CP = [ 1 0 0 ± 1 ] t ,
I RCP = A 0 + A 3 , I LCP = A 0 - A 3 ,
A 3 = ( 1 / 2 ) ( I RCP - I LCP ) ,
A 0 = ( 1 / 2 ) ( I RCP + I LCP ) .
S = [ 1 cos 2 cos 2 θ cos 2 sin 2 θ sin 2 ] t .
β = 45 ° -
S ENCS ( θ ) = [ 1 2 β cos 2 θ 2 β sin 2 θ 1 ] t ,
S ENCS ( θ + 90 ° ) = [ 1 - 2 β cos 2 θ - 2 β sin 2 θ 1 ] t
I av = ( 1 / 2 ) [ I ENCS ( θ ) + I ENCS ( θ + 90 ° ) ] = ( 1 / 2 ) [ A S ENCS ( θ ) + A S ENCS ( θ + 90 ° ) ] = A { ( 1 / 2 ) [ S ENCS ( θ ) + S ENCS ( θ + 90 ° ) ] } .
( 1 / 2 ) [ S ENCS ( θ ) + S ENCS ( θ + 90 ° ) ] = [ 1 0 0 1 ] t = S RCP ,
I av = A S RCP = I RCP .
A 3 = [ - 0.001 0.447 - 0.405 - 0.429 ] t .
A = [ 0.746 0.157 - 0.071 - 0.001 2.433 - 1.180 0.516 0.447 0.694 - 0.244 0.137 - 0.405 2.597 - 1.934 - 0.296 - 0.429 ] .
I = A int S p 0 s 0 ,
S p 0 s 0 = R ( α 0 ) S t e .
R ( α 0 ) = [ 1 0 0 0 0 cos 2 α 0 sin 2 α 0 0 0 - sin 2 α 0 cos 2 α 0 0 0 0 0 1 ]
I = [ A int R ( α 0 ) ] S t e = A ext S t e ,
A ext = A int R ( α 0 ) .
A int = A ext R ( - α 0 ) ,
a 02 i = 0 ,
a 01 e ( - sin 2 α 0 ) + a 02 e ( cos 2 α 0 ) = 0 ,
α 0 = ( 1 / 2 ) arctan ( a 02 e / a 01 e ) .
A int = [ 0.746 0.172 0.000 - 0.001 2.433 - 1.288 - 0.016 0.447 0.694 - 0.279 0.024 - 0.405 2.597 - 1.640 - 1.067 - 0.429 ] ,
τ = T - 1 , δ = Δ - π / 2 ,
γ = C true - C ,
M QWR = [ 1 - τ 0 0 - τ 1 0 0 0 0 - δ 1 0 0 1 - δ ] ,
S = R ( - C - γ ) M QWR R ( C + γ ) S LP ( P ) .
s 1 = ( ½ + ½ cos 4 C ) + Δ s 1 ,
s 2 = ( ½ sin 4 C ) + Δ s 2 ,
s 3 = ( sin 2 C ) + Δ s 3 ,
Δ s 1 = a 0 + a 2 cos 2 C + a 4 cos 4 C + b 4 sin 4 C + a 6 cos 6 C ,
Δ s 2 = d 2 sin 2 C + c 4 cos 4 C + d 4 sin 4 C + d 6 sin 6 C ,
Δ s 3 = e 2 cos 2 C + f 4 sin 4 C .
a 0 = - a 4 = - d 4 = - δ / 2 ,
a 2 = - a 6 = d 2 / 3 = - d 6 = - f 4 / 2 = - τ / 4 ,
b 4 = - c 4 = e 2 = - 2 γ .
S = A - 1 I ,
A - 1 = [ 0.917 0.099 0.106 0.001 1.385 - 0.102 0.285 - 0.378 - 1.389 0.804 1.766 - 0.826 0.267 0.503 - 1.862 - 0.050 ] .
δ = - 0.061 ( ± 0.003 ) ,
τ = 0.016 ( ± 0.002 ) .
s 1 = ( cos 2 P ) + Δ s 1 ,
s 2 = ( 0 ) + Δ s 2 ,
s 3 = ( - sin 2 P ) + Δ s 3 ;
Δ s 1 = a 0 + b 2 sin 2 P + a 4 cos 4 P ,
Δ s 2 = c 2 cos 2 P + d 2 sin 2 P ,
Δ s 3 = e 2 cos 2 P + f 4 sin 4 P
a 0 = - a 4 = f 4 = - τ / 2 ,
b 2 = c 2 = e 2 = 2 γ ,
d 2 = - δ ,
γ = - 0.108 ° ( ± 0.013 ° ) .

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