Abstract

A division-of-amplitude photopolarimeter that uses a parallel-slab multiple-reflection beam splitter was described recently [Opt. Lett. 21, 1709 (1996)]. We provide a general analysis and an optimization of a specific design that uses a fused-silica slab that is uniformly coated with a transparent thin film of ZnS on the front surface and with an opaque Ag or Au reflecting layer on the back. Multiple internal reflections within the slab give rise to a set of parallel, equispaced, reflected beams numbered 0, 1, 2, and 3 that are intercepted by photodetectors D 0, D 1, D 2, and D 3, respectively, to produce output electrical signals i 0, i 1, i 2, and i 3, respectively. The instrument matrix A, which relates the output-signal vector I to the input Stokes vector S by I = AS, and its determinant D are analyzed. The instrument matrix A is nonsingular; hence all four Stokes parameters can be measured simultaneously over a broad spectral range (UV–VIS–IR). The optimum film thickness, the optimum angle of incidence, and the effect of light-beam deviation on the measured input Stokes parameters are considered.

© 1999 Optical Society of America

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  1. A. M. El-Saba, R. M. A. Azzam, M. A. G. Abushagur, “Parallel-slab division-of-amplitude photopolarimeter,” Opt. Lett. 21, 1709–1711 (1996).
    [CrossRef] [PubMed]
  2. R. M. A. Azzam, “Multichannel polarization state detectors for time-resolved ellipsometry,” Thin Solid Films 234, 371–374 (1993).
    [CrossRef]
  3. R. M. A. Azzam, K. A. Giardina, “Polarization analysis based on grating conical diffraction,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 2–13 (1992).
    [CrossRef]
  4. R. M. A. Azzam, “Multidetector photopolarimeter for industrial optical sensing and metrology,” in Industrial Applications of Optical Inspection, Metrology, and Sensing, G. M. Brown, K. G. Harding, H. P. Stahl, eds., Proc. SPIE1821, 270–283 (1993).
    [CrossRef]
  5. E. Collett, “Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser pulses,” Surf. Sci. 96, 156–167 (1980).
    [CrossRef]
  6. R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).
  7. A. S. Siddiqui, “Real time measuring of polarization,” Photonics Spectra 26, 120–124 (1992).
  8. R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 1407–1412 (1985).
    [CrossRef]
  9. K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
    [CrossRef]
  10. S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A 9, 1615–1622 (1992).
    [CrossRef]
  11. R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
    [CrossRef]
  12. R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on a planar grating diffraction,” J. Opt. Soc. Am A 10, 1190–1196 (1993).
    [CrossRef]
  13. 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]
  14. R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
    [CrossRef]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometery and Polarized Light (North Holand, Amsterdam, 1977).
  16. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  17. J. Liu, R. M. A. Azzam, “Effect of light-beam deviation on the instrument matrix of the four-detector photopolarimeter,” Opt. Eng. 36, 943–951 (1997).
    [CrossRef]

1997 (1)

J. Liu, R. M. A. Azzam, “Effect of light-beam deviation on the instrument matrix of the four-detector photopolarimeter,” Opt. Eng. 36, 943–951 (1997).
[CrossRef]

1996 (1)

1993 (2)

R. M. A. Azzam, “Multichannel polarization state detectors for time-resolved ellipsometry,” Thin Solid Films 234, 371–374 (1993).
[CrossRef]

R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on a planar grating diffraction,” J. Opt. Soc. Am A 10, 1190–1196 (1993).
[CrossRef]

1992 (2)

1991 (2)

R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

1989 (1)

1988 (1)

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)

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

1980 (1)

E. Collett, “Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser pulses,” Surf. Sci. 96, 156–167 (1980).
[CrossRef]

Abushagur, M. A. G.

Azzam, R. M. A.

J. Liu, R. M. A. Azzam, “Effect of light-beam deviation on the instrument matrix of the four-detector photopolarimeter,” Opt. Eng. 36, 943–951 (1997).
[CrossRef]

A. M. El-Saba, R. M. A. Azzam, M. A. G. Abushagur, “Parallel-slab division-of-amplitude photopolarimeter,” Opt. Lett. 21, 1709–1711 (1996).
[CrossRef] [PubMed]

R. M. A. Azzam, “Multichannel polarization state detectors for time-resolved ellipsometry,” Thin Solid Films 234, 371–374 (1993).
[CrossRef]

R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on a planar grating diffraction,” J. Opt. Soc. Am A 10, 1190–1196 (1993).
[CrossRef]

R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
[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, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

R. M. A. Azzam, “Multidetector photopolarimeter for industrial optical sensing and metrology,” in Industrial Applications of Optical Inspection, Metrology, and Sensing, G. M. Brown, K. G. Harding, H. P. Stahl, eds., Proc. SPIE1821, 270–283 (1993).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometery and Polarized Light (North Holand, Amsterdam, 1977).

R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
[CrossRef]

R. M. A. Azzam, K. A. Giardina, “Polarization analysis based on grating conical diffraction,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 2–13 (1992).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometery and Polarized Light (North Holand, Amsterdam, 1977).

Brudzewski, K.

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

Collett, E.

E. Collett, “Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser pulses,” Surf. Sci. 96, 156–167 (1980).
[CrossRef]

Cross, R.

R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).

Elminyawi, I. M.

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.

A. M. El-Saba, R. M. A. Azzam, M. A. G. Abushagur, “Parallel-slab division-of-amplitude photopolarimeter,” Opt. Lett. 21, 1709–1711 (1996).
[CrossRef] [PubMed]

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]

Giardina, K. A.

R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on a planar grating diffraction,” J. Opt. Soc. Am A 10, 1190–1196 (1993).
[CrossRef]

R. M. A. Azzam, K. A. Giardina, “Polarization analysis based on grating conical diffraction,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 2–13 (1992).
[CrossRef]

Heffner, B.

R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).

Hernday, P.

R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).

Krishnan, S.

Liu, J.

J. Liu, R. M. A. Azzam, “Effect of light-beam deviation on the instrument matrix of the four-detector photopolarimeter,” Opt. Eng. 36, 943–951 (1997).
[CrossRef]

Lopez, A. G.

Siddiqui, A. S.

A. S. Siddiqui, “Real time measuring of polarization,” Photonics Spectra 26, 120–124 (1992).

J. Mod. Opt. (1)

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

J. Opt. Soc. Am A (1)

R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on a planar grating diffraction,” J. Opt. Soc. Am A 10, 1190–1196 (1993).
[CrossRef]

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

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]

J. Opt. Soc. Am. A (2)

Lasers and Optronics (1)

R. Cross, B. Heffner, P. Hernday, “Polarization measurement goes automatic,” Lasers and Optronics 10(11), 25–26 (1991).

Opt. Acta (1)

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

Opt. Eng. (1)

J. Liu, R. M. A. Azzam, “Effect of light-beam deviation on the instrument matrix of the four-detector photopolarimeter,” Opt. Eng. 36, 943–951 (1997).
[CrossRef]

Opt. Lett. (1)

Photonics Spectra (1)

A. S. Siddiqui, “Real time measuring of polarization,” Photonics Spectra 26, 120–124 (1992).

Surf. Sci. (1)

E. Collett, “Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser pulses,” Surf. Sci. 96, 156–167 (1980).
[CrossRef]

Thin Solid Films (1)

R. M. A. Azzam, “Multichannel polarization state detectors for time-resolved ellipsometry,” Thin Solid Films 234, 371–374 (1993).
[CrossRef]

Other (5)

R. M. A. Azzam, K. A. Giardina, “Polarization analysis based on grating conical diffraction,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 2–13 (1992).
[CrossRef]

R. M. A. Azzam, “Multidetector photopolarimeter for industrial optical sensing and metrology,” in Industrial Applications of Optical Inspection, Metrology, and Sensing, G. M. Brown, K. G. Harding, H. P. Stahl, eds., Proc. SPIE1821, 270–283 (1993).
[CrossRef]

R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometery and Polarized Light (North Holand, Amsterdam, 1977).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (16)

Fig. 1
Fig. 1

Diagram of the PS-DOAP.

Fig. 2
Fig. 2

Ellipsometric angle ψ k (k = 0, 1, 2, 3) for the first four reflected orders as functions of the incidence angle ϕ0 obtained by use of an uncoated SiO2–Ag parallel slab at λ = 633 nm.

Fig. 3
Fig. 3

Ellipsometric parameter Δ1 - Δ3 as a function of the angle of incidence ϕ0 obtained by use of an uncoated SiO2–Ag parallel slab at λ = 633 nm.

Fig. 4
Fig. 4

Power reflectance R k (k = 0, 1, 2, 3) for the first four reflected orders as functions of the incidence angle ϕ0 obtained by use of an uncoated SiO2–Ag parallel slab at λ = 633 nm.

Fig. 5
Fig. 5

Normalized determinant D N as a function of the incidence angle ϕ0 obtained by use of an uncoated SiO2–Ag parallel slab at λ = 633 nm.

Fig. 6
Fig. 6

Power reflectance R k (k = 0, 1, 2, 3) for the first four reflected orders as functions of the incidence angle ϕ0 obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm. The thickness d of the ZnS thin-film coating is 70 nm.

Fig. 7
Fig. 7

Normalized determinant D N as a function of the incidence angle ϕ0 obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm. The thickness d of the ZnS thin-film coating is 70 nm.

Fig. 8
Fig. 8

Power reflectance R k (k = 2, 3) for the second and the third reflected orders as functions of the coating thickness d obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm and an angle of incidence of ϕ0 = 45°.

Fig. 9
Fig. 9

Power reflectance R k (k = 0, 1, 2, 3) for the first four reflected orders as functions of the coating thickness d obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm. The angles of incidence are ϕ0 = 45°, 47.5°, 50°.

Fig. 10
Fig. 10

Normalized determinant D N as a function of the coating thickness d obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm. The angles of incidence are ϕ0 = 45°, 47.5°, 50°.

Fig. 11
Fig. 11

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the longitude angle θ obtained by used of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm and an angle of incidence of ϕ0 = 45°. The thickness of the ZnS thin-film coating is 70 nm.

Fig. 12
Fig. 12

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the latitude angle ∊ obtained by used of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm, an angle of incidence of ϕ0 = 45°, and a longitude angle of θ = -45°. The thickness of the ZnS thin-film coating is 70 nm.

Fig. 13
Fig. 13

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the latitude angle ∊ obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm, an angle of incidence of ϕ0 = 45°, and a longitude angle of θ = 0. The thickness of the ZnS thin-film coating is 70 nm.

Fig. 14
Fig. 14

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the latitude angle ∊ obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm, an angle of incidence of ϕ0 = 45°, and a longitude angle of θ = 45°. The thickness of the ZnS thin-film coating is 70 nm.

Fig. 15
Fig. 15

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the latitude angle ∊ obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm, an angle of incidence of ϕ0 = 45°, and a longitude angle of θ = 90°. The thickness of the ZnS thin-film coating is 70 nm.

Fig. 16
Fig. 16

Stokes parameters ΔS k (k = 1, 2, 3) as functions of the longitude angle θ obtained by use of a coated ZnS–SiO2–Ag parallel slab at λ = 633 nm and angles of incidence of ϕ0 = 40°, 50°. The thickness of the ZnS thin-film coating is 70 nm.

Equations (16)

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i k = k = 0 3   a mk S k ,     m = 0 ,   1 ,   2 ,   3 ,   .
I = AS ,
S = A - 1 I .
M k = R k 1 - cos   2 ψ k 0 0 - cos   2 ψ k 1 0 0 0 0 sin   2 ψ k   cos   Δ k sin   2 ψ k   sin   Δ k 0 0 - sin   2 ψ k   cos   Δ k sin   2 ψ k   cos   Δ k .
P k = 1 / 2 1 cos   2 α k sin   2 α k 0 cos   2 α k cos 2   2 α k sin   2 α k   cos   2 α k 0 - sin   2 α k sin   2 α k   cos   2 α k sin 2   2 α k 0 0 0 0 0 .
D = W 1 W 2 / 16 a 00 a 11 - a 01 a 10 a 22 a 33 - a 23 a 32 + a 00 a 21 - a 01 a 20 a 13 a 32 - a 12 a 33 + a 00 a 31 - a 01 a 30 a 12 a 23 - a 13 a 22 + a 02 a 33 - a 03 a 32 a 10 a 21 - a 11 a 20 + a 02 a 23 - a 03 a 22 a 11 a 30 - a 10 a 31 + a 02 a 13 - a 03 a 12 a 20 a 31 - a 21 a 30 ,
W 1 = k 0 k 1 k 2 k 3 ,   W 2 = R 0 R 1 R 2 R 3 ,   a k 0 = 1 - cos   2 α k   cos   2 ψ k ,   a k 1 = cos   2 α k - cos   2 ψ k ,   a k 2 = sin   2 α k   sin   2 ψ k   cos   Δ k ,   a k 3 = sin   2 α k   sin   2 ψ k   sin   Δ k ,     k = 0 ,   1 ,   2 ,   3 .
D = W 1 W 2 / 16 1 + cos   2 ψ 0 1 - cos   2 ψ 2 sin   2 ψ 1 × sin   2 ψ 3 sin Δ 1 - Δ 3 .
S = A - 1 A S .
Δ S = S - S
A = 0.4845 - 0.445 0.0000 0.0000 0.4368 0.1808 - 0.3905 0.0750 0.1143 0.1143 0.0000 0.0000 0.0262 0.0099 0.0204 - 0.0132 .
K = 1.0000 0.0000 0.0000 0.0000 0.0000 1.1092 0.0000 0.0000 0.0000 0.0000 4.2383 0.0000 0.0000 0.0000 0.0000 18.4665 ,
A = 0.4845 - 0.4845 0.000 0.0000 0.4845 0.2005 - 0.4332 0.0832 0.4845 0.4845 0.0000 0.0000 0.4845 0.1826 0.3771 - 0.2434 .
A = 0.4873 - 0.4873 0.000 0.0000 0.4853 0.2043 - 0.4321 0.0842 0.4808 0.4808 0.0000 0.0000 0.4774 0.1751 0.3710 - 0.2440 .
S = 1 cos   2   cos   2 θ cos   2   sin   2 θ sin   2 .
S = 1 cos   2 θ sin   2 θ 0 .

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