Abstract

A broadband division-of-amplitude polarimeter (DOAP) is presented. It can provide the real-time measurement of any state of polarization of light, described by its Stokes vector, in large spectral windows. The light is split first into two beams by a prism and then into four beams by means of any polarizer device that will separate the two linear orthogonal states of polarization. Finally, the Stokes vector is directly deduced from the four measured intensities. To avoid interference effects, the splitting of light into four beams is induced only by refractive-index contrast effects between semi-infinite media that are weakly dependent on the wavelength. An experimental setup working from 0.4 to 2 μm is described. It provides similar sensitivities for all the states of polarization, and its characteristics are constant, on a scale of a few percent, within the spectral window. Calibrations performed at 458 and 633 nm display good agreement between theoretical and experimental values. The accuracy of the prism DOAP, evaluated by measurement of the Stokes vector produced by a rotating Glan polarizer, is better than 1%. An infrared extension of this polarimeter is also presented.

© 1998 Optical Society of America

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References

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  1. X. Huard, Polarisation de la Lumière (Masson, Paris, 1994).
  2. E. Collett, Polarized Light—Fundamentals and Application (Dekker, New York, 1993).
  3. J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Bellingham, Wash., 1995).
    [CrossRef]
  4. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
    [CrossRef]
  5. D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
    [CrossRef]
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  7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
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  8. E. Compain, B. Drevillon, “High frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
    [CrossRef]
  9. R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
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  10. R. Anderson, “Measurement of Mueller matrices,” Appl. Opt. 31, 11–13 (1992), and references therein.
    [CrossRef] [PubMed]
  11. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
    [CrossRef]
  12. 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).
    [CrossRef]
  13. K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
    [CrossRef]
  14. 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]
  15. R. M. A. Azzam, K. A. Giardina, “Photopolarimeter based on planar grating diffraction,” J. Opt. Soc. Am. A 10, 1190–1196 (1993).
    [CrossRef]
  16. A. M. El-Saba, R. M. A. Azzam, M. A. G. Abushagur, “Parallel-slab division-of-amplitude polarimeter,” Opt. Lett. 21, 1709–1711 (1996).
    [CrossRef] [PubMed]
  17. F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36, 5388–5395 (1997), and references therein.
    [CrossRef] [PubMed]
  18. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  19. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 1, Sect. 5, Part 4.
  20. E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
    [CrossRef]
  21. E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
    [CrossRef]

1998

E. Compain, B. Drevillon, “High frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

1997

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36, 5388–5395 (1997), and references therein.
[CrossRef] [PubMed]

1996

1993

1992

1991

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

1985

1984

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

1982

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).
[CrossRef]

1980

1978

1976

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Abushagur, M. A. G.

Anderson, R.

Aspnes, D. E.

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Azzam, R. M. A.

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Bashara, N. M.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 1, Sect. 5, Part 4.

Bottiger, J. R.

Bouree, J. E.

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

Brudzewski, K.

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

Collett, E.

E. Collett, Polarized Light—Fundamentals and Application (Dekker, New York, 1993).

Compain, E.

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

E. Compain, B. Drevillon, “High frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Delplancke, F.

Drevillon, B.

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

E. Compain, B. Drevillon, “High frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

El-Saba, A. M.

Fry, E. S.

Giardina, K. A.

Goldstein, D. H.

Hauge, P. S.

Huard, X.

X. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

Hui, J.

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

Krishnan, S.

Parey, J. Y.

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

Stover, J. C.

J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Bellingham, Wash., 1995).
[CrossRef]

Thompson, R. C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 1, Sect. 5, Part 4.

Am. J. Phys.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

E. Compain, B. Drevillon, “High frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Surf. Sci.

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Thin Solid Films

E. Compain, B. Drevillon, J. Hui, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314, 47–52 (1998).
[CrossRef]

Other

X. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

E. Collett, Polarized Light—Fundamentals and Application (Dekker, New York, 1993).

J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Bellingham, Wash., 1995).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 1, Sect. 5, Part 4.

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Figures (5)

Fig. 1
Fig. 1

Principle of a visible–near-infrared DOAP with an uncoated dielectric prism. The light beam is separated first into two by a prism then into four by two Wollaston prisms (W1 and W2) oriented at 45° with respect to the plane of incidence. Stokes vector S is determined from the four intensities i 1i 4.

Fig. 2
Fig. 2

Phase shift Δ r of the internal reflection(s) as a function of the internal angle of reflection ϕ for various prism refractive indices n p .

Fig. 3
Fig. 3

Experimental realization of the visible–near-infrared prism DOAP. The beam splitter is an uncoated orthogonal prism made with a high-refractive-index glass (Corning Cr46, n p = 1.81). The parasitic backreflection inside the prism is stopped by a roughened black edge. The angle of the prism χ and the angle of incidence ϕ1 are optimized to provide a uniform sensitivity to all the states of polarization. The intensities of the four beams reflected on mirrors M1–4 are successively measured by a photomultiplier tube (PM). D is a diffuser that scatters the beam selected by the mechanical multiplexer (Mplx) onto the photomultiplier photocathode (PC) to decrease the sensitivity to the angle of incidence. W1 and W2 are Wollaston prisms.

Fig. 4
Fig. 4

ZnS-prism beam splitter for infrared division of the amplitude polarimeter. n p = 2.2 is a middle-range refractive index, S is the Stokes vector of incident light, and the angles correspond to the second configuration described in the text.

Fig. 5
Fig. 5

Experimental measurements of known Stokes vectors (I, Q, U, V) of a rotating Glan polarizer as a function of the orientation of the polarizer. Filled squares, Q/ I; filled triangles, U/ I; stars, V/ I; and open circles, the degree of polarization are compared with their theoretical values (solid curves).

Tables (1)

Tables Icon

Table 1 Parameters of the Visible–Near-Infrared Prism-DOAP (at 488 nm)

Equations (24)

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S = I Q U V = I x + I y I x - I y I + 45 - I - 45 I R - I L ,
S = A - 1 i 1 ,   i 2 ,   i 3 ,   i 4 T ,
T 2 = ½ t p 2 + t s 2 ,     R 2 = ½ r p 2 + r s 2 ,
exp - α L .
tan Ψ r exp i Δ r = r p / r s   reflection ,
tan Ψ t exp i Δ t = t p / t s   transmission .
A = D 1 - cos   2 Ψ r sin   2 Ψ r cos   Δ r sin   2 Ψ r sin   Δ r 1 - cos   2 Ψ r - sin   2 Ψ r cos   Δ r - sin   2 Ψ r sin   Δ r 1 - cos   2 Ψ t sin   2 Ψ t cos   Δ t sin   2 Ψ t sin   Δ t 1 - cos   2 Ψ t - sin   2 Ψ t cos   Δ t - sin   2 Ψ t sin   Δ t ,
D = 1 2 G 1 R 0 0 0 0 G 2 R 0 0 0 0 G 3 T 0 0 0 0 G 4 T
s A = λ 1 / λ 4 ,
A q 0   optimal     s A q 0   maximum .
A q 0   optimal     R + T   maximum .
R = T 1 / 2 , Ψ r = π / 2 - Ψ t = π / 8   or   3 π / 8 , Δ r - Δ t = π / 2   modulo   π .
ϕ c = arcsin n 1 n 0 .
r s , p 2 + n t cos   ϕ t n i cos   ϕ i   t s , p 2 = 1 .
R = 1 / 2 Ψ r = π / 8 T = 1 / 2 Ψ t = 3 π / 8 .
cos   ϕ = n p 2 - 1 n p 2 + 1 1 / 2     Δ r = π + 4   arctan   n p Δ r ϕ = 0 .
A the 488   nm = 187 0 0 0 0 187 0 0 0 0 195 0 0 0 0 195 × 1 - 0.575 0.818 0 1 - 0.575 - 0.818 0 1 0.617 - 0.003 0.787 1 0.617 0.003 - 0.787 .
δ A = i , j = 1 4 A ij ϕ 1 2 1 / 2 .
A exp 633   nm = 185 0 0 0 0 164 0 0 0 0 169 0 0 0 0 151 × 1 - 0.565 0.827 0.005 1 - 0.589 - 0.826 - 0.010 1 0.611 0.243 0.727 1 0.614 - 0.260 - 0.715 .
A the 633   nm = 187 0 0 0 0 187 0 0 0 0 196 0 0 0 0 196 × 1 - 0.569 0.822 0.000 1 - 0.569 - 0.822 0.000 1 0.610 0.213 0.763 1 0.610 - 0.213 - 0.763 .
A exp 458   nm = 194 0 0 0 0 172 0 0 0 0 147 0 0 0 0 142 × 1 - 0.537 0.822 0.019 1 - 0.557 - 0.820 - 0.025 1 0.617 - 0.011 0.779 1 0.615 - 0.015 - 0.776 .
A the 458   nm = 188 0 0 0 0 188 0 0 0 0 179 0 0 0 0 179 × 1 - 0.577 0.816 0.000 1 - 0.577 - 0.816 0.000 1 0.619 - 0.065 0.782 1 0.619 0.065 - 0.782 .
p S = Q 2 + U 2 + V 2 1 / 2 I ,
S a = I 1 Q / I U / I V / I = I 1 cos   2 a sin   2 a 0 .

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