Abstract

A new scatterometer–polarimeter is described. It measures the angular distribution of intensity and of the complete Mueller matrix of light scattered by rough surfaces and particle suspensions. The measurement time is 1 s/scattering angle in the present configuration but can be reduced to a few milliseconds with modified electronics. The instrument uses polarization modulation and a Fourier analysis of four detected signals to obtain the 16 Mueller matrix elements. This method is particularly well suited to online, real time, industrial process control involving rough surfaces and large particle suspensions (an arithmetic roughness or particle diameter of >1 µm). Some results are given.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  17. R. M. A. Azzam, K. A. Giardina, A. G. Lopez, “Conventional and generalized Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Eng. 30, 1583–1588 (1991).
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  18. R. M. A. Azzam, “Mueller matrix measurement using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 (1986).
    [CrossRef]
  19. S. Krishnan, P. C. Nordine, “Mueller-matrix ellipsometry using the DOAP: a study of depolarization effects,” Appl. Opt. 33, 4184–4192 (1994).
    [CrossRef] [PubMed]

1995 (2)

B. Griffiths, R. Middleton, B. Wilkie, “Three-dimensional surface measurement using light scattering,” Int. J. Mach. Tools Manufact. 35, 141–145 (1995).
[CrossRef]

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

1994 (1)

1992 (3)

1991 (3)

C. Gorecki, “Caractérisation ou qualification optique de surface en production,” Bull. Soc. Fr. Mécan. 1991–2, 125–131 (1991).

E. Bois, “Mesure en continu de profil de rugosité sur ligne de production de tôle d’acier,” Bull. Soc. Fr. Mécan. 1991–2, 119–123 (1991).

R. M. A. Azzam, K. A. Giardina, A. G. Lopez, “Conventional and generalized Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Eng. 30, 1583–1588 (1991).
[CrossRef]

1990 (1)

1987 (2)

1986 (1)

1985 (1)

1982 (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all 4 Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

1980 (1)

1978 (1)

1973 (1)

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Anderson, R.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), pp. 490–492.

Bois, E.

E. Bois, “Mesure en continu de profil de rugosité sur ligne de production de tôle d’acier,” Bull. Soc. Fr. Mécan. 1991–2, 119–123 (1991).

Bottiger, J. R.

Chipman, R. A.

Fry, E. S.

Giardina, K. A.

R. M. A. Azzam, K. A. Giardina, A. G. Lopez, “Conventional and generalized Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Eng. 30, 1583–1588 (1991).
[CrossRef]

Goldstein, D. H.

Gorecki, C.

C. Gorecki, “Caractérisation ou qualification optique de surface en production,” Bull. Soc. Fr. Mécan. 1991–2, 125–131 (1991).

Griffiths, B.

B. Griffiths, R. Middleton, B. Wilkie, “Three-dimensional surface measurement using light scattering,” Int. J. Mach. Tools Manufact. 35, 141–145 (1995).
[CrossRef]

Huffman, D. R.

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Hunt, A. J.

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Jellison, G. E.

Krishnan, S.

Le Bosse, J. C.

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

Lopez, A. G.

R. M. A. Azzam, K. A. Giardina, A. G. Lopez, “Conventional and generalized Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Eng. 30, 1583–1588 (1991).
[CrossRef]

Mathia, T.

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

Middleton, R.

B. Griffiths, R. Middleton, B. Wilkie, “Three-dimensional surface measurement using light scattering,” Int. J. Mach. Tools Manufact. 35, 141–145 (1995).
[CrossRef]

Nordine, P. C.

Rousseau, J.

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

Thompson, R. C.

Wilkie, B.

B. Griffiths, R. Middleton, B. Wilkie, “Three-dimensional surface measurement using light scattering,” Int. J. Mach. Tools Manufact. 35, 141–145 (1995).
[CrossRef]

Zahouani, H.

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

Appl. Opt. (5)

Bull. Soc. Fr. Mécan. (2)

C. Gorecki, “Caractérisation ou qualification optique de surface en production,” Bull. Soc. Fr. Mécan. 1991–2, 125–131 (1991).

E. Bois, “Mesure en continu de profil de rugosité sur ligne de production de tôle d’acier,” Bull. Soc. Fr. Mécan. 1991–2, 119–123 (1991).

Int. J. Mach. Tools Manufact. (2)

B. Griffiths, R. Middleton, B. Wilkie, “Three-dimensional surface measurement using light scattering,” Int. J. Mach. Tools Manufact. 35, 141–145 (1995).
[CrossRef]

T. Mathia, H. Zahouani, J. Rousseau, J. C. Le Bosse, “Functional significance of different techniques for surface morphology measurements,” Int. J. Mach. Tools Manufact. 35, 195–202 (1995).
[CrossRef]

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

Opt. Acta (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all 4 Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

Opt. Eng. (1)

R. M. A. Azzam, K. A. Giardina, A. G. Lopez, “Conventional and generalized Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Eng. 30, 1583–1588 (1991).
[CrossRef]

Opt. Lett. (5)

Rev. Sci. Instrum. (1)

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Other (1)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), pp. 490–492.

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Figures (5)

Fig. 1
Fig. 1

Principle of Mueller-matrix scatterometry: the incident polarized light, with Stokes vector Sinc¯, is scattered and the polarization of the light scattered at angle θ, Sscatθ¯, is analyzed; by varying the incident polarization and, for each state, measuring the scattered polarization, one can obtain the sample Mueller matrix, Mθ¯¯.

Fig. 2
Fig. 2

Scheme of the automated photopolarimeter: BS’s, cube beam splitters; PT’s, polarization transformers; PD’s, photodiodes.

Fig. 3
Fig. 3

Sinc and cosc functions around the FFT harmonic frequency f 0 = 20 Hz for a pure sine original function analyzed during a T 0 = 0.482 s period. The solid and dashed curves are the real and the imaginary part of the FFT, respectively. The frequency domain discretization by a 1/T 0 period comb (dotted vertical lines) does not correspond to maxima nor to zero of those functions.

Fig. 4
Fig. 4

Sixteen Mueller-matrix elements (classified as they appear in the matrix) of a Glan–Thompson prism polarizer drawn as a function of its azimuthal angle (in degrees). The solid curves are the experimental results (including device and sample imperfections); the dotted ones are the theoretical curves of an ideal polarizer. The Mueller elements are all normalized by the very first one. This one, the total detected intensity, given relative to the intensity detected without any sample, is constant and equal to a little less than half as a result of slight absorption in the polarizer.

Fig. 5
Fig. 5

16 Mueller-matrix elements as a function of the scattering angle (in degrees). The solid and dashed curves are the EBT steel sheet; the dotted and dashed–dotted ones are the ground sheet. Data from two orthogonal orientations of the samples are shown: laminating directions perpendicular (solid and dotted curves) and parallel (dashed and dashed–dotted curves) to the scattering plane. The angle of incidence was 80° relative to the normal. The first element is the scattered intensity in log scale. All other elements are normalized relative to it and drawn in linear scale from -1 to +1.

Equations (18)

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S¯=IQUV=ElEl*+ErEr*ElEl*-ErEr*ElEr*+ErEl*iElEr*-ErEl*,
Sscatθ, φ¯=Mθ, φ¯¯Sinc¯.
P0°=121100110000000000
R45°=C110000cosδ1-sinδ100sinδ1cosδ100001
R0°=C21000010000cosδ2-sinδ200sinδ2cosδ2
Sinc¯=B1, cosδ1, sinδ1sinδ2, sinδ1cosδ2t,
δ1=A01+A1 sinω1t,
δ2=A02+A2 sinω2t.
Sscati=j=03 Mi,jSincj.
Iθ¯=D¯¯Sscatθ¯.
cosAk sinωkt=J0Ak+2n=1J2nAkcos2nωkt,
sinAk sinωkt=2n=0J2n+1Akcos2n+1ωkt,
Iij=03Di,jMj,0+Mj,1J0A1+j=03Di,jMj,1×2J2A1cos2ω1t++j=03Di,jMj,2×2J1A1J1A2cosω1-ω2t++j=03Di,jMj,32J1A1J0A2sinω1t+.
Ff=-A2T01-cos2πf-f0T02πf-f0T0-iA2T0sin2πf-f0T02πf-f0T0.
j=03Di,jMj,3=vi,32J1A1J0A2,
j=03Di,jMj,2=vi,22J1A1J1A2,
j=03Di,jMj,1=vi,12J2A1,
j=03Di,jMj,0=vi,0-vi,1J0A12J2A1.

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