Abstract

A fully automated Mueller-matrix ellipsometer with a division-of-amplitude photopolarimeter as the polarization-state detector is described. This device achieves Mueller-matrix ellipsometry by measuring the Stokes parameters of reflected light as a function of the fast axis C of a quarter-wave retarder, which, in combination with a fixed linear polarizer, determines the polarization state of incident light. The reflected Stokes parameters were Fourier analyzed to give the 16 elements of the Mueller matrix. We investigated depolarization of polarized light on reflection from rough, heterogeneous, and anisotropic surfaces by obtaining measurements on rolled aluminum and plant leaves. The results demonstrate (1) a variation of degree of polarization of reflected light with the input polarization state, (2) the precision with which the measured matrices describe the depolarization results, (3) effects of surface anisotropy (rolling direction) on depolarization and cross polarization by reflection from aluminum surfaces, and (4) large values and differences in the depolarization effects from conifer and deciduous leaves. Depolarization of light reflected by the aluminum surfaces was most sensitive to the angle between the plane of incidence and the rolling direction when the incident Stokes parameters S 1, S 2, and S 3 were equal.

© 1994 Optical Society of America

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References

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  1. O. Hunderi, “Optics of rough surfaces, discontinuous films and heterogeneous materials,” Surf. Sci. 96, 1–31 (1980).
    [CrossRef]
  2. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization,” Opt. Lett. 10, 309–311 (1985).
    [CrossRef] [PubMed]
  3. R. M. A. Azzam, E. Masetti, I. M. Elminyawi, A. M. El-Saba, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
    [CrossRef]
  4. 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]
  5. R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
    [CrossRef]
  6. 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]
  7. D. A. Ramsey, “Mueller-matrix ellipsometry involving extremely rough surfaces,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1985).
  8. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
    [CrossRef] [PubMed]
  9. C. Brosseau, “Analysis of experimental data for Mueller polarization matrices,” Optik 85, 83–86 (1990).
  10. S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 177–185 (1989).
  11. W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).
  12. R. M. A. Azzam, “Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 (1986).
    [CrossRef] [PubMed]
  13. 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]
  14. R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
    [CrossRef]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chaps. 1–3.
  16. P. S. Hauge, “Mueller-matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  17. A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
    [CrossRef]
  18. P. S. Hauge, “Automated Mueller-matrix ellipsometry,” Opt. Commun. 17, 74–76 (1976).
    [CrossRef]
  19. R. M. A. Azzam, Department of Electrical Engineering, University of New Orleans, New Orleans, La. 70148 (personal communication, March1993).
  20. J. E. Kalshoven, P. W. Dabney, “An airborne laser polarimeter system (ALPS) for terrestrial physics research,” in Recent Advances in Sensors, Radiometer, and Data Processing for Remote Sensing, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 924, 33–35 (1988).
  21. J. E. Kalshoven, P. W. Dabney, “Remote sensing of the Earth’s surface with an airborne polarized laser,” IEEE Trans. Geosci. Remote Sensing 31, 438–446 (1993).
    [CrossRef]

1993

J. E. Kalshoven, P. W. Dabney, “Remote sensing of the Earth’s surface with an airborne polarized laser,” IEEE Trans. Geosci. Remote Sensing 31, 438–446 (1993).
[CrossRef]

1992

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

C. Brosseau, “Analysis of experimental data for Mueller polarization matrices,” Optik 85, 83–86 (1990).

1989

1988

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

1986

1985

R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization,” Opt. Lett. 10, 309–311 (1985).
[CrossRef] [PubMed]

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[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

O. Hunderi, “Optics of rough surfaces, discontinuous films and heterogeneous materials,” Surf. Sci. 96, 1–31 (1980).
[CrossRef]

1978

1976

P. S. Hauge, “Automated Mueller-matrix ellipsometry,” Opt. Commun. 17, 74–76 (1976).
[CrossRef]

1973

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

Abromson, D.

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

Azzam, R. M. 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]

R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
[CrossRef]

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

R. M. A. Azzam, “Mueller-matrix ellipsometry using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 (1986).
[CrossRef] [PubMed]

R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization,” Opt. Lett. 10, 309–311 (1985).
[CrossRef] [PubMed]

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[CrossRef]

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]

R. M. A. Azzam, Department of Electrical Engineering, University of New Orleans, New Orleans, La. 70148 (personal communication, March1993).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chaps. 1–3.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chaps. 1–3.

Bickel, W. S.

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

Brosseau, C.

C. Brosseau, “Analysis of experimental data for Mueller polarization matrices,” Optik 85, 83–86 (1990).

Chiao, S. C.

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 177–185 (1989).

Dabney, P. W.

J. E. Kalshoven, P. W. Dabney, “Remote sensing of the Earth’s surface with an airborne polarized laser,” IEEE Trans. Geosci. Remote Sensing 31, 438–446 (1993).
[CrossRef]

J. E. Kalshoven, P. W. Dabney, “An airborne laser polarimeter system (ALPS) for terrestrial physics research,” in Recent Advances in Sensors, Radiometer, and Data Processing for Remote Sensing, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 924, 33–35 (1988).

Elminyawi, I. M.

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

El-Saba, A. M.

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

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]

Hauge, P. S.

P. S. Hauge, “Mueller-matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
[CrossRef]

P. S. Hauge, “Automated Mueller-matrix ellipsometry,” Opt. Commun. 17, 74–76 (1976).
[CrossRef]

Hsu, J. Y.

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

Huffman, D. R.

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

Hunderi, O.

O. Hunderi, “Optics of rough surfaces, discontinuous films and heterogeneous materials,” Surf. Sci. 96, 1–31 (1980).
[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]

Iafelice, V.

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

Kalshoven, J. E.

J. E. Kalshoven, P. W. Dabney, “Remote sensing of the Earth’s surface with an airborne polarized laser,” IEEE Trans. Geosci. Remote Sensing 31, 438–446 (1993).
[CrossRef]

J. E. Kalshoven, P. W. Dabney, “An airborne laser polarimeter system (ALPS) for terrestrial physics research,” in Recent Advances in Sensors, Radiometer, and Data Processing for Remote Sensing, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 924, 33–35 (1988).

Krishnan, S.

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]

R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
[CrossRef]

Masetti, E.

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

Ramsey, D. A.

D. A. Ramsey, “Mueller-matrix ellipsometry involving extremely rough surfaces,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1985).

Williams, M. W.

Appl. Opt.

IEEE Trans. Geosci. Remote Sensing

J. E. Kalshoven, P. W. Dabney, “Remote sensing of the Earth’s surface with an airborne polarized laser,” IEEE Trans. Geosci. Remote Sensing 31, 438–446 (1993).
[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]

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter (DOAP),” Opt. Acta 32, 767–777 (1985).
[CrossRef]

Opt. Commun.

P. S. Hauge, “Automated Mueller-matrix ellipsometry,” Opt. Commun. 17, 74–76 (1976).
[CrossRef]

Opt. Eng.

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.

Optik

C. Brosseau, “Analysis of experimental data for Mueller polarization matrices,” Optik 85, 83–86 (1990).

Rev. Sci. Instrum.

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

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

Surf. Sci.

O. Hunderi, “Optics of rough surfaces, discontinuous films and heterogeneous materials,” Surf. Sci. 96, 1–31 (1980).
[CrossRef]

Other

D. A. Ramsey, “Mueller-matrix ellipsometry involving extremely rough surfaces,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1985).

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 177–185 (1989).

W. S. Bickel, J. Y. Hsu, S. C. Chiao, D. Abromson, V. Iafelice, “The Mueller matrix–Stokes vector representation of surface scattering,” in Polarization Considerations for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 891, 32–39 (1988).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chaps. 1–3.

R. M. A. Azzam, Department of Electrical Engineering, University of New Orleans, New Orleans, La. 70148 (personal communication, March1993).

J. E. Kalshoven, P. W. Dabney, “An airborne laser polarimeter system (ALPS) for terrestrial physics research,” in Recent Advances in Sensors, Radiometer, and Data Processing for Remote Sensing, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 924, 33–35 (1988).

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the system used for Mueller-matrix ellipsometry.

Fig. 2
Fig. 2

Plots of the normalized Stokes parameters: (a) S 1, (b) S 2, (c) S 3 versus the retarder fast axis C in the polarization-state generator. The three curves in each panel were obtained for indicated angles of 0°, 45°, and 90° between the rolling direction and the plane of incidence.

Fig. 3
Fig. 3

Plots of the degree of polarization P versus the retarder fast axis C in the polarization-state generator for angles of 0° and 180° between the rolling direction and the plane of incidence. MM is the Mueller matrix.

Fig. 4
Fig. 4

Plots of the degree of polarization P versus the retarder fast axis C in the polarization-state generator for angles of 90° and 270° between the rolling direction and the plane of incidence.

Fig. 5
Fig. 5

Plots of the degree of polarization P versus the retarder fast axis C in the polarization-state generator for angles of 45° and 315° between the rolling direction and the plane of incidence.

Fig. 6
Fig. 6

Plots of the degree of polarization P versus the retarder fast axis C in the polarization-state generator for angles of 135° and 225° between the rolling direction and the plane of incidence.

Fig. 7
Fig. 7

Plot of the degree of polarization P versus the angle between the rolling direction and the plane of incidence with the retarder fast axis C set to 110°.

Fig. 8
Fig. 8

Plot of the degree of polarization P versus the angle between the rolling direction and the plane of incidence with the retarder fast axis C set to 0°.

Fig. 9
Fig. 9

Plot of the degree of p-s cross polarization X ps versus the angle between the rolling direction and the plane of incidence.

Fig. 10
Fig. 10

Three dimensional plot of the apparent values for ψ and Δ and the rolling angle projected in three planes.

Fig. 11
Fig. 11

Plots of the degree of polarization P versus the retarder fast axis C in the polarization-state generator for maple and yew leaves.

Equations (16)

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I = F S ,
S = F - 1 I .
S r = M S i .
M = [ 1 - cos ( 2 ψ ) 0 0 - cos ( 2 ψ ) 1 0 0 0 0 sin ( 2 ψ ) cos ( Δ ) sin ( 2 ψ ) sin ( Δ ) 0 0 - sin ( 2 ψ ) sin ( Δ ) sin ( 2 ψ ) cos ( Δ ) ]
S r , i = A i 0 + A i 2 cos 2 C + B i 2 sin 2 C + A i 4 cos 4 C + B i 4 sin 4 C .
M i 0 = A i 0 - A i 4 , M i 1 = 2 A i 4 , M i 2 = 2 B i 4 , M i 3 = B i 2 .
P = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 S 0 .
X p s = ( S 0 - S 1 ) 1 / 2 ( S 0 + S 1 ) 1 / 2 .
ψ = 1 2 cos - 1 ( - M 01 - M 10 2 ) ,
Δ = tan - 1 ( M 23 - M 32 M 22 + M 33 ) .
M 0 = [ 1.0000 - 0.1505 0.0009 0.0203 - 0.1416 0.9890 - 0.0049 0.0009 0.0044 0.0035 - 0.7840 0.5762 - 0.0155 0.0297 - 0.5443 - 0.7865 ] .
M 180 = [ 1.0000 - 0.1550 - 0.0016 0.0200 - 0.1374 0.9838 - 0.0156 0.0016 0.0088 - 0.0065 - 0.7879 0.5488 - 0.0105 0.0207 - 0.5065 - 0.8277 ] .
M 90 = [ 1.0000 - 0.3718 0.0259 0.0280 - 0.3772 1.0001 0.0066 - 0.0100 - 0.0009 0.0131 - 0.6275 0.6437 - 0.0237 0.0213 - 0.6425 - 0.6603 ] .
M 45 = [ 1.0000 - 0.2572 - 0.1018 0.0310 - 0.2612 0.9898 0.0194 0.0379 0.0930 - 0.0195 - 0.7375 0.5989 0.0271 0.0544 - 0.5904 - 0.7452 ] .
M maple = [ 1.0000 - 0.5555 - 0.0239 - 0.0277 - 0.5805 0.7655 - 0.0196 0.0012 0.0636 - 0.0726 - 0.4061 0.1270 - 0.0679 0.0141 - 0.1715 - 0.3476 ] .
M yew = [ 1.0000 - 0.1521 - 0.0019 - 0.0199 - 0.2172 0.2230 - 0.0043 0.0205 0.0262 0.0053 - 0.0868 0.0435 - 0.0795 0.0104 - 0.0543 - 0.0298 ] .

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