Abstract

In ellipsometry of rough surfaces, the commonly used parameters, psi and delta, are insufficient to characterize completely the changes in polarization state which occur when light is reflected from a rough surface. When an experimentally determined Mueller matrix is available, parameters indicative of depolarization, cross polarization, and change in ellipticity can be found. When the Mueller matrix is regarded as an operator mapping input polarization states depicted on a Poincaré sphere to output states in a similar coordinate system, these new parameters can be illustrated in terms of their effects on the Poincaré sphere. The depolarization and cross polarization parameters correlate with specimen roughness and reflectance.

© 1986 Optical Society of America

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References

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  1. A complete overview of ellipsometry is contained in R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. P. S. Hauge, “Mueller Matrix Ellipsometry with Imperfect Compensators,” J. Opt. Soc. Am. 68, 1519 (1978).
    [CrossRef]
  3. R. M. A. Azzam, “Photopolarimetric Measurement of the Mueller Matrix by Fourier Analysis of a Single Detected Signal,” Opt. Lett. 5, 148 (1978).
    [CrossRef]
  4. D. E. Aspnes, “Fourier Transform Detection System for Rotating Analyzer Ellipsometers,” Opt. Commun. 8, 222 (1973).
    [CrossRef]
  5. D. A. Ramsey, “Mueller Matrix Ellipsometry Involving Extremely Rough Surfaces,” Doctoral Dissertation, U. Michigan, Ann Arbor (1985).
  6. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
    [CrossRef]
  7. P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
    [CrossRef]
  8. Ref. 1, p. 491.
  9. Introductory explanations of the Poincaré sphere are contained in the following three books: D. Clark, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971); W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, NJ, 1964); W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, MA, 1962).
  10. All the experimental Mueller matrices used in this paper were originally published in Ref. 5.

1980

P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

1978

P. S. Hauge, “Mueller Matrix Ellipsometry with Imperfect Compensators,” J. Opt. Soc. Am. 68, 1519 (1978).
[CrossRef]

R. M. A. Azzam, “Photopolarimetric Measurement of the Mueller Matrix by Fourier Analysis of a Single Detected Signal,” Opt. Lett. 5, 148 (1978).
[CrossRef]

1973

D. E. Aspnes, “Fourier Transform Detection System for Rotating Analyzer Ellipsometers,” Opt. Commun. 8, 222 (1973).
[CrossRef]

1969

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Fourier Transform Detection System for Rotating Analyzer Ellipsometers,” Opt. Commun. 8, 222 (1973).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Photopolarimetric Measurement of the Mueller Matrix by Fourier Analysis of a Single Detected Signal,” Opt. Lett. 5, 148 (1978).
[CrossRef]

A complete overview of ellipsometry is contained in R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

A complete overview of ellipsometry is contained in R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Clark, D.

Introductory explanations of the Poincaré sphere are contained in the following three books: D. Clark, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971); W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, NJ, 1964); W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, MA, 1962).

Grainger, J. F.

Introductory explanations of the Poincaré sphere are contained in the following three books: D. Clark, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971); W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, NJ, 1964); W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, MA, 1962).

Hauge, P. S.

P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

P. S. Hauge, “Mueller Matrix Ellipsometry with Imperfect Compensators,” J. Opt. Soc. Am. 68, 1519 (1978).
[CrossRef]

Muller, R. H.

P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
[CrossRef]

Ramsey, D. A.

D. A. Ramsey, “Mueller Matrix Ellipsometry Involving Extremely Rough Surfaces,” Doctoral Dissertation, U. Michigan, Ann Arbor (1985).

Smith, C. G.

P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

D. E. Aspnes, “Fourier Transform Detection System for Rotating Analyzer Ellipsometers,” Opt. Commun. 8, 222 (1973).
[CrossRef]

Opt. Lett.

R. M. A. Azzam, “Photopolarimetric Measurement of the Mueller Matrix by Fourier Analysis of a Single Detected Signal,” Opt. Lett. 5, 148 (1978).
[CrossRef]

Surf. Sci.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
[CrossRef]

P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

Other

Ref. 1, p. 491.

Introductory explanations of the Poincaré sphere are contained in the following three books: D. Clark, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971); W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, NJ, 1964); W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, MA, 1962).

All the experimental Mueller matrices used in this paper were originally published in Ref. 5.

A complete overview of ellipsometry is contained in R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

D. A. Ramsey, “Mueller Matrix Ellipsometry Involving Extremely Rough Surfaces,” Doctoral Dissertation, U. Michigan, Ann Arbor (1985).

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Figures (11)

Fig. 1
Fig. 1

Horizontal specimen surface and vertical plane of incidence showing electrical vectors Ep and Es of the p and s waves.

Fig. 2
Fig. 2

Poincaré sphere showing polarization states P (the p wave, linearly polarized vertically) and R and L (right and left circularly polarized light).

Fig. 3
Fig. 3

Poincaré sphere illustrating the effect of the Δ matrix which causes rotation of the sphere about the S(1) axis by an angle Δ and the Ψ matrix, which moves points along a meridian either nearer to or farther from the pole {1,1,0,0}. Drawn for Ψ = 30° and Δ = 60°.

Fig. 4
Fig. 4

Poincaré surface which results from operation of Matrix 4 on the Poincaré sphere in Fig. 2 representing input states. Those lines originally in the first octant are shown as solid lines.

Fig. 5
Fig. 5

Cross section of Poincaré surface showing the point P′ resulting from transformation of point P, and point R′, from transformation of point R and their respective distances from the origin L P and L R .

Fig. 6
Fig. 6

Transformed point P′ showing angles α and β.

Fig. 7
Fig. 7

Poincaré surface resulting from application of Matrix 8 shown for comparison with Fig. 4.

Fig. 8
Fig. 8

(a) Polarization fraction P1, (b) cross-polarization parameter α; and (c) ellipticity parameter β, plotted as functions of surface roughness for four different angles of incidence ϕ.

Fig. 9
Fig. 9

Ellipticity parameter β increases with cross-polarization parameter α.

Fig. 10
Fig. 10

Cross polarization is the greatest in low reflectance specimens.

Fig. 11
Fig. 11

Greatest asymmetry in depolarization is associated with the most depolarization.

Tables (1)

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Table I Parameters (Defined in Text) used to Generate Matrix (8)

Equations (20)

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S ( 1 ) = ( k 2 - 1 ) / ( k 2 + 1 ) .
tan ( Ψ ) = ( E p / E p ) / ( E s / E s ) .
k = E p / E s = k tan ( Ψ ) .
A 1 = ( L s - L p ) / ( L s + L p ) ;
P 1 = 2 L p L s / ( L s + L p ) .
Ψ = arctan { [ 1 + cos ( A ) ] / [ 1 - cos ( A ) ] } 1 / 2
[ L R cos ( A ) ] 2 ( L P ) 2 + [ L R sin ( A ) ] 2 ( D R ) 2 = 1.
A 2 = ( D - 45 - D + 45 ) / ( D - 45 + D + 45 ) ;
P 2 = 2 D - 45 D + 45 ) / ( D - 45 + D + 45 )
A 3 = ( D L - D R ) / ( D L + D R ) ,
P 3 = 2 D L D R ) / ( D L + D R ) .
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 )
1 0 0 0 0 1 0 0 0 0 cos ( Δ ) sin ( Δ ) 0 0 - sin ( Δ ) cos ( Δ )
1 - cos ( 2 Ψ ) 0 0 - cos ( 2 Ψ ) 1 0 0 0 0 sin ( 2 Ψ ) 0 0 0 0 sin ( 2 Ψ )
1.0000 0.1631 - 0.0322 0.0802 0.0083 0.4038 0.2555 - 0.2158 - 0.0026 0.4297 - 0.1376 0.2016 - 0.0116 0.0597 - 0.3175 - 0.3690
1 0 0 0 0 cos ( β ) 0 - sin ( β ) 0 0 1 0 0 sin ( β ) 0 cos ( β )
1 0 0 0 0 cos ( α ) - sin ( α ) 0 0 sin ( α ) cos ( α ) 0 0 0 0 1
1 A 1 A 2 A 3 0 P 1 0 0 0 0 P 2 0 0 0 0 P 3
1.0000 0.1633 - 0.0655 0.0725 0.0018 0.4042 0.2324 - 0.2324 0.0019 0.4302 - 0.1745 0.2624 0.0003 0.0598 - 0.3149 - 0.3170
M ( 0 , 0 ) = 1 , M ( 0 , 1 ) = [ A 1 - cos ( 2 Ψ ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 0 , 2 ) = [ A 2 - sin ( 2 Ψ ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 0 , 3 ) = [ A 3 sin ( 2 Ψ ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 1 , 0 ) = - P 1 cos ( 2 Ψ ) cos ( β ) cos ( α ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 1 , 1 ) = P 1 cos ( β ) cos ( α ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 1 , 2 ) = [ P 2 sin ( 2 Ψ ) sin ( Δ ) sin ( β ) cos ( α ) - P 2 sin ( 2 Ψ ) cos ( Δ ) sin ( α ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 1 , 3 ) = - [ P 3 sin ( 2 Ψ ) cos ( Δ ) sin ( β ) cos ( α ) + P 3 sin ( 2 Ψ ) sin ( Δ ) sin ( α ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 2 , 0 ) = - P 1 cos ( 2 Ψ ) cos ( β ) sin ( α ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 2 , 1 ) = P 1 cos ( β ) sin ( α ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 2 , 2 ) = [ P 2 sin ( 2 Ψ ) sin ( Δ ) sin ( β ) cos ( α ) + P 2 sin ( 2 Ψ ) cos ( α ) cos ( Δ ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 2 , 3 ) = - [ P 3 sin ( 2 Ψ ) cos ( Δ ) sin ( β ) sin ( α ) + P 3 sin ( 2 Ψ ) sin ( Δ ) cos ( α ) ] / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 3 , 0 ) = - P 1 cos ( 2 Ψ ) sin ( β ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 3 , 1 ) = P 1 sin ( β ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 3 , 2 ) = - P 2 sin ( 2 Ψ ) sin ( Δ ) cos ( β ) / [ 1 - A 1 cos ( 2 Ψ ) ] , M ( 3 , 3 ) = P 3 sin ( 2 Ψ ) cos ( Δ ) cos ( β ) / [ 1 - A 1 cos ( 2 Ψ ) ] .

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