Abstract

We investigate the systematic errors at the second order for a Mueller matrix ellipsometer in the dual rotating compensator configuration. Starting from a general formalism, we derive explicit second- order errors in the Mueller matrix coefficients of a given sample. We present the errors caused by the azimuthal inaccuracy of the optical components and their influences on the measurements. We demonstrate that the methods based on four-zone or two-zone averaging measurement are effective to vanish the errors due to the compensators. For the other elements, it is shown that the systematic errors at the second order can be canceled only for some coefficients of the Mueller matrix. The calibration step for the analyzer and the polarizer is developed. This important step is necessary to avoid the azimuthal inaccuracy in such elements. Numerical simulations and experimental measurements are presented and discussed.

© 2010 Optical Society of America

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References

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  1. R. W. Collins, “Dual rotating compensator” in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew Publishing, 2005), Chap. 7.3.3, pp. 546–566.
  2. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [CrossRef] [PubMed]
  3. M. H. Smith, “Optimization of dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488–2493 (2002).
    [CrossRef] [PubMed]
  4. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
    [CrossRef]
  5. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeter in the presence of error sources,” Opt. Express 16, 11589–11603 (2008).
    [CrossRef] [PubMed]
  6. M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Systematic errors specific to a snapshot Mueller matrix polarimeter,” Appl. Opt. 48, 1135–1142 (2009).
    [CrossRef]
  7. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  8. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  9. E. Bahar, “Road maps for use of Mueller matrix measurements to detect and identify biological and chemical materials through their optical activity: potential applications in biomedicine, biochemistry, security and industry,” J. Opt. Soc. Am. B 26, 364–370 (2009).
    [CrossRef]
  10. L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16, 8814–8824 (2008).
    [CrossRef] [PubMed]
  11. G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Phys. Status Solidi C 5, 1027–1030(2008).
    [CrossRef]
  12. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
    [CrossRef]
  13. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
    [CrossRef]

2009 (3)

2008 (3)

2006 (1)

2002 (1)

1999 (1)

1997 (1)

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

1996 (1)

1990 (1)

Bahar, E.

Broch, L.

L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16, 8814–8824 (2008).
[CrossRef] [PubMed]

G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Phys. Status Solidi C 5, 1027–1030(2008).
[CrossRef]

Cariou, J.

Chenault, D. B.

Chipman, R. A.

Collins, R. W.

R. W. Collins, “Dual rotating compensator” in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew Publishing, 2005), Chap. 7.3.3, pp. 546–566.

Compain, E.

Drevillon, B.

Dubreuil, M.

En Naciri, A.

Goldstein, D. H.

Goldstein, D. L.

Johann, L.

L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16, 8814–8824 (2008).
[CrossRef] [PubMed]

G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Phys. Status Solidi C 5, 1027–1030(2008).
[CrossRef]

Le Jeune, B.

M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Systematic errors specific to a snapshot Mueller matrix polarimeter,” Appl. Opt. 48, 1135–1142 (2009).
[CrossRef]

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Lu, S.-Y.

Ossikovski, R.

Piller, G.

G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Phys. Status Solidi C 5, 1027–1030(2008).
[CrossRef]

Poirier, S.

Rivet, S.

Shaw, J. A.

Smith, M. H.

Twietmeyer, K. M.

Tyo, J. S.

Appl. Opt. (4)

J. Opt. Soc. Am. A (3)

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

Opt. Express (2)

Phys. Status Solidi C (1)

G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Phys. Status Solidi C 5, 1027–1030(2008).
[CrossRef]

Prog. Quantum Electron. (1)

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Other (1)

R. W. Collins, “Dual rotating compensator” in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew Publishing, 2005), Chap. 7.3.3, pp. 546–566.

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Figures (5)

Fig. 1
Fig. 1

Diagram of the MME in P.C1.Sample.C2.A arrangement.

Fig. 2
Fig. 2

Experimental calibration (square points) and fit (solid curve) of the polarizer with an isotropic 106 nm Si O 2 / Si with angle of incidence θ = 70 ° and wavelength λ = 633 nm ( Ψ = 47 ° and Δ = 79 ° ).

Fig. 3
Fig. 3

Simulation of H P as a function of the angle Δ of the sample used. The white curve represents the best conditions, where Δ = 90 ° .

Fig. 4
Fig. 4

Simulation of H P as a function of the angle Ψ of the sample used. The best conditions are achieved for Ψ 60 ° .

Fig. 5
Fig. 5

Systematic errors due to a bad position of the analyzer. The calculated curves (solid curves, dashed curves, and dotted curves) use Table 4. The measurements are performed at A = P = 0 ° (× exes) for all the coefficients, A = 90 ° ; P = 0 ° (+ pluses) for M 13 and M 14 and A = 0 ° ; P = 90 ° (* asterisks) for M 41 with C S 1 = 0.69 ° and C S 2 = 1.66 ° .

Tables (5)

Tables Icon

Table 1 Second-Order Errors Due to Azimuthal Mispositioning of Element

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Table 2 Calculated Elements of Matrix δ 2 M [Eq. (17)] for the First Compensator a

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Table 3 Calculated Elements of Matrix δ 2 M [Eq. (17)] for the Second Compensator a

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Table 4 Systematic Errors in Mueller Matrix Ellipsometer for Isotropic Sample if C S 1 = C S 2 = 0 ° and δ 1 = δ 2 = 90 ° when the Four-Zone Averaging Measurement Method is Performed

Tables Icon

Table 5 Statistical Study of the Influence of Element Mispositioning on a Mueller Matrix if C S 1 = C S 2 = 0 ° and δ 1 = δ 2 = 90 ° a

Equations (41)

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M = [ M T L M T R M B L M B R ] ,
S f = [ M A . R ( A ) ] . [ R 1 ( C 2 ) . M C 2 . R ( C 2 ) ] . M . [ R 1 ( C 1 ) . M C 1 . R ( C 1 ) ] . [ R 1 ( P ) . M P ] . S i ,
I = I 0 [ a 0 + n ( a 2 n cos 2 n C + b 2 n sin 2 n C ) ] ,
( M i j ) sample = 1 4 A ; A + π 2 P ; P + π 2 ( M i j ) measured .
H P = a 2 m 2 2 + b 2 m 2 2 a 4 m 2 2 + b 4 m 2 2 ,
a 2 m 2 = c 1 sin δ 2 sin 2 Ψ sin Δ sin 2 A sin 2 P ,
b 2 m 2 = c 1 sin δ 2 sin 2 Ψ sin Δ cos 2 A sin 2 P ,
a 4 m 2 = c 1 s 2 ( cos 2 P cos 2 A sin 2 Ψ cos Δ sin 2 P sin 2 A ) + s 2 cos 2 Ψ cos 2 A ,
b 4 m 2 = c 1 s 2 ( cos 2 P sin 2 A + sin 2 Ψ cos Δ sin 2 P cos 2 A ) + s 2 cos 2 Ψ sin 2 A ,
H A = a 2 m 1 2 + b 2 m 1 2 a 4 m 1 2 + b 4 m 1 2 ,
a 2 m 1 = c 2 sin δ 1 sin 2 Ψ sin Δ sin 2 P sin 2 A ,
b 2 m 1 = c 2 sin δ 1 sin 2 Ψ sin Δ cos 2 P sin 2 A ,
a 4 m 1 = s 1 c 2 ( cos 2 A cos 2 P sin 2 Ψ cos Δ sin 2 A sin 2 P ) + s 1 cos 2 Ψ cos 2 P ,
b 4 m 1 = s 1 c 2 ( cos 2 A sin 2 P + sin 2 Ψ cos Δ sin 2 A cos 2 P ) + s 1 cos 2 Ψ sin 2 P ,
M = M 0 + δ M = M 0 + M x δ x + 1 2 2 M x 2 δ x 2 + ,
δ 2 R ( Θ ) = 4 ( 0 0 0 0 0 cos 2 Θ sin 2 Θ 0 0 sin 2 Θ cos 2 Θ 0 0 0 0 0 ) δ Θ 2 ,
δ M = 2 ( 0 M 13 δ P M 12 δ P 2 M 12 δ P M 13 δ P 2 M 14 δ P 2 0 M 23 δ P M 22 δ P 2 M 22 δ P M 23 δ P 2 M 34 δ P M 24 δ P 2 0 M 33 δ P M 32 δ P 2 M 32 δ P M 33 δ P 2 M 24 δ P M 34 δ P 2 0 M 43 δ P M 42 δ P 2 M 42 δ P M 43 δ P 2 M 44 δ P 2 ) .
1 2 2 M C S i 2 δ C S i 2 = 8 ( 0 δ 2 M 12 δ 2 M 13 δ 2 M 14 δ 2 M 21 δ 2 M 22 δ 2 M 23 δ 2 M 24 δ 2 M 31 δ 2 M 32 δ 2 M 33 δ 2 M 34 δ 2 M 41 δ 2 M 42 δ 2 M 43 δ 2 M 44 ) δ C S i 2 .
f ( x , y , ρ ) = x sin 2 ρ + y cos 2 ρ ,
δ 2 M C i = ( 0 0 0 0 0 0 0 0 0 0 cos δ i sin δ i 0 0 sin δ i cos δ i ) δ δ i 2 ,
1 2 2 M δ 1 2 δ δ 1 2 = ( 0 τ 1 c M 12 τ 1 c M 13 1 2 M 14 f ( τ 1 s M 23 , τ 1 c M 22 , P ) τ 1 c M 22 τ 1 c M 23 1 2 M 24 f ( τ 1 s M 33 , τ 1 c M 32 , P ) τ 1 c M 32 τ 1 c M 33 1 2 M 34 f ( τ 1 s M 43 , τ 1 c M 22 , P ) τ 1 c M 42 τ 1 c M 43 1 2 M 44 ) δ δ 1 2 ,
1 2 2 M δ 2 2 δ δ 2 2 = ( 0 f ( τ 2 s M 32 , τ 2 c M 22 , A ) f ( τ 2 s M 33 , τ 2 c M 23 , A ) f ( τ 2 s M 34 , τ 2 c M 24 , A ) τ 2 c M 21 τ 2 c M 22 τ 2 c M 23 τ 2 c M 24 τ 2 c M 31 τ 2 c M 32 τ 2 c M 33 τ 2 c M 34 1 2 M 41 1 2 M 42 1 2 M 43 1 2 M 44 ) δ δ 2 2 ,
M 0 = ( 1 N 0 0 N 1 0 0 0 0 C S 0 0 S C ) .
M = ( 1.0 0.0077 0.0247 0.0108 0.006 1.0023 0.0207 0.0196 0.0259 0.0259 1 , 0014 0.0151 0.0115 0.0229 0.0159 0.9857 ) .
δ M = 2 ( 0 0 2 c 2 C δ A 2 c 2 S δ A 0 N δ A 2 δ A 2 C δ A N δ A δ A C δ A 2 S δ A 2 c 1 S δ A S δ A S δ A 2 C δ A 2 ) .
2 ( 0 0 0 0 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 )
2 ( 0 M 12 M 13 M 14 0 M 22 M 23 M 24 0 M 32 M 33 M 34 0 M 42 M 43 M 44 )
2 ( 0 4 M 12 4 M 13 M 14 0 4 M 22 4 M 23 M 24 0 4 M 32 4 M 33 M 34 0 4 M 42 4 M 43 M 44 )
2 ( 0 0 0 0 4 M 21 4 M 22 4 M 23 4 M 24 4 M 31 4 M 32 4 M 33 4 M 34 M 41 M 42 M 43 M 44 )
2 ( 0 0 0 0 N ( δ A ) 2 ( δ A ) 2 C δ A S δ A N δ A δ A C ( δ A ) 2 S ( δ A ) 2 0 S δ A S ( δ A ) 2 C ( δ A ) 2 )
2 ( 0 N ( δ P ) 2 N δ P 0 0 ( δ P ) 2 δ P S δ P 0 C δ P C ( δ P ) 2 S ( δ P ) 2 0 S δ P S ( δ P ) 2 C ( δ P ) 2 )
2 ( 0 4 N ( δ C S 1 ) 2 2 N δ C S 1 0 0 4 ( δ C S 1 ) 2 2 δ C S 1 S δ C S 1 0 2 C δ C S 1 4 C ( δ C S 1 ) 2 S ( δ C S 1 ) 2 0 2 S δ C S 1 4 S ( δ C S 1 ) 2 C ( δ C S 1 ) 2 )
2 ( 0 0 0 0 4 N ( δ C S 2 ) 2 4 ( δ C S 2 ) 2 2 C δ C S 2 2 S δ C S 2 2 N δ C S 2 2 δ C S 2 4 C ( δ C S 2 ) 2 4 S ( δ C S 2 ) 2 0 S δ C S 2 S ( δ C S 2 ) 2 C ( δ C S 2 ) 2 )
( 0 N δ δ 1 0 0 0 δ δ 1 0 0 0 0 C δ δ 1 S ( δ δ 1 ) 2 / 2 0 0 S δ δ 1 C ( δ δ 1 ) 2 / 2 )
( 0 0 0 0 N δ δ 2 δ δ 2 0 0 0 0 C δ δ 2 S δ δ 2 0 0 S ( δ δ 2 ) 2 / 2 C ( δ δ 2 ) 2 / 2 )
( 0 0 ± 0.011 0 ± 0.030 0 0 ± 0.011 0.002 ± 0.015 0 ± 0.067 0 0 ± 0.021 0 ± 0.067 0.002 ± 0.015 0 0 0 0 0.001 ± 0.0004 )
( 0 0 0 0 0 0.002 ± 0.015 0 ± 0.067 0 0 0 ± 0.067 0.001 ± 0.015 0 0 0 0 0 ± 0.0004 )
( 0 0 ± 0.011 0 ± 0.007 0 ± 0.037 0 ± 0.011 0.001 ± 0.015 0 ± 0.048 0 ± 0.055 0 ± 0.007 0 ± 0.048 0 ± 0.003 0.001 ± 0.010 0 ± 0.033 0 ± 0.055 0.001 ± 0.010 0 ± 0 )
( 0 0 ± 0.001 0 ± 0.003 0 0 ± 0.001 0.002 ± 0.015 0 ± 0.048 0 ± 0.055 0 ± 0.003 0 ± 0.048 0 ± 0 0.002 ± 0.010 0 0 ± 0.055 0.002 ± 0.010 0 ± 0 )
( 0 0 ± 0.011 0 ± 0.030 0 ± 0.0006 0 ± 0.011 0.002 ± 0.015 0 ± 0.067 0 ± 0.001 0 ± 0.030 0 ± 0.067 0.002 ± 0.015 0 ± 0.001 0 ± 0.0005 0 ± 0.001 0 ± 0.001 0.001 ± 0.004 )
( 0 0 ± 0.001 0 ± 0.0004 0 ± 0 0 ± 0.001 0 ± 0.015 0 ± 0.067 0 ± 0.001 0 ± 0 0 ± 0.067 0.003 ± 0.015 0 ± 0.0011 0 ± 0 0 ± 0.001 0 ± 0.001 0.0015 ± 0.0006 )

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