Abstract

The characterization of anisotropic materials and complex systems by ellipsometry has pushed the design of instruments to require the measurement of the full reflection Mueller matrix of the sample with a great precision. Therefore Mueller matrix ellipsometers have emerged over the past twenty years. The values of some coefficients of the matrix can be very small and errors due to noise or systematic errors can induce distored analysis. We present a detailed characterization of the systematic errors for a Mueller Matrix Ellipsometer in the dual-rotating compensator configuration. Starting from a general formalism, we derive explicit first-order expressions for the errors on all the coefficients of the Mueller matrix of the sample. The errors caused by inaccuracy of the azimuthal arrangement of the optical components and residual ellipticity introduced by imperfect optical elements are shown. A new method based on a four-zone averaging measurement is proposed to vanish the systematic errors.

© 2008 Optical Society of America

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References

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  1. Q1. E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-1680 (1997).
    [CrossRef]
  2. E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
    [CrossRef]
  3. R. W. Collins, Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds., (William Andrew Publishing & Springer-Verlag, 2005), Chap. 7.3.3, p 546-566.
  4. G. E. JellisonJr, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
    [CrossRef]
  5. M. Kildemo, I. S. Nerbø, E. Søndergaard, L. Holt, I. Simonsen, and M. Stchakovsky, "Optical response of nanostructured GaSb," Physica Status Solidi (c), in press.
  6. J. S. Tyo, D. L. Golstein, D. B. Chenault, and J. A. Shaw, "Review of passive imaging polarimetry for remote sensing applications," Appl. Opt. 45, 5453-5469 (2006).
    [CrossRef] [PubMed]
  7. M. H. Smith, "Optimization of dual-rotating-retarder Mueller matrix polarimeter," Appl. Opt. 41, 2488-2493 (2002).
    [CrossRef] [PubMed]
  8. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  9. R. Kleim, L. Kuntzler and A. El Ghemmaz, "Systematic errors in rotating-compensator ellipsometry," J. Opt. Soc. Am. A 11, 2550-2558 (1994).
    [CrossRef]
  10. S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
    [CrossRef]
  11. A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
    [CrossRef]
  12. G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
    [CrossRef]
  13. R. M. A Azzam and N. M. Bashara, ellipsometry and Polarized Light (North-holland, Amsterdam, 1977).
  14. J. M. M. De Nijs and A. Van Siflhout, "Systematic and random errors in rotating-analyzer ellipsometry," J. Opt. Soc. Am. A 5, 773-781 (1988).
    [CrossRef]

2008 (1)

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
[CrossRef]

2006 (1)

2004 (1)

E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

2002 (2)

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

M. H. Smith, "Optimization of dual-rotating-retarder Mueller matrix polarimeter," Appl. Opt. 41, 2488-2493 (2002).
[CrossRef] [PubMed]

1998 (2)

G. E. JellisonJr, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

1997 (1)

Q1. E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-1680 (1997).
[CrossRef]

1994 (1)

1990 (1)

1988 (1)

Bertucci, S.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

Broch, L.

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
[CrossRef]

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

Chenault, D. B.

Chipman, R. A.

Compain, E.

Q1. E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-1680 (1997).
[CrossRef]

De Martino, A.

E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

De Nijs, J. M. M.

Drevillon, B.

E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

Q1. E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-1680 (1997).
[CrossRef]

El Ghemmaz, A.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

R. Kleim, L. Kuntzler and A. El Ghemmaz, "Systematic errors in rotating-compensator ellipsometry," J. Opt. Soc. Am. A 11, 2550-2558 (1994).
[CrossRef]

En Naciri, A.

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

Garcia-Caurel, E.

E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

Goldstein, D. H.

Golstein, D. L.

Jellison, G. E.

G. E. JellisonJr, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

Johann, L.

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
[CrossRef]

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

Kleim, R.

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

R. Kleim, L. Kuntzler and A. El Ghemmaz, "Systematic errors in rotating-compensator ellipsometry," J. Opt. Soc. Am. A 11, 2550-2558 (1994).
[CrossRef]

Kuntzler, L.

Nicolas, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

Pawlowski, A.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

Piller, G.

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
[CrossRef]

Shaw, J. A.

Smith, M. H.

Stein, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

Tyo, J. S.

Van Siflhout, A.

Appl. Opt. (2)

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

Physica Status Solidi (1)

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Physica Status Solidi (c)  5, 1027-1030 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

Q1. E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-1680 (1997).
[CrossRef]

Thin Solid Films (4)

E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

G. E. JellisonJr, "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, and R. Kleim, "Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry," Thin Solid Films 313-314, 73-78 (1998).
[CrossRef]

A. En Naciri, L. Broch, L. Johann, and R. Kleim, "Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing," Thin Solid Films 406, 103-112 (2002).
[CrossRef]

Other (3)

R. M. A Azzam and N. M. Bashara, ellipsometry and Polarized Light (North-holland, Amsterdam, 1977).

M. Kildemo, I. S. Nerbø, E. Søndergaard, L. Holt, I. Simonsen, and M. Stchakovsky, "Optical response of nanostructured GaSb," Physica Status Solidi (c), in press.

R. W. Collins, Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds., (William Andrew Publishing & Springer-Verlag, 2005), Chap. 7.3.3, p 546-566.

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Figures (2)

Fig. 1.
Fig. 1.

Diagram of the Mueller Matrix Ellipsometer in PC1SC2A arrangement.

Fig. 2.
Fig. 2.

δM 21 versus the azimuth of the compensators C S1 and C S2 for an isotropic sample with Ψ=46.14° and Δ=79.61° and for P=0 and δ 1=90°. The value of δM 21 ranges from -1 (black area), up to +1 (white area). The blue lines represents δM 21=0

Tables (3)

Tables Icon

Table 1. The calculated elements of the matrix δ M (Eq. 12) for the first compensator. The functions B 1=c 2[cos(2A+4C S2)-cos2A] and B 2=c 2[sin(2A+4C S2)-sin2A] are null if a two-zone measurement in A is performed independently the angle C S2.

Tables Icon

Table 2. The calculated elements of the matrix δ M (Eq. 12) for the second compensator. The functions B 3=c 1[cos(2P+4C S1)-cos2P] and B 4=c 1[sin(2P+4C S1)-sin2P] are null if a two-zone measurement in P is performed independently the angle C S1.

Tables Icon

Table 3. Systematic errors in MME if C S1=C S2=0° and δ 1=δ 2=90° when the four-zone averaging measurement method is performed.

Equations (23)

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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 ,
R ( Θ ) = ( 1 0 0 0 0 cos 2 Θ sin 2 Θ 0 0 sin 2 Θ cos 2 Θ 0 0 0 0 1 ) .
I = I 0 [ a 0 + n ( a 2 n cos 2 n C + b 2 n sin 2 n C ) ] ,
δ M ij = δ a T . J M ij ,
δ M = δ a T . M a = ( δ a T . M 11 a δ a T . M 12 a δ a T . M 13 a δ a T . M 14 a δ a T . M 21 a δ a T . M 22 a δ a T . M 23 a δ a T . M 24 a δ a T . M 31 a δ a T . M 32 a δ a T . M 33 a δ a T . M 34 a δ a T . M 41 a δ a T . M 42 a δ a T . M 43 a δ a T . M 44 a ) .
S f = S f 0 + k S f x k δ x k = S f 0 + k ( δ S f ) x k ,
f ( x , y , ρ ) = x sin 2 ρ + y cos 2 ρ ,
( δ M ij ) ρ + ( δ M ij ) ρ + π 2 = 0 .
δ R ( Θ ) = 2 ( 0 0 0 0 0 sin 2 Θ cos 2 Θ 0 0 cos 2 Θ sin 2 Θ 0 0 0 0 0 ) δ Θ ,
δ M = 2 .
( 0 2 c 2 f ( M 22 , M 32 , A ) 2 c 2 f ( M 23 , M 33 , A ) 2 c 2 f ( M 24 , M 34 , A ) M 31 M 32 M 33 M 34 M 21 M 22 M 23 M 24 c 1 f ( M 42 , M 43 , P ) M 43 M 42 0 ) δ A
δ M = 2 ( 0 M 13 M 12 c 2 f ( M 24 , M 34 , A ) 2 c 1 f ( M 22 , M 23 , P ) M 23 M 22 M 34 2 c 1 f ( M 32 , M 33 , P ) M 33 M 32 M 24 2 c 1 f ( M 42 , M 43 , P ) M 43 M 42 0 ) δ P .
δ M = 2 ( 0 δ M 12 δ M 13 δ M 14 δ M 21 δ M 22 δ M 23 δ M 24 δ M 31 δ M 32 δ M 33 δ M 34 δ M 41 δ M 42 δ M 43 δ M 44 ) δ C Si
δ M = 2 ( 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 ) γ x ,
δ M = 2 ( 0 M 42 cos δ 2 M 43 cos δ 2 M 44 cos δ 2 0 0 0 0 0 0 0 0 δ M 41 δ M 42 δ M 43 f ( M 24 , M 34 , A ) ) δ γ A ,
δ M = 2 ( 0 0 0 δ M 14 M 24 cos δ 1 0 0 f ( M 23 M 32 , M 22 + M 33 , P ) M 34 cos δ 1 0 0 f ( M 22 + M 33 , M 23 + M 32 , P ) M 44 cos δ 1 0 0 f ( M 42 , M 43 , P ) ) δ γ P
δ M C = 2 ( 0 0 0 0 0 0 sin δ C 1 cos δ C 0 sin δ C 0 0 0 1 cos δ C 0 0 ) γ C .
δ M = 2 sin δ 1 ( 0 0 0 c 2 2 c 1 f ( M 24 , M 34 , A ) f ( M 22 , M 23 , P ) 0 0 1 2 c 1 M 34 f ( M 32 , M 33 , P ) 0 0 1 2 c 1 M 24 f ( M 42 , M 43 , P ) 0 0 0 ) δ γ C 1
δ M = 2 sin δ 2
( 0 f ( M 22 , M 32 , A ) f ( M 23 , M 33 , A ) f ( M 24 , M 34 , A ) 0 0 0 0 0 0 0 0 c 1 2 c 2 f ( M 42 , M 43 , P ) 1 2 c 2 M 43 1 2 c 2 M 42 0 ) δ γ C 2
δ M = 1 t 1 ( 0 M 12 M 13 t 1 M 14 tan δ 1 f ( M 23 , M 22 , P ) M 22 M 23 t 1 M 24 tan δ 1 f ( M 33 , M 32 , P ) M 32 M 33 t 1 M 34 tan δ 1 f ( M 43 , M 42 , P ) M 42 M 43 t 1 M 44 tan δ 1 ) δ δ 1 ,
δ M = 1 t 2 ( 0 f ( M 32 , M 22 , A ) f ( M 33 , M 23 , A ) f ( M 34 , M 24 , A ) M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 t 2 M 41 tan δ 2 t 2 M 42 tan δ 2 t 2 M 43 tan δ 2 t 2 M 44 tan δ 2 ) δ δ 2 ,
( M ij ) sample = 1 4 A ; A + π 2 P ; P + π 2 ( M ij ) measured .

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