Abstract

The two ellipsometric parameters of an isotropic sample can be measured with simplified polarimetry setups that acquire at least four intensity measurements. However, these measurements are perturbed by noise and the measurement strategy has to be optimized, in order to limit noise propagation. We determine two different measurement strategies that are optimal for both white Gaussian additive noise and Poisson shot noise. The first one involves a polarization state generator (PSG) with a single state of polarization and a polarization state analyzer (PSA) with four states. The second one involves both PSG and PSA having two states. The total estimation variances obtained with both strategies are demonstrated to be minimal, of equal values, and independent of the ellipsometric parameters to be measured. They are based on simple optical elements and could simplify and accelerate ellipsometric measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2017 (2)

2016 (5)

2015 (2)

2013 (1)

2012 (2)

2010 (2)

2009 (1)

2006 (1)

1994 (1)

1993 (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

1975 (1)

Aiello, A.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Phys. Rev. Lett. 115(26), 263901 (2015).
[Crossref] [PubMed]

Alali, S.

Anna, G.

Aspnes, D. E.

Azzam, R. M. A.

Bénière, A.

Boffety, M.

Chen, X.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

Dolfi, D.

Du, B.

Duan, Q. Y.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Favaro, A.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Phys. Rev. Lett. 115(26), 263901 (2015).
[Crossref] [PubMed]

Foreman, M. R.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Phys. Rev. Lett. 115(26), 263901 (2015).
[Crossref] [PubMed]

Goudail, F.

Gribble, A.

Gu, H.

Gupta, V. K.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Hagen, N.

Hoover, B. G.

Hu, H.

Huang, B.

Jiang, H.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

Johnson, S. J.

Krishnan, S.

Li, W.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

Li, X.

Liu, H.

Liu, S.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

Liu, T.

Nordine, P. C.

Ossikovski, R.

Otani, Y.

Roussel, S.

Sauer, H.

Song, Z.

Sorooshian, S.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Studna, A. A.

Tyo, J. S.

Vitkin, I. A.

Wang, H.

X. Li, H. Hu, H. Wang, L. Wu, and T. Liu, “Influence of noise statistics on optimizing the distribution of integration time for degree of linear polarization polarimetry,” Opt. Eng. 57(6), 1 (2018).
[Crossref]

Wang, K.

Wang, Z.

Wei, H.

Wu, L.

X. Li, H. Hu, H. Wang, L. Wu, and T. Liu, “Influence of noise statistics on optimizing the distribution of integration time for degree of linear polarization polarimetry,” Opt. Eng. 57(6), 1 (2018).
[Crossref]

X. Li, H. Hu, L. Wu, and T. Liu, “Optimization of instrument matrix for Mueller matrix ellipsometry based on partial elements analysis of the Mueller matrix,” Opt. Express 25(16), 18872–18884 (2017).
[Crossref] [PubMed]

Zhang, C.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

Zhu, J.

Appl. Opt. (8)

J. Opt. (1)

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Optim. Theory Appl. (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Opt. Eng. (1)

X. Li, H. Hu, H. Wang, L. Wu, and T. Liu, “Influence of noise statistics on optimizing the distribution of integration time for degree of linear polarization polarimetry,” Opt. Eng. 57(6), 1 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Phys. Rev. Lett. 115(26), 263901 (2015).
[Crossref] [PubMed]

Other (9)

D. Goldstein, Polarized Light (Dekker, 2003).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

H. Tompkins and E. Irene, Handbook of Ellipsometry (William Andrew, 2005).

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, 1982).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science Publishing, 1987).

E. Garcia-Caurel, R. Ossikovski, M. Foldyna, A. Pierangelo, B. Drévillon, and A. D. Martino, “Advanced Mueller ellipsometry instrumentation and data analysis,” in Ellipsometry at the Nanoscale, M. Losurdo and K. Hingerl, ed. (Springer, 2013), Chap. 2.

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Figures (3)

Fig. 1
Fig. 1 Typical system configuration for complete Mueller matrix polarimetry.
Fig. 2
Fig. 2 Optimal system configuration for polarimetry of “1×4 type”.
Fig. 3
Fig. 3 Optimal system configuration for polarimetry of “2×2 type”.

Tables (3)

Tables Icon

Table 1 Optimal parameters and eigenstate vectors of PSG/PSA for “1×4 type”.

Tables Icon

Table 2 Optimal parameters and eigenstate vectors of PSG/PSA for “2×2 type”.

Tables Icon

Table 3 The performance for the two types of optimal strategies in the presence of white Gaussian additive noise and Poisson shot noise.

Equations (54)

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ρtanψexp( iΔ ) r p r s .
M=r[ 1 a 0 0 a 1 0 0 0 0 b c 0 0 -c b ],
S( α S , ε S )= [ 1, s 1 , s 2 , s 3 ] T = [ 1, cos( 2 α S )cos( 2 ε S ), sin( 2 α S )cos( 2 ε S ), sin( 2 ε S ) ] T , T i ( α i T , ε i T )= [ 1, t i1 , t i 2 , t i 3 ] T = [ 1, cos( 2 α i T )cos( 2 ε i T ), sin( 2 α i T )cos( 2 ε i T ), sin( 2 ε i T ) ] T ,
I i = 1 2 T i T MS, i={ 1,2,3,4 },
I i = 1 2 A i T V M I= 1 2 Q V M , i{ 1,2,3,4 },
V M Ω = [ r, ra, ra, r, rb, rc, rc, rd ] T .
I= 1 2 Q Ω V M Ω ,
I= 1 2 [ 1 t 11 s 1 s 1 t 11 1 t 21 s 1 s 1 t 21 1 t 31 s 1 s 1 t 31 1 t 41 s 1 s 1 t 41 s 2 t 12 s 2 t 13 s 3 t 12 s 3 t 13 s 2 t 22 s 2 t 23 s 3 t 22 s 3 t 23 s 2 t 32 s 2 t 33 s 3 t 32 s 3 t 33 s 2 t 42 s 2 t 43 s 3 t 42 s 3 t 43 ][ r ra ra r rb rc rc rb ],
I= 1 2 [ 1+ s 1 t 11 t 11 + s 1 s 2 t 12 + s 3 t 13 s 3 t 12 s 2 t 13 1+ s 1 t 21 t 21 + s 1 s 2 t 22 + s 3 t 23 s 3 t 22 s 2 t 23 1+ s 1 t 31 t 31 + s 1 s 2 t 32 + s 3 t 33 s 3 t 32 s 2 t 33 1+ s 1 t 41 t 41 + s 1 s 2 t 42 + s 3 t 43 s 3 t 42 s 2 t 43 ][ r ra rb rc ].
W=[ W 11 W 12 W 13 W 14 W 21 W 22 W 23 W 24 W 31 W 32 W 33 W 34 W 41 W 42 W 43 W 44 ]= 1 2 [ 1+ s 1 t 11 t 11 + s 1 s 2 t 12 + s 3 t 13 s 3 t 12 s 2 t 13 1+ s 1 t 21 t 21 + s 1 s 2 t 22 + s 3 t 23 s 3 t 22 s 2 t 23 1+ s 1 t 31 t 31 + s 1 s 2 t 32 + s 3 t 33 s 3 t 32 s 2 t 33 1+ s 1 t 41 t 41 + s 1 s 2 t 42 + s 3 t 43 s 3 t 42 s 2 t 43 ].
I=W V M 4 .
W i1 2 = W i2 2 + W i3 2 + W i4 2 , i{ 1,2,3,4 }.
V ^ M 4 = W 1 I.
ψ= 1 2 cos 1 [ a ]= 1 2 cos 1 { [ V ^ M 4 ] 2 [ V ^ M 4 ] 1 }, Δ= tan 1 [ c b ]= tan 1 { [ V ^ M 4 ] 4 [ V ^ M 4 ] 3 },
Γ V ^ M 4 = W 1 Γ I ( W 1 ) T ,
W opt =arg min W { i=1 4 γ i }.
Γ V ^ M 4 =[ W 1 ( W 1 ) T ] σ 2 ,
γ i = [ Γ V ^ M 4 ] ii = [ W 1 ( W 1 ) T ] ii σ 2 .
i=1 4 γ i = σ 2 trace[ W 1 ( W 1 ) T ],
W opt Gau =arg min W { trace[ W 1 ( W 1 ) T ] }.
γ 1 = σ 2 , γ 2 =3 σ 2 , γ 3 =3 σ 2 , γ 4 =3 σ 2 ,
[ Γ I ] ii = I i = k=1 4 W ik [ V M 4 ] k , i{ 1,2,3,4 },
γ i = j=1 4 [ V M 4 ] j k=1 4 [ W ik 1 ] 2 W kj , i{ 1,2,3,4 }.
F( W )= max V M 4 { i=1 4 γ i }= max V M 4 { j=1 4 u j [ V M 4 ] j },
F( W )= max u { r( u 1 +uv ) },
F( W )=r( u 1 + u ),
W opt Poi =arg min W F( W ).
u 1 = 1 2 i=1 4 k=1 4 [ W ik 1 ] 2 = 1 2 trace[ W 1 ( W 1 ) T ]= 1 2 F Gau ( W ),
F( W )=r( 1 2 F Gau ( W )+ u ),
W opt Poi = 1 2 [ 1 1/ 3 1/ 3 1/ 3 1 1/ 3 1/ 3 1/ 3 1 1/ 3 1/ 3 1/ 3 1 1/ 3 1/ 3 1/ 3 ],
γ 1 = 1 2 r, γ i = 3 2 r i{ 2,3,4 },
W 2×2 = 1 2 [ 1+ s 11 t 11 t 11 + s 11 s 12 t 12 + s 13 t 13 s 13 t 12 s 12 t 13 1+ s 11 t 21 t 21 + s 11 s 12 t 22 + s 13 t 23 s 13 t 22 s 12 t 23 1+ s 21 t 11 t 11 + s 21 s 22 t 12 + s 23 t 13 s 23 t 12 s 22 t 13 1+ s 21 t 21 t 21 + s 21 s 22 t 22 + s 23 t 23 s 23 t 22 s 22 t 23 ],
W 2×2opt Gau =arg min W 2×2 { trace[ W 2×2 1 ( W 2×2 1 ) T ] }.
γ 1 = σ 2 , γ 2 = γ 3 = γ 4 =3 σ 2 , i=1 4 γ i =10 σ 2 ,
W 2×2opt Poi =arg min W 2×2 { r( u _ 1 + u _ ) }.
VAR[ I ]=η σ Poi 2 + η 2 σ Gau 2 ,
W i2 2 + W i3 2 + W i4 2 = 1 4 [ ( t i1 + s 1 ) 2 + ( s 2 t i2 + s 3 t i3 ) 2 + ( s 3 t i2 s 2 t i3 ) 2 ] = 1 4 [ t i1 2 + s 1 2 +2 t i1 s 1 + s 2 2 t i2 2 + s 3 2 t i3 2 + s 3 2 t i2 2 + s 2 2 t i3 2 ] = 1 4 [ t i1 2 + s 1 2 +2 t i1 s 1 +( s 2 2 + s 3 2 )( t i2 2 + t i3 2 ) ].
W i2 2 + W i3 2 + W i4 2 = 1 4 [ t i1 2 + s 1 2 +2 t i1 s 1 +( 1 s 1 2 )( 1 t i1 2 ) ] = 1 4 [ 1+2 t i1 s 1 + s 1 2 t i1 2 ] = 1 4 ( t i1 + s 1 ) 2 = W i1 2 .
W=TQ= 1 2 [ 1 t 11 t 12 t 13 1 t 21 t 22 t 23 1 t 31 t 32 t 33 1 t 41 t 42 t 43 ][ 1 s 1 s 1 1 s 2 s 3 s 3 s 2 ].
trace [ ( TQ ) T ( TQ ) ] 1 =trace[ ( T T T ) 1 ( Q T Q ) 1 ].
F( A,B )=trace( A 1 B 1 ),
( A,B )=arg min A,B { F( A,B ) },
trace( A )=2 and s =1,
trace( A 1 B 1 )trace( A 1 )trace( B 1 ).
( A,B )=arg min A,B { G( A,B ) } =arg min A { trace( A 1 ) }arg min B { trace( B 1 ) }.
A 0 1 =arg min A { trace( A 1 ) }=[ 1 3 3 3 ],
trace( B 1 )= 4 ( 1 s 1 2 ) 2 ,
B 0 1 =arg min B { trace( B 1 ) }=[ 1 1 1 1 ].
trace( A 0 1 B 0 1 )=trace( A 0 1 )trace( B 0 1 ).
G( X 0 ) X λ g( X 0 ) A μ g( X 0 ) B =0,
F( X 0 +δX )=F( X 0 )+ ( F X ) T ( X 0 )δX+ o 1 ( δX ), G( X 0 +δX )=G( X 0 )+ ( G X ) T ( X 0 )δX+ o 2 ( δX ),
Δ( δX )=G( X+δX )F( X+δX ),
Δ( δX )= ( G X ) T ( X 0 )δX ( F X ) T ( X 0 )δX+o( δX ) = [ λ g( X 0 ) A +μ g( X 0 ) B F X ( X 0 ) ] T δX+o( δX ),
F X ( X 0 )λ g( X 0 ) A μ g( X 0 ) B =0.

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