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Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators
Thin Solid Films 519 (2011) 2601–2603
Contents lists available at ScienceDirect
Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators Laurent Broch ⁎, Aotmane En Naciri, Luc Johann Laboratoire de Physique des Milieux Denses, Universite Paul Verlaine, Metz, 1 Boulevard Arago CP 87811, F-57078 METZ Cedex 3, France
a r t i c l e
i n f o
Available online 17 December 2010 Keywords: Systematic errors Mueller matrix ellipsometry
a b s t r a c t A dual rotating compensator ellipsometer based on the optical PC1SC2A configuration described by Collins [1, chap. 7.3] has been developed. The systematic errors for this configuration if the compensators are quarterwave plates have been already studied [2, 3, 4]. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In this condition random errors are optimized. The aim of this work is to used such retarders and verify if the systematic errors due to uncertainties of the optical elements (i.e. analyzer, polarizer, first and second compensators) are improved too. For each optical element in different configurations like single or 4-zone average measurements, the systematic errors are given and compared according to the compensators. It is demonstrated that using a 127° instead of quarter-wave retarders coupled with 4-zone averaging measurement is the best configuration for this instrument. These results were confirmed by a statistical study. © 2010 Elsevier B.V. All rights reserved.
Mueller matrix, M, is also related to the real Mueller matrix of the sample, M0 by:
1. Introduction The MME is based on the optical PC1SC2A arrangement described by Collins [1]. C1 and C2 are the synchronized rotating compensators. The polarizer and analyzer are characterized, respectively by the azimuthal angles P and A. The sample is denoted by S. The optical signal is the dot-product of the first row of the total Mueller matrices of the configuration with the input Stokes vector. The detected irradiance as the form I = I0 a0 + ∑ða2n cos2nC + b2n sin2nC Þ ; n
ð1Þ
where C = ωt is related to C1 and C2 by the relations C1 = ω1t = m1(C − CS1)ωt and C2 = ω2t = m2(C − CS2)ωt respectively. C1 and C2 are the angles of the fast axes of the first and second compensators at time t. The frequency of the two compensators is synchronized and related to the base mechanical frequency ω by the two integers m1 and m2. The fixed azimuth CS1 and CS2 are imposed by the mechanical assembly of the compensators and correspond to the azimuth of their fast axis with respect to the plane of incidence at the beginning of the acquisition. All errors, independently of their origins, propagate through the coefficients a2n and b2n to the normalized coefficients Mij of the matrix of the sample. These errors are introduced by the component imperfection, azimuth-angle errors and windows. The measured ⁎ Corresponding author. E-mail address: [email protected] (L. Broch). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.12.046
0
0
M = M + δM = M +
∂M 1 ∂2 M 2 δx + ::: δx + 2 ∂x2 ∂x
ð2Þ
where x is the perturbation, δM represents the systematic error matrix of the perturbation. The second and the third terms represent the first and the second order respectively. The systematic errors have been already studied [2–4]. The 4-zone averaging measurement has been developed and used to eliminate some systematic errors. This method is defined as: Mij
sample
=
1 ∑ 4 A;A+ π2
∑
P;P + π2
Mij
measured
:
ð3Þ
Usually a 4-zone averaging measurement is performed with A and P = ±45°. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In these conditions random errors are optimized. When applied to our configuration, we have detailed in the following sections if such retarders are adapted to minimize the systematic errors due to uncertainties of the analyzer, the polarizer and the two compensators.
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L. Broch et al. / Thin Solid Films 519 (2011) 2601–2603
2. Systematic errors due to the analyzer and the polarizer Errors to the first and the second orders due to these optical elements are presented on Table 1. This table shows that all coefficients of the Mueller matrix of the sample are not affected by the retardance of the compensators. For the analyzer, the errors on M12, M13, M14 and M41 are δ proportional to ci = cos2 i , where i = 1, 2 according to the compen2 sator considered. Thus, they are smaller if δi = 127∘ than δi = 90∘. These errors are eliminated if a zone averaging measurement is performed (i.e. f(x, y, 45∘) = − f(x, y, − 45∘)). For the polarizer, the errors on M14,M21,M31 and M41 are smaller if δi = 127∘ than δi = 90∘. These errors are eliminated if a zone averaging measurement is performed according to Eq. (3). The errors on the other coefficients are independent of δi. These conclusions are summaries in Tables 2 and 3. 127° as retarder is a better choice than the quarter-wave with a single measurement. Nevertheless, there is no difference between δi = 127∘ and δi = 90∘ if a 4-zone averaging measurement is performed. 3. Systematic errors due to the compensators Errors due to uncertainties on the azimuthal and the retardance of the compensators are detailed in previous papers [2,4] and presented on Tables 2 and 3 for single zone and 4-zone averaging measurements respectively. 3.1. Azimuthal error Errors on CS1: δM14 is dependent on the value of the second compensator. δM21, δM31 and δM41 are dependent on the value of the first compensator. Together, the errors are smaller if δi = 127∘ than δi = 90∘ (see Table 2). The other coefficients are not affected by the compensators. δM21 = δM31 = δM41 = 0 if the zone averaging measurement is performed on the analyzer independently of the value of the compensators (see Table 3). For δM14 a zone-averaging on the analyzer will eliminate the effect of the value of the compensator. But an error will still exist to the second order. Errors on CS2: M12, M13 and M14 are dependent on the value of the second compensator. M41 is dependent on the value of the first compensator. They are smaller if δi = 127∘ than 90∘ (see Table 2). M12, M13 and M14 will become null if a zone averaging on the polarizer is performed (see Table 3). An error on M41 will still exist to the second order if a zone averaging measurement on the polarizer is performed. As for the analyzer and the polarizer, 127° as retarder is the best configuration with a single measurement. But, there is no difference
Table 1 Systematic errors due to the analyzer and the polarizer. f(x, y, ρ) is defined as: f(x, y, ρ) = δ x sin 2ρ + y cos 2ρ and ci = cos2 i . 2 Coefficient
Analyzer mispositioning
Polarizer mispositioning
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
4c2f(− M22, M32, A)δA 4c2f(− M23, M33, A)δA 4c2f(− M24, M34, A)δA − 2M31δA − 2M21δA2 − 2M32δA − 2M22δA2 − 2M33δA − 2M23δA2 − 2M34δA − 2M24δA2 2M21δA − 2M31δA2 2M22δA − 2M32δA2 2M23δA − 2M33δA2 2M24δA − 2M34δA2 2c1f(M42, − M43, P)δA − 2M41δA2 2M43δA − 2M42δA2 − 2M42δA − 2M43δA2 − 2M44δA2
− 2M13δP 2M12δP 2c2f(M24, − M34, A)δP 4c1f(− M22, M23, P)δP − 2M21δP2 − 2M23δP − 2M22δP2 2M22δP − 2M23δP2 2M34δP − 2M24δP2 4c1f(− M32, M33, P)δP − 2M31δP2 − 2M33δP − 2M32δP2 2M32δP − 2M33δP2 − 2M24δP − 2M34δP2 4c1f(− M42, M43, P)δP − 2M41δP2 − 2M43δP − 2M42δP2 2M42δP − 2M43δP2 − 2M44δP2
Table 2 Best conditions for the compensators with a single zone measurement (A = P = 0∘). 0 → errors are null; and ≡ → errors are independent of δ1, δ2. First order
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
Second order
δA
δP
δCS1
δCS2
δδ1
δδ2
δA2
δP2
δC2S1
δC2S2
δδ21
δδ22
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 127° ≡ ≡ ≡
≡ ≡ 127° 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
≡ ≡ 127° 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 90° 127° 127° 127° 90° 127° 127° 127° 90° 127° 127° 127° 90°
127° 127° 127° 127° 127° 127° 127° 127° 127° 127° 127° 90° 90° 90° 90°
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
90° 90° ≡ 90° 90° 90° ≡ 90° 90° 90° ≡ 90° 90° 90° ≡
90° 90° 90° 90° 90° 90° 90° 90° 90° 90° 90° ≡ ≡ ≡ ≡
between δi = 127∘ or 90° if a 4-zone averaging measurement is performed. 3.2. Birefringence error Errors on δ1: with a single zone measurement (Table 2), four coefficients are in the best condition if δ1 = 90∘ to the first order, δ1 = 127∘ is better for the others. To the second order, all the coefficients are in the best condition if δ1 = 90∘ to the second order. With a 4-zone averaging measurement (Table 3), the same comments can be made except for δM21, δM31 and δM41 where the errors are null. Errors on δ2: we can make the same comments as for the other compensator. Only the four coefficients involved are different. For the birefringence error types, a simple conclusion cannot be made. Some errors are minimized with δi = 127∘ and others with δi = 90∘. The 4-zone averaging measurement eliminated only some errors. To avoid systematic errors it is necessary to know with high accuracy the birefringence of these elements. 4. Numerical study Numerical values of the errors can be evaluated for a perfect isotropic sample with Ψ = 47∘ and Δ = 79∘. We have simulated for all optical elements an error between − 1∘ and + 1∘ with a step of 0.1∘. The results are presented in two configurations according to the value Table 3 Best conditions for the compensators with a 4-zone averaging measurement (A = ± 45∘ and P = ± 45∘). 0 → errors are null; and ≡ → the errors are independent of δ1, δ2. First order
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
Second order
δA
δP
δCS1
δCS2
δδ1
δδ2
δA2
δP2
δC2S1
δC2S2
δδ21
δδ22
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ 0 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ 0 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 0 ≡ ≡ ≡
127° 127° 90° 0° 127° 127° 90° 0° 127° 127° 90° 0° 127° 127° 90°
0° 0° 0° 127° 127° 127° 127° 127° 127° 127° 127° 90° 90° 90° 90°
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
90° 90° ≡ 0° 90° 90° ≡ 0° 90° 90° ≡ 0° 90° 90° ≡
0° 0° 0° 90° 90° 90° 90° 90° 90° 90° 90° ≡ ≡ ≡ ≡
L. Broch et al. / Thin Solid Films 519 (2011) 2601–2603
2603
Table 4 Statistical study of the errors on the Mueller matrix of an isotropic sample (Ψ = 47 ∘ and Δ = 79 ∘ ) if A = P = CS1 = CS2 = 0∘. All errors have been simulated between − 1∘ to + 1∘ (step 0.1°). Configuration
Error (average error ± its standard deviation) 1 0 ≈0 0:011 0 0:007 0 0:037 C B ≈0 0:011 −0:001 0:015 0 0:048 0 0:055 C B @ ≈0 0:007 0 0:048 ≈−0 0:003 −0:001 0:010 A 0 0:033 0 0:055 0:001 0:010 ≈−0 0:0001 1 0 0 ≈0 0:005 0 0:004 0 0:015 C B ≈0 0:005 −0:001 0:007 0 0:048 0 0:055 C B @ ≈0 0:004 0 0:048 ≈−0 0:001 −0:001 0:009 A ≈−0 0:013 0 0:055 0:001 0:009 −0:001 0:002 0
δ1 =δ2 =90°
δ1 =δ2 =127°
of the compensators on Table 4. For each coefficient the average and its standard deviation are computed. All coefficients of the matrix are affected by the errors. The average errors are generally closed to zero and are identical if δi = 90∘ or δi = 127∘. The only exception is for M44 where the average error is greater (×10). The standard deviations are lower with 127∘ than 90∘ except for M23, M24, M32 and M42. For these coefficients the errors are unchanged. It is necessary to remember that the errors are not only based on the value of the compensators and the values, Mij, of the sample must be considered too. 5. Conclusion We have characterized the systematic error effects on the measurement of all the Mueller matrix coefficients of a sample for a double rotating compensator ellipsometer. Smith has shown that 127∘ retarders are the best value for optimized random errors in a DRR polarimeter. We have applied it and studied the influence of the
compensators on the acquired parameters. The study of the systematic errors shows that 127∘ retarders are more appropriate than the quarter-wave compensators. But, this is not the same conclusion for the errors δδi and δδ2i which some coefficients are minimized with a quarter wave retarder particularly for errors to the second order. This study demonstrates that using a 4-zone averaging measurement eliminates the influence of the compensator except for the errors due to δδi. Thus, the optimized measurements in ellipsometry are obtained with 127∘ retarders coupled with a 4-zone averaging method. References [1] R.W. Collins, Handbook of ellipsometry, William Andrew Publishing & SpringerVerlag, 2005. [2] L. Broch, A. En Naciri, L. Johann, Opt. Express 16 (12) (2008) 8814. [3] G. Piller, L. Broch, L. Johann, Phys. Status Solidi (c) 5 (5) (2008) 1027. [4] L. Broch, A. En Naciri, L. Johann, Appl. Opt. 49 (17) (2010) 3250. [5] M.H. Smith, Appl. Opt. 41 (2002) 2488.
Contents lists available at ScienceDirect
Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators Laurent Broch ⁎, Aotmane En Naciri, Luc Johann Laboratoire de Physique des Milieux Denses, Universite Paul Verlaine, Metz, 1 Boulevard Arago CP 87811, F-57078 METZ Cedex 3, France
a r t i c l e
i n f o
Available online 17 December 2010 Keywords: Systematic errors Mueller matrix ellipsometry
a b s t r a c t A dual rotating compensator ellipsometer based on the optical PC1SC2A configuration described by Collins [1, chap. 7.3] has been developed. The systematic errors for this configuration if the compensators are quarterwave plates have been already studied [2, 3, 4]. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In this condition random errors are optimized. The aim of this work is to used such retarders and verify if the systematic errors due to uncertainties of the optical elements (i.e. analyzer, polarizer, first and second compensators) are improved too. For each optical element in different configurations like single or 4-zone average measurements, the systematic errors are given and compared according to the compensators. It is demonstrated that using a 127° instead of quarter-wave retarders coupled with 4-zone averaging measurement is the best configuration for this instrument. These results were confirmed by a statistical study. © 2010 Elsevier B.V. All rights reserved.
Mueller matrix, M, is also related to the real Mueller matrix of the sample, M0 by:
1. Introduction The MME is based on the optical PC1SC2A arrangement described by Collins [1]. C1 and C2 are the synchronized rotating compensators. The polarizer and analyzer are characterized, respectively by the azimuthal angles P and A. The sample is denoted by S. The optical signal is the dot-product of the first row of the total Mueller matrices of the configuration with the input Stokes vector. The detected irradiance as the form I = I0 a0 + ∑ða2n cos2nC + b2n sin2nC Þ ; n
ð1Þ
where C = ωt is related to C1 and C2 by the relations C1 = ω1t = m1(C − CS1)ωt and C2 = ω2t = m2(C − CS2)ωt respectively. C1 and C2 are the angles of the fast axes of the first and second compensators at time t. The frequency of the two compensators is synchronized and related to the base mechanical frequency ω by the two integers m1 and m2. The fixed azimuth CS1 and CS2 are imposed by the mechanical assembly of the compensators and correspond to the azimuth of their fast axis with respect to the plane of incidence at the beginning of the acquisition. All errors, independently of their origins, propagate through the coefficients a2n and b2n to the normalized coefficients Mij of the matrix of the sample. These errors are introduced by the component imperfection, azimuth-angle errors and windows. The measured ⁎ Corresponding author. E-mail address: [email protected] (L. Broch). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.12.046
0
0
M = M + δM = M +
∂M 1 ∂2 M 2 δx + ::: δx + 2 ∂x2 ∂x
ð2Þ
where x is the perturbation, δM represents the systematic error matrix of the perturbation. The second and the third terms represent the first and the second order respectively. The systematic errors have been already studied [2–4]. The 4-zone averaging measurement has been developed and used to eliminate some systematic errors. This method is defined as: Mij
sample
=
1 ∑ 4 A;A+ π2
∑
P;P + π2
Mij
measured
:
ð3Þ
Usually a 4-zone averaging measurement is performed with A and P = ±45°. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In these conditions random errors are optimized. When applied to our configuration, we have detailed in the following sections if such retarders are adapted to minimize the systematic errors due to uncertainties of the analyzer, the polarizer and the two compensators.
2602
L. Broch et al. / Thin Solid Films 519 (2011) 2601–2603
2. Systematic errors due to the analyzer and the polarizer Errors to the first and the second orders due to these optical elements are presented on Table 1. This table shows that all coefficients of the Mueller matrix of the sample are not affected by the retardance of the compensators. For the analyzer, the errors on M12, M13, M14 and M41 are δ proportional to ci = cos2 i , where i = 1, 2 according to the compen2 sator considered. Thus, they are smaller if δi = 127∘ than δi = 90∘. These errors are eliminated if a zone averaging measurement is performed (i.e. f(x, y, 45∘) = − f(x, y, − 45∘)). For the polarizer, the errors on M14,M21,M31 and M41 are smaller if δi = 127∘ than δi = 90∘. These errors are eliminated if a zone averaging measurement is performed according to Eq. (3). The errors on the other coefficients are independent of δi. These conclusions are summaries in Tables 2 and 3. 127° as retarder is a better choice than the quarter-wave with a single measurement. Nevertheless, there is no difference between δi = 127∘ and δi = 90∘ if a 4-zone averaging measurement is performed. 3. Systematic errors due to the compensators Errors due to uncertainties on the azimuthal and the retardance of the compensators are detailed in previous papers [2,4] and presented on Tables 2 and 3 for single zone and 4-zone averaging measurements respectively. 3.1. Azimuthal error Errors on CS1: δM14 is dependent on the value of the second compensator. δM21, δM31 and δM41 are dependent on the value of the first compensator. Together, the errors are smaller if δi = 127∘ than δi = 90∘ (see Table 2). The other coefficients are not affected by the compensators. δM21 = δM31 = δM41 = 0 if the zone averaging measurement is performed on the analyzer independently of the value of the compensators (see Table 3). For δM14 a zone-averaging on the analyzer will eliminate the effect of the value of the compensator. But an error will still exist to the second order. Errors on CS2: M12, M13 and M14 are dependent on the value of the second compensator. M41 is dependent on the value of the first compensator. They are smaller if δi = 127∘ than 90∘ (see Table 2). M12, M13 and M14 will become null if a zone averaging on the polarizer is performed (see Table 3). An error on M41 will still exist to the second order if a zone averaging measurement on the polarizer is performed. As for the analyzer and the polarizer, 127° as retarder is the best configuration with a single measurement. But, there is no difference
Table 1 Systematic errors due to the analyzer and the polarizer. f(x, y, ρ) is defined as: f(x, y, ρ) = δ x sin 2ρ + y cos 2ρ and ci = cos2 i . 2 Coefficient
Analyzer mispositioning
Polarizer mispositioning
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
4c2f(− M22, M32, A)δA 4c2f(− M23, M33, A)δA 4c2f(− M24, M34, A)δA − 2M31δA − 2M21δA2 − 2M32δA − 2M22δA2 − 2M33δA − 2M23δA2 − 2M34δA − 2M24δA2 2M21δA − 2M31δA2 2M22δA − 2M32δA2 2M23δA − 2M33δA2 2M24δA − 2M34δA2 2c1f(M42, − M43, P)δA − 2M41δA2 2M43δA − 2M42δA2 − 2M42δA − 2M43δA2 − 2M44δA2
− 2M13δP 2M12δP 2c2f(M24, − M34, A)δP 4c1f(− M22, M23, P)δP − 2M21δP2 − 2M23δP − 2M22δP2 2M22δP − 2M23δP2 2M34δP − 2M24δP2 4c1f(− M32, M33, P)δP − 2M31δP2 − 2M33δP − 2M32δP2 2M32δP − 2M33δP2 − 2M24δP − 2M34δP2 4c1f(− M42, M43, P)δP − 2M41δP2 − 2M43δP − 2M42δP2 2M42δP − 2M43δP2 − 2M44δP2
Table 2 Best conditions for the compensators with a single zone measurement (A = P = 0∘). 0 → errors are null; and ≡ → errors are independent of δ1, δ2. First order
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
Second order
δA
δP
δCS1
δCS2
δδ1
δδ2
δA2
δP2
δC2S1
δC2S2
δδ21
δδ22
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 127° ≡ ≡ ≡
≡ ≡ 127° 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
≡ ≡ 127° 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 90° 127° 127° 127° 90° 127° 127° 127° 90° 127° 127° 127° 90°
127° 127° 127° 127° 127° 127° 127° 127° 127° 127° 127° 90° 90° 90° 90°
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡ 127° ≡ ≡ ≡
127° 127° 127° ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
90° 90° ≡ 90° 90° 90° ≡ 90° 90° 90° ≡ 90° 90° 90° ≡
90° 90° 90° 90° 90° 90° 90° 90° 90° 90° 90° ≡ ≡ ≡ ≡
between δi = 127∘ or 90° if a 4-zone averaging measurement is performed. 3.2. Birefringence error Errors on δ1: with a single zone measurement (Table 2), four coefficients are in the best condition if δ1 = 90∘ to the first order, δ1 = 127∘ is better for the others. To the second order, all the coefficients are in the best condition if δ1 = 90∘ to the second order. With a 4-zone averaging measurement (Table 3), the same comments can be made except for δM21, δM31 and δM41 where the errors are null. Errors on δ2: we can make the same comments as for the other compensator. Only the four coefficients involved are different. For the birefringence error types, a simple conclusion cannot be made. Some errors are minimized with δi = 127∘ and others with δi = 90∘. The 4-zone averaging measurement eliminated only some errors. To avoid systematic errors it is necessary to know with high accuracy the birefringence of these elements. 4. Numerical study Numerical values of the errors can be evaluated for a perfect isotropic sample with Ψ = 47∘ and Δ = 79∘. We have simulated for all optical elements an error between − 1∘ and + 1∘ with a step of 0.1∘. The results are presented in two configurations according to the value Table 3 Best conditions for the compensators with a 4-zone averaging measurement (A = ± 45∘ and P = ± 45∘). 0 → errors are null; and ≡ → the errors are independent of δ1, δ2. First order
δM12 δM13 δM14 δM21 δM22 δM23 δM24 δM31 δM32 δM33 δM34 δM41 δM42 δM43 δM44
Second order
δA
δP
δCS1
δCS2
δδ1
δδ2
δA2
δP2
δC2S1
δC2S2
δδ21
δδ22
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ 0 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ 0 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 0 ≡ ≡ ≡
127° 127° 90° 0° 127° 127° 90° 0° 127° 127° 90° 0° 127° 127° 90°
0° 0° 0° 127° 127° 127° 127° 127° 127° 127° 127° 90° 90° 90° 90°
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡ 0 ≡ ≡ ≡
0 0 0 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
90° 90° ≡ 0° 90° 90° ≡ 0° 90° 90° ≡ 0° 90° 90° ≡
0° 0° 0° 90° 90° 90° 90° 90° 90° 90° 90° ≡ ≡ ≡ ≡
L. Broch et al. / Thin Solid Films 519 (2011) 2601–2603
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Table 4 Statistical study of the errors on the Mueller matrix of an isotropic sample (Ψ = 47 ∘ and Δ = 79 ∘ ) if A = P = CS1 = CS2 = 0∘. All errors have been simulated between − 1∘ to + 1∘ (step 0.1°). Configuration
Error (average error ± its standard deviation) 1 0 ≈0 0:011 0 0:007 0 0:037 C B ≈0 0:011 −0:001 0:015 0 0:048 0 0:055 C B @ ≈0 0:007 0 0:048 ≈−0 0:003 −0:001 0:010 A 0 0:033 0 0:055 0:001 0:010 ≈−0 0:0001 1 0 0 ≈0 0:005 0 0:004 0 0:015 C B ≈0 0:005 −0:001 0:007 0 0:048 0 0:055 C B @ ≈0 0:004 0 0:048 ≈−0 0:001 −0:001 0:009 A ≈−0 0:013 0 0:055 0:001 0:009 −0:001 0:002 0
δ1 =δ2 =90°
δ1 =δ2 =127°
of the compensators on Table 4. For each coefficient the average and its standard deviation are computed. All coefficients of the matrix are affected by the errors. The average errors are generally closed to zero and are identical if δi = 90∘ or δi = 127∘. The only exception is for M44 where the average error is greater (×10). The standard deviations are lower with 127∘ than 90∘ except for M23, M24, M32 and M42. For these coefficients the errors are unchanged. It is necessary to remember that the errors are not only based on the value of the compensators and the values, Mij, of the sample must be considered too. 5. Conclusion We have characterized the systematic error effects on the measurement of all the Mueller matrix coefficients of a sample for a double rotating compensator ellipsometer. Smith has shown that 127∘ retarders are the best value for optimized random errors in a DRR polarimeter. We have applied it and studied the influence of the
compensators on the acquired parameters. The study of the systematic errors shows that 127∘ retarders are more appropriate than the quarter-wave compensators. But, this is not the same conclusion for the errors δδi and δδ2i which some coefficients are minimized with a quarter wave retarder particularly for errors to the second order. This study demonstrates that using a 4-zone averaging measurement eliminates the influence of the compensator except for the errors due to δδi. Thus, the optimized measurements in ellipsometry are obtained with 127∘ retarders coupled with a 4-zone averaging method. References [1] R.W. Collins, Handbook of ellipsometry, William Andrew Publishing & SpringerVerlag, 2005. [2] L. Broch, A. En Naciri, L. Johann, Opt. Express 16 (12) (2008) 8814. [3] G. Piller, L. Broch, L. Johann, Phys. Status Solidi (c) 5 (5) (2008) 1027. [4] L. Broch, A. En Naciri, L. Johann, Appl. Opt. 49 (17) (2010) 3250. [5] M.H. Smith, Appl. Opt. 41 (2002) 2488.