F. A. Modine, G. E. Jellison, and G. R. Gruzalski, "Errors in ellipsometry measurements made with a photoelastic modulator," J. Opt. Soc. Am. 73, 892-900 (1983)
The equations governing ellipsometry measurements made with a photoelastic modulator are presented in a simple but general form. These equations are used to study the propagation of both systematic and random errors, and an assessment of the accuracy of the ellipsometer is made. A basis is provided for choosing among various ellipsometer configurations, measurement procedures, and methods of data analysis. Several new insights into the performance of this type of ellipsometer are supplied.
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Averaging the Results Obtained for Two Orthogonal Orientations of the Polarizer, Modulator, Analyzer, or the Combination of Polarizer and Modulator as a Unit, Will Reduce Errorsa
C-I
C-II
C-III
Error
Z
X
Y
Y
Z
δR
C or P
C, P, or A
C, P, or A
C or P
C or P
δK
0
C or M
C or M
C, M or A
C, M, or A
δP
0
C or P
C or P
C, P, or A
C, P, or A
δM
0
A
0
0
A
δA
0
A
A
A
A
The polarizer, modulator, analyzer, and combination of polarizer and modulator as a unit are denoted by P, M, A, and A, respectively. The table indicates which components may be rotated by ±90° in order to eliminate an error to first order. δP, δM, and δA denote errors in angular alignment. Note that because P is defined relative to the modulator, the error δM implies a corresponding real rotation of the polarizer, but M as used in the table refers to the modulator alone. The δR and δK denote an error in the amplitude of the oscillating retardation and a static retardation, respectively. A 0 indicates that there is no first-order error.
Table 2
Root-Sum-Squared Errors δψ and δΔ for Cases That Simulate Representative Measurements on Si (ψ = 18.3° and Δ = 173°) and TaC (ψ = 39° and Δ = 139°)a
Analysis
Averaging
Parameters
XY
XZ
YZ
〈XYZ〉
〈XY〉Z
Si at 500 nm
None
δψ
0.648
0.135
0.135
0.467
0.135
δΔ
0.246
17.24
0.211
0.246
0.246
C and A
δψ
0.005
0.005
0.005
0.001
0.005
δΔ
0.017
0.197
0.017
0.017
0.017
TaC at 1000 nm None
None
δψ
2.61
0.086
0.086
0.223
0.086
δΔ
0.560
1.97
0.669
0.560
0.560
C and A
δψ
0.143
0.002
0.002
0.006
0.002
δΔ
0.005
0.000
0.015
0.005
0.005
The results are for X and Y data taken in configuration II and Z data taken in configuration I. The method of analysis is identified by naming the variables used to compute ψ and Δ. Brackets are used to indicate a least-squares analysis of the variables. The assumed alignment and calibration errors are 2.0° in modulation amplitude and azimuth of the calibration retardation plate, 0.2° in other azimuth settings, and 0.2° modulator static retardation.
Table 3
The rms Errors δψ and δΔ Computed by a Monte Carlo Analysis for Cases That Simulate Measurements on Si (ψ = 18.3° and Δ = 173°) and TaC (ψ = 39° and Δ = 139°)a
Analysis
Configuration
Parameters
XY
XZ
YZ
〈XYZ〉
〈XY〉Z
Si at 500 nm
Y(II)
δψ
0.029
0.043
0.043
0.024
0.043
Z(I)
δΔ
0.058
1.18
0.059
0.058
0.058
Y(III)
δψ
0.029
0.041
0.041
0.024
0.041
Z(I)
δΔ
0.056
1.11
0.056
0.056
0.056
TaC at 1000 nm
Y(III)
δψ
0.275
0.023
0.023
0.025
0.023
Z(I)
δΔ
0.127
0.071
0.215
0.127
0.127
Y(III)
δψ
0.277
0.022
0.022
0.024
0.022
Z(III)
δΔ
0.130
0.073
0.218
0.130
0.130
The methods of analysis are identified by naming the variables used to compute ψ and Δ. Brackets are used to indicate a least-squares analysis. Errors are assumed to be normally distributed with standard deviations as follows: 2.0° for modulation amplitude and azimuth of the calibration retardation plate; 0.2° for other azimuthal settings and modulator static retardation; additive random noise equivalent to one part per thousand of the calibration signals; and 0.02° reproducibility in averaging over the analyzer and the polarizer and modulator as a unit.
Tables (3)
Table 1
Averaging the Results Obtained for Two Orthogonal Orientations of the Polarizer, Modulator, Analyzer, or the Combination of Polarizer and Modulator as a Unit, Will Reduce Errorsa
C-I
C-II
C-III
Error
Z
X
Y
Y
Z
δR
C or P
C, P, or A
C, P, or A
C or P
C or P
δK
0
C or M
C or M
C, M or A
C, M, or A
δP
0
C or P
C or P
C, P, or A
C, P, or A
δM
0
A
0
0
A
δA
0
A
A
A
A
The polarizer, modulator, analyzer, and combination of polarizer and modulator as a unit are denoted by P, M, A, and A, respectively. The table indicates which components may be rotated by ±90° in order to eliminate an error to first order. δP, δM, and δA denote errors in angular alignment. Note that because P is defined relative to the modulator, the error δM implies a corresponding real rotation of the polarizer, but M as used in the table refers to the modulator alone. The δR and δK denote an error in the amplitude of the oscillating retardation and a static retardation, respectively. A 0 indicates that there is no first-order error.
Table 2
Root-Sum-Squared Errors δψ and δΔ for Cases That Simulate Representative Measurements on Si (ψ = 18.3° and Δ = 173°) and TaC (ψ = 39° and Δ = 139°)a
Analysis
Averaging
Parameters
XY
XZ
YZ
〈XYZ〉
〈XY〉Z
Si at 500 nm
None
δψ
0.648
0.135
0.135
0.467
0.135
δΔ
0.246
17.24
0.211
0.246
0.246
C and A
δψ
0.005
0.005
0.005
0.001
0.005
δΔ
0.017
0.197
0.017
0.017
0.017
TaC at 1000 nm None
None
δψ
2.61
0.086
0.086
0.223
0.086
δΔ
0.560
1.97
0.669
0.560
0.560
C and A
δψ
0.143
0.002
0.002
0.006
0.002
δΔ
0.005
0.000
0.015
0.005
0.005
The results are for X and Y data taken in configuration II and Z data taken in configuration I. The method of analysis is identified by naming the variables used to compute ψ and Δ. Brackets are used to indicate a least-squares analysis of the variables. The assumed alignment and calibration errors are 2.0° in modulation amplitude and azimuth of the calibration retardation plate, 0.2° in other azimuth settings, and 0.2° modulator static retardation.
Table 3
The rms Errors δψ and δΔ Computed by a Monte Carlo Analysis for Cases That Simulate Measurements on Si (ψ = 18.3° and Δ = 173°) and TaC (ψ = 39° and Δ = 139°)a
Analysis
Configuration
Parameters
XY
XZ
YZ
〈XYZ〉
〈XY〉Z
Si at 500 nm
Y(II)
δψ
0.029
0.043
0.043
0.024
0.043
Z(I)
δΔ
0.058
1.18
0.059
0.058
0.058
Y(III)
δψ
0.029
0.041
0.041
0.024
0.041
Z(I)
δΔ
0.056
1.11
0.056
0.056
0.056
TaC at 1000 nm
Y(III)
δψ
0.275
0.023
0.023
0.025
0.023
Z(I)
δΔ
0.127
0.071
0.215
0.127
0.127
Y(III)
δψ
0.277
0.022
0.022
0.024
0.022
Z(III)
δΔ
0.130
0.073
0.218
0.130
0.130
The methods of analysis are identified by naming the variables used to compute ψ and Δ. Brackets are used to indicate a least-squares analysis. Errors are assumed to be normally distributed with standard deviations as follows: 2.0° for modulation amplitude and azimuth of the calibration retardation plate; 0.2° for other azimuthal settings and modulator static retardation; additive random noise equivalent to one part per thousand of the calibration signals; and 0.02° reproducibility in averaging over the analyzer and the polarizer and modulator as a unit.