The effects of generalized component imperfections, azimuth-angle errors, and errors of the normalized Fourier coefficients of the detected photoelectric current on the measured ratio of reflection coefficients ρ in rotating-analyzer ellipsometers (RAE) are determined. The problem is formulated in such a way that much of the earlier work done on error analysis for null ellipsometers (NE) can be adapted to RAE. The results are conveniently expressed in terms of coupling coefficients that determine the extent to which a given source of error couples to an error of the measured value of ρ. The optical properties of the compensator (if used) and of the surface can be simultaneously obtained from a set of two measurements using RAE, in a manner similar to two-zone measurements in NE. In addition, novel methods of obtaining and combining the results from two measurements are examined, with the objective of cancelling the effect of many of the systematic sources of errors. One such method employs two incident polarizations of the same ellipticity but with orthogonal azimuths, in which case the measured value of ρ is almost independent of the input optics. If the two incident polarizations are chosen, instead, to have equal but opposite ellipticity and azimuth, the effects of the polarizer imperfection, off-diagonal elements in the compensator, entrance-window, surface, and exit-window imperfection matrices, as well as polarizer and compensator azimuth-angle errors, all disappear upon such two-measurement averaging; the effects of analyzer imperfection or azimuth-angle error and errors of the normalized Fourier coefficients are only partially cancelled. Finally, use of RAE in generalized ellipsometry and its attendant problems are examined.

D. E. Aspnes J. Opt. Soc. Am. A 21(3) 403-410 (2004)

References

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These coefficients determine the extent to which small component imperfections δρ_{P}, δT_{Z} (Z = C, W, S, W′), δρ_{A} of the polarizer P, compensator C, entrance window W, surface S, exit window W′, and analyzer A, respectively, couple into errors of the ratio ρ of the p → p and s → s complex reflection coefficients of a surface S in a PCW SW′ A rotating-analyzer ellipsometer (RAE) according to δρ = γ_{Z}δρ_{Z} (Z = P,A) and δρ = ∑_{ij}γ_{Zij}δT_{Zij} (Z = C, W, S, W′) [Eqs. (13), (17)]. χ_{i} and χ_{r} define the states of polarization of the light beam incident on and reflected from the surface S, respectively [Eq. (1)]. P, C are the azimuth angles of the polarizer and compensator; ρ_{C} is the slow(l) to fast(f) complex relative transmittance of the compensator; and α, β are the normalized Fourier coefficients of the photoelectric current detected after the rotating analyzer [Eq. (44)]. In case no compensator is used (i.e., in the PW SW′ A ellipsometer arrangement), set C = 0, ρ_{C} = 1. Furthermore, if the substitution χ_{r} = −cotA is made, the coupling coefficients for component imperfections appropriate to a null ellipsometer are obtained, where (P, C, A) represents a set of nulling angles.
Because it is the ratio of the diagonal elements of the reflection matrix that is measured by the ellipsometer, the elements in the (1,1) and (2,2) diagonal-array positions are of no significance.

Table II

Coupling coefficients for azimuth-angle and normalized-Fourier-coefficient errors in RAE.^{a}

Q, χ_{r}, (∂χ_{r}/∂α), and (∂χ_{r}/∂β) are as given in Table I.

These coefficients determine the extent to which small azimuth-angle errors δZ (Z = P, C, A) and small normalized-Fourier-coefficient errors δα, δβ couple to cause errors of the ratio of reflection coefficients ρ in a PCWSW′A rotating-analyzer ellipsometer (RAE) according to δρ = γ_{Z}′δZ (Z = P, C, A; α, β) [Eqs. (27), (55)]. The different quantities that appear in this table have the same meanings as are indicated in footnote a to Table I. In case no compensator is used (i.e., in the PW SW′ A ellipsometer arrangement), set C = 0, ρ_{C} = 1. γ_{P}′ and γ_{C}′ as given here also apply to null ellipsometers if the substitution χ_{r} = −cotA is made, in which case (P, C, A) represents a set of nulling angles.

Tables (2)

Table I

Coupling coefficients for component imperfections in RAE.^{a}

These coefficients determine the extent to which small component imperfections δρ_{P}, δT_{Z} (Z = C, W, S, W′), δρ_{A} of the polarizer P, compensator C, entrance window W, surface S, exit window W′, and analyzer A, respectively, couple into errors of the ratio ρ of the p → p and s → s complex reflection coefficients of a surface S in a PCW SW′ A rotating-analyzer ellipsometer (RAE) according to δρ = γ_{Z}δρ_{Z} (Z = P,A) and δρ = ∑_{ij}γ_{Zij}δT_{Zij} (Z = C, W, S, W′) [Eqs. (13), (17)]. χ_{i} and χ_{r} define the states of polarization of the light beam incident on and reflected from the surface S, respectively [Eq. (1)]. P, C are the azimuth angles of the polarizer and compensator; ρ_{C} is the slow(l) to fast(f) complex relative transmittance of the compensator; and α, β are the normalized Fourier coefficients of the photoelectric current detected after the rotating analyzer [Eq. (44)]. In case no compensator is used (i.e., in the PW SW′ A ellipsometer arrangement), set C = 0, ρ_{C} = 1. Furthermore, if the substitution χ_{r} = −cotA is made, the coupling coefficients for component imperfections appropriate to a null ellipsometer are obtained, where (P, C, A) represents a set of nulling angles.
Because it is the ratio of the diagonal elements of the reflection matrix that is measured by the ellipsometer, the elements in the (1,1) and (2,2) diagonal-array positions are of no significance.

Table II

Coupling coefficients for azimuth-angle and normalized-Fourier-coefficient errors in RAE.^{a}

Q, χ_{r}, (∂χ_{r}/∂α), and (∂χ_{r}/∂β) are as given in Table I.

These coefficients determine the extent to which small azimuth-angle errors δZ (Z = P, C, A) and small normalized-Fourier-coefficient errors δα, δβ couple to cause errors of the ratio of reflection coefficients ρ in a PCWSW′A rotating-analyzer ellipsometer (RAE) according to δρ = γ_{Z}′δZ (Z = P, C, A; α, β) [Eqs. (27), (55)]. The different quantities that appear in this table have the same meanings as are indicated in footnote a to Table I. In case no compensator is used (i.e., in the PW SW′ A ellipsometer arrangement), set C = 0, ρ_{C} = 1. γ_{P}′ and γ_{C}′ as given here also apply to null ellipsometers if the substitution χ_{r} = −cotA is made, in which case (P, C, A) represents a set of nulling angles.