The problem of relating the specimen ψ and Δ to the instrument readings and component imperfections is solved by a general technique without any restriction on the nature of the imperfections. Thus, for the first time, small incoherent effects in the ellipsometer can be treated. The solution is explicit and is given in terms of the properties of the ideal ellipsometer and the Mueller imperfection matrices of the optical devices. After deriving the general solution, we consider a conventional ellipsometric arrangement. Arrays of coupling constants are introduced which clearly show the effect of imperfections on ψ and Δ. Also, we indicate in a schematic way the elements that remain effective after averaging over two and four zones. Component depolarization is discussed and the matrix elements contributing to it are found. Besides allowing all previous results to be obtained, some new conclusions of this analysis are: a small depolarization of the polarizer light output affects ψ (0.15° error in ψ for 1% depolarization) and this effect remains after two- and four-zone averaging. The effect of both coherent and incoherent p ↔ s cross scattering by the cell windows cancels if a two-zone average is taken. The same applies for the coherent and incoherent cross scattering by the specimen-surface roughness or by surface optical activity. For the compensator, cross scattering caused by birefringence and optical activity cancels only if a four-zone average is taken.
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SkI0 represents the Stokes vector incident on the kth element in an ideal ellipsometer. Starting with SPI0 = {1,0,0,μ) and using the Mueller matrices of the ideal elements in Appendix A, we proceed to determine the Stokes vectors incident on the successive elements encountered by the beam along its path by making use of the relation S(k+1)I0 = Tk0SkI0. Simple differentiation gives ∂SkI0/∂P and use of the ideal nulling angles of Eq. (13) provides the third and fourth columns of this table.
Γk0 is a 1×4 row vector that determines the contribution to the detected flux arising from the Stokes-vector perturbation generated at the output of the kth device due to its imperfection [Eqs. (1)–(3)]. Starting with ΓA0 = (1,0,0,0) and using the Mueller matrices of the ideal elements in Appendix A, we work backwards to determine Γk0(k = W′, S, W, C, and P) using the relation Γk−10 = Γk0Tk0 (e.g., ΓW′0 = ΓA0TA0) which follows from simple rules of matrix algebra. Differentiation gives ∂Γk0/∂A and use of the ideal nulling angles in Eq. (13) provides the third and fourth columns of this table.
These arrays determine the effect on the nulling angles in zones 1 and 3 (and hence on ψ and Δ) of the imperfection matrices (tij) of the respective elements according to the relations δP = Σ βijtij and δA = Σ αijtij. For the polarizer, compensator, analyzer, and specimen, the imperfection matrices are referenced to their principal frames, whereas for the windows they are given in the ps frame. To get the arrays appropriate to zones 2 and 4, multiply the elements having a bar on the top by −1. The multiplier on the left multiplies each of the elements of the array inside the vertical lines.
Table IV
Location of effective elements—two-zone averages.a
P
C
W
S
W′
A
(a)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(b)
X
0
0
X
X
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
X
0
X
0
X
0
0
0
X
0
0
0
X
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
X
0
0
X
X
0
0
X
0
0
0
0
0
0
0
0
X
0
0
0
X
X
0
0
0
0
0
0
0
0
0
0
0
0
This table gives the location of those elements of the Mueller imperfection matrices of the different ellipsometer components that affect (a) Δ, (b) ψ when a two-zone average is taken. X (or
) and 0 denote that the corresponding matrix elements are effective and not effective, respectively. The bar over X has the meaning explained in the text.
Table V
Location of effective elements—four-zone averages.a
Location of the elements contributing to the depolarization.a
P
C
W
S
W′
A
X
0
0
X
X
X
X
0
X
0
X
X
X
0
X
X
X
X
X
0
X
X
0
0
X
0
0
X
X
X
X
0
0
0
0
0
X
0
X
X
X
X
X
0
X
X
0
0
0
0
0
0
0
0
0
0
X
0
X
X
X
0
X
X
X
X
X
0
X
X
0
0
0
0
0
0
X
X
X
0
X
0
X
X
0
0
0
0
0
0
0
0
X
X
0
0
The above table indicates the location of the elements in the Mueller imperfection matrices of the different ellipsometer components that contribute to the depolarization of light when the system is set for minimum detected flux.
SkI0 represents the Stokes vector incident on the kth element in an ideal ellipsometer. Starting with SPI0 = {1,0,0,μ) and using the Mueller matrices of the ideal elements in Appendix A, we proceed to determine the Stokes vectors incident on the successive elements encountered by the beam along its path by making use of the relation S(k+1)I0 = Tk0SkI0. Simple differentiation gives ∂SkI0/∂P and use of the ideal nulling angles of Eq. (13) provides the third and fourth columns of this table.
Γk0 is a 1×4 row vector that determines the contribution to the detected flux arising from the Stokes-vector perturbation generated at the output of the kth device due to its imperfection [Eqs. (1)–(3)]. Starting with ΓA0 = (1,0,0,0) and using the Mueller matrices of the ideal elements in Appendix A, we work backwards to determine Γk0(k = W′, S, W, C, and P) using the relation Γk−10 = Γk0Tk0 (e.g., ΓW′0 = ΓA0TA0) which follows from simple rules of matrix algebra. Differentiation gives ∂Γk0/∂A and use of the ideal nulling angles in Eq. (13) provides the third and fourth columns of this table.
These arrays determine the effect on the nulling angles in zones 1 and 3 (and hence on ψ and Δ) of the imperfection matrices (tij) of the respective elements according to the relations δP = Σ βijtij and δA = Σ αijtij. For the polarizer, compensator, analyzer, and specimen, the imperfection matrices are referenced to their principal frames, whereas for the windows they are given in the ps frame. To get the arrays appropriate to zones 2 and 4, multiply the elements having a bar on the top by −1. The multiplier on the left multiplies each of the elements of the array inside the vertical lines.
Table IV
Location of effective elements—two-zone averages.a
P
C
W
S
W′
A
(a)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(b)
X
0
0
X
X
0
0
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
0
0
X
0
X
0
X
0
0
0
X
0
0
0
X
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
X
0
0
X
X
0
0
X
0
0
0
0
0
0
0
0
X
0
0
0
X
X
0
0
0
0
0
0
0
0
0
0
0
0
This table gives the location of those elements of the Mueller imperfection matrices of the different ellipsometer components that affect (a) Δ, (b) ψ when a two-zone average is taken. X (or
) and 0 denote that the corresponding matrix elements are effective and not effective, respectively. The bar over X has the meaning explained in the text.
Table V
Location of effective elements—four-zone averages.a
Location of the elements contributing to the depolarization.a
P
C
W
S
W′
A
X
0
0
X
X
X
X
0
X
0
X
X
X
0
X
X
X
X
X
0
X
X
0
0
X
0
0
X
X
X
X
0
0
0
0
0
X
0
X
X
X
X
X
0
X
X
0
0
0
0
0
0
0
0
0
0
X
0
X
X
X
0
X
X
X
X
X
0
X
X
0
0
0
0
0
0
X
X
X
0
X
0
X
X
0
0
0
0
0
0
0
0
X
X
0
0
The above table indicates the location of the elements in the Mueller imperfection matrices of the different ellipsometer components that contribute to the depolarization of light when the system is set for minimum detected flux.