Universal evaluations and expressions of measuring uncertainty for rotating-element spectroscopic ellipsometers

Yong Jai Cho; Won Chegal; Jeong Pyo Lee; Hyun Mo Cho

doi:10.1364/OE.23.016481

1. Introduction

One of the main issues in ellipsometry has been always the improvement of uncertainty measurement for spectroscopic ellipsometers (SEs), which was discussed at the 3rd international conference on spectroscopic ellipsometry (ICSE) in 2003 [1]. It is also predicted that this issue will continue to be focused on in the future [2,3]. It is well known that the limits of optics-based measurements are primarily due to not only the signal-to-noise ratio of the measurement system but also the types of measurements being made [4]. In ellipsometry, the latter may be related with the optical configuration, data acquisition method, data-reduction process, type of ellipsometric sample parameters, and measurement setting parameters, where the measurement setting parameters of the azimuthal angles of fixed elements, wavelengths, angles of incidence, etc. have generally user optional attribute [5–9]. Changing the type of the sample under study into the other may influence the measuring uncertainty, since the uncertainty is strongly related with the optical properties of the sample. Thus, not only the SE developers, but also the SE users need a universal method for evaluating and expressing uncertainty of the result of a measurement.

If there is a universal method of evaluating and expressing the measurement uncertainty for SEs, one can choose the type of SE suitable for a sample, optimize the measurement settings, and possibly predict the practical limit for obtaining information of a sample from measurements. In particular, the method might be helpful to not only develop the Mueller-matrix SEs, but also find their optimum measurement conditions since the Mueller-matrix SEs have a huge number of influence quantities, which contribute to the uncertainty of the measurement, compared with the common rotating-element SEs [10–18]. The method can also be used for instrumentation, simulation, standardization, laboratory accreditation, and metrology. Many of the papers presented at the 6th ICSE in 2014 dealt with complex materials that require a number of nontraditional parameters to describe them, increasing the need for measuring uncertainty [2]. However, from a study of earlier works [19–34], there appears to be no generally accepted method for characterizing SE measuring uncertainty [35].

In this work, we focus on measuring uncertainty for multichannel rotating-element spectroscopic ellipsometers (RE-SEs) because various types of multichannel RE-SEs using an integrating photodetector have been developed and widely used in real-time diagnostics [2,3]. We first introduce a general theory for the data-reduction process to derive the universal theoretical expressions for the measuring uncertainties of the ellipsometric sample parameters. To solve the incompleteness of the uncertainty expressions, we derive analytic functions for quantifying the estimated covariance between the Fourier coefficient (FC) means extracted from a detected radiant flux waveform using a new Fourier analysis algorithm [36]. Finally, we compare the data obtained with the functions with the experimental data.

2. Universal expressions of measuring uncertainty

The universal method for evaluating and expressing the uncertainty of the result of a measurement should be applicable to all kinds of measurements and all types of input data used in measurements. In ellipsometry, the radiant fluxes or exposures measured by a photodetector element are determined on the basis of a series of observations. Then, the measured data are transformed into the ellipsometric sample parameters using the mathematical model, i.e., data-reduction process. Therefore, the universal data-reduction process is of critical importance because, in addition to uncertainty in the observations, it generally includes various influence quantities that are inexactly known. This lack of knowledge contributes to the uncertainty of the measurement, as do the variations of the repeated observations and any uncertainty associated with the mathematical model itself.

In general RE-SE ellipsometric configurations [5–9,37–41], the collimated light beam from the light source passes through a polarization state generator (PSG), is reflected from (or transmitted through) the sample, and passes through the polarization state analyzer (PSA) to impinge the photodetector element (PDE), which transforms the detected radiant flux into electrical signals. The rotatable elements composed of the linearly polarizing and compensating elements are appropriately positioned at the PSG and PSA depending on the optical configurations of the RE-SE type. At least one of the rotatable elements must rotate with a constant angular frequency with others fixed at selected azimuthal angles. For proper measurements, the azimuthal angle positions of the characteristic axes of the rotatable elements should be measured from a real origin defined by an axis that is parallel to the plane of incidence and normal to the sample surface. These azimuth angles are measured from the real origin using the right-hand rule: the thumb is pointed in the direction of light propagation and the other fingers then point toward increasing angle. For brevity, the elements are assumed to be ideal while the angle of incidence $ϕ$ , wavelength $λ$ , and other system parameters are known to arbitrary uncertainty.

In the Stokes representation [42,43], the Stokes vectors of $S^{(LS)} = {(L_{0}, L_{1}, L_{2}, L_{3})}^{T}$ and $S^{(PDE)} = {(S_{0}^{(PDE)}, S_{1}^{(PDE)}, S_{2}^{(PDE)}, S_{3}^{(PDE)})}^{T}$ can be described for a light wave incident along the path through the PSG and PDE, respectively, and the Mueller matrices of $M^{(SP)} = {(M_{j k})}_{4 \times 4}$ , $T_{PSG} M^{(PSG)}$ , $T_{PSA} M^{(PSA)}$ , and $M^{(DOS)} = {(D_{j k})}_{4 \times 4}$ for the sample, the PSG, the PSA, and the detector optic system (deployed between the PSA and PDE), respectively. Here, $T_{PSG}$ and $T_{PSA}$ denote PSG and PSA effective transmittances, respectively, and the superscript $T$ is the transpose of a matrix. The Stokes vector of the monochromatic light wave incident on a PDE is described as

S^{(PDE)} = T_{PSA} T_{PSG} M^{(DOS)} M^{(PSA)} M^{(SP)} M^{(PSG)} S^{(LS)}

The radiant flux waveform detected by a PDE with active area

A_{PDE}

and quantum efficiency

μ_{QE}

is:

I (θ_{r}) = μ_{QE} A_{PDE} S_{0}^{(PDE)} = I_{0} + \sum_{n = 1}^{N_{ho}} [A_{n} \cos (n θ_{r}) + B_{n} \sin (n θ_{r})]

where

θ_{r}

denotes the azimuthal angle of the constantly rotating element with respect to the real origin,

I_{0}

the dc FC, and

A_{n}

and

B_{n}

the ac FCs. Using a column vector

V^{(SP)} = {(v_{j})}_{u v \times 1} = {(M_{11}, \dots, M_{1 v}, \dots, M_{u 1}, \dots, M_{u v})}^{T}

with the entries of

M^{(SP)}

, where the integers

u

and

v

depend on the RE-SE type used, the FCs can be represented as the scalar product given by

I_{0} = γ \sum_{j = 1}^{u} \sum_{k = 1}^{v} i_{0, j k} M_{j k} = γ i_{0} \cdot V^{(SP)},

A_{n} = γ \sum_{j = 1}^{u} \sum_{k = 1}^{v} a_{n, j k} M_{j k} = γ a_{n} \cdot V^{(SP)},

B_{n} = γ \sum_{j = 1}^{u} \sum_{k = 1}^{v} b_{n, j k} M_{j k} = γ b_{n} \cdot V^{(SP)},

where

γ

is a common factor,

i_{0} = {(i_{0, j k})}_{1 \times u v}

,

i_{0, j k} = (\partial I_{0} / \partial M_{j k}) / γ

,

a_{n} = {(a_{n, j k})}_{1 \times u v}

,

a_{n, j k} = (\partial A_{n} / \partial M_{j k}) / γ

,

b_{n} = {(b_{n, j k})}_{1 \times u v}

, and

b_{n, j k} = (\partial B_{n} / \partial M_{j k}) / γ

. If an RE-SE has a part of or an entire of a fixed polarizer as the first element in the PSG and a fixed analyzer as the last element in the PSA along the light wave train starting from the light source, then

γ

is given as

u_{c} (N_{SP})

where

κ = μ_{QE} A_{PDE} T_{PSA} T_{PSG} D_{11} L_{0}

,

d_{12 (13)} = D_{12 (13)} / D_{11}

,

l_{1 (2)} = L_{1 (2)} / L_{0}

, and the azimuthal angles

A

and

P

of the fixed analyzer and the fixed polarizer, respectively, are measured with respect to the real origin. Therefore, if the value of the common factor

γ

is determined for a RE-SE used, the values of the FCs for a given sample can be calculated theoretically using Eqs. (3)–(5).

Upon setting $X = {(x_{j})}_{(2 N_{ho} + 1) \times 1} = {(I_{0}, A_{1}, B_{1}, \dots, A_{N_{ho}}, B_{N_{ho}})}^{T}$ and $Ω = γ {(i_{0}, a_{1}, b_{1}, \dots, a_{N_{ho}}, b_{N_{ho}})}^{T}$ , Eqs. (3)-(5) can be expressed as $X = Ω V^{(SP)}$ . If $Ω^{T} Ω$ is nonsingular, the sample vector can be obtained as

V^{(SP)} = C X,

where the matrix of the sensitivity coefficients is

C = {(c_{j k})}_{u v \times (2 N_{ho} + 1)} = {(Ω^{T} Ω)}^{- 1} Ω^{T}

. Thus, each entry of the sample vector of Eq. (7) is not measured directly, but determined from

2 N_{ho} + 1

other quantities through the functional relationship of

v_{j} = \sum_{n = 1}^{2 N_{ho} + 1} c_{j n} x_{n}

, where

x_{n}

is obtained by a series of observations.

v_{j}

is also a function of

N_{u}

different uncertainty sources (not only the repeated observations) described as

U = {(u_{k})}_{N_{u} \times 1}^{T}

of which the variables may be the wavelength of the light detected in a PDE, the angle of incidence, the azimuthal angles of the fixed elements, and so on. Then, the combined standard uncertainty of

v_{j}

for a sample is given by [44]

u_{c} (v_{j}) = {[\sum_{n, m = 1}^{2 N_{ho} + 1} c_{j n} c_{j m} s ({\bar{x}}_{n}, {\bar{x}}_{m}) + \sum_{k = 1}^{N_{u}} {({\frac{\partial v_{j}}{\partial u_{k}} |}_{U})}^{2} u^{2} (u_{k})]}^{1 / 2},

where

s ({\bar{x}}_{n}, {\bar{x}}_{m})

is the estimated covariance associated with the FC means,

u (u_{k})

is the standard uncertainty of

u_{k}

, the partial derivatives are evaluated at

U

, and the over-bar denotes the arithmetic mean. The first and second terms on the right-hand side in Eq. (8) may be called the type A and B evaluations of standard uncertainty, respectively [44].

For isotropic samples with a uniform reflecting surface, $N_{SP} = - M_{12} / M_{11} (or - M_{21} / M_{11})$ , $C_{SP} = M_{33} / M_{11} (or M_{44} / M_{11})$ , and $S_{SP} = M_{34} / M_{11} (or - M_{43} / M_{11})$ have been used as the sample parameters [42,43]. Thus, if a sample parameter $q_{j}$ is described as a function of the entries of $V^{(SP)}$ , its combined standard uncertainty of $u_{c} (q_{j})$ is given by [44]

\begin{array}{l} u_{c} (q_{j}) = [\sum_{p, q = 1}^{u v} \sum_{n, m = 1}^{2 N_{ho} + 1} ({\frac{\partial q_{j}}{\partial v_{p}} |}_{V^{(SP)}}) ({\frac{\partial q_{j}}{\partial v_{q}} |}_{V^{(SP)}}) c_{p n} c_{q m} s ({\bar{x}}_{n}, {\bar{x}}_{m}) \\ + {\sum_{k = 1}^{N_{u}} {({\frac{\partial q_{j}}{\partial u_{k}} |}_{U})}^{2} u^{2} (u_{k})]}^{1 / 2} . \end{array}

Therefore, one may evaluate the uncertainty of an RE-SE theoretically using Eq. (8) or Eq. (9). However, the theoretical expressions require the analytic functions for the estimated covariance between the FC means, which have not previously been calculated [45].

3. Modeling of covariance between Fourier coefficients

To obtain the theoretical expressions for the covariance between the FC means, we want more detailed analysis of the data acquisition method for the RE-SEs. For measurement using an RE-SE, the output signals of the radiant flux values of the quasi-monochromatic light collected by the PDE at time $t$ can be expressed using Eq. (2) as:

I (t) = {I^{'}}_{0} + \sum_{n = 1}^{N_{ho}} [{A^{'}}_{n} \cos (n ω t) + {B^{'}}_{n} \sin (n ω t)] + δ (t),

where

{I^{'}}_{0}

denotes the experimental dc FC,

{A^{'}}_{n}

and

{B^{'}}_{n}

the experimental ac FCs,

ω

the angular frequency of the constantly rotating element with revolution period

T

, and

δ (t)

the fluctuations due to random noises. In state-of-the-art real-time RE-SEs, each pixel or each binning pixel group in the CCD or photodiode arrays acts as a PDE. When the PDE is read out

J

times during the interval

T

with equal scanning times of

T / J

, the measured exposure data for an arbitrarily given integration time

T_{i}

are generated as

S_{j} = \int_{(j - 1) T / J + T_{d}}^{(j - 1) T / J + T_{d} + T_{i}} I (t) d t

, where

T_{d}

denotes the time delay prior to beginning the integration after receiving a trigger pulse (one of the clock pulses equidistantly generated from the optical encoder in the rotational stage for the constantly rotating element). The measured exposures can be expressed as

\begin{array}{l} S_{j} = T_{i} {I^{'}}_{0} + \sum_{n = 1}^{N_{ho}} \frac{T_{i}}{ξ_{n}} \sin ξ_{n} [\cos (n θ_{j}) ({A^{'}}_{n} \cos χ_{n} + {B^{'}}_{n} \sin χ_{n}) \\ - \sin (n θ_{j}) ({A^{'}}_{n} \sin χ_{n} - {B^{'}}_{n} \cos χ_{n})] + δ S_{j}, \end{array}

where

θ_{j} = 2 π (j - 1) / J

,

χ_{n} = ξ_{n} (1 + 2 T_{d} / T_{i})

, and

ξ_{n} = n π T_{i} / T

. A full set of exposures

S_{j}

can be represented as

S = Ξ X^{'} + δ S

, where

S = {(S_{1}, \dots, S_{J})}^{T}

,

δ S = {(δ S_{1}, \dots, δ S_{J})}^{T}

,

X^{'} = {({x^{'}}_{j})}_{(2 N_{ho} + 1) \times 1} = {({I^{'}}_{0}, {A^{'}}_{1}, {B^{'}}_{1}, \dots, {A^{'}}_{N_{ho}}, {B^{'}}_{N_{ho}})}^{T}

,

Ξ = {(Ξ_{j k})}_{J \times (2 N_{ho} + 1)}

, and

Ξ_{j k} = \partial S_{j} / \partial {x^{'}}_{k}

. If

Ξ^{T} Ξ

is nonsingular the solutions can be obtained by the least-squares estimation given as

\bar{X^{'}} = {(Ξ^{T} Ξ)}^{- 1} Ξ^{T} \bar{S} .

To obtain the solutions to the abovementioned equations, we recently introduced a concise theory using a new method based on the discrete Fourier transform [36], which is applied to the measured $S_{j}$ as

{\bar{H}}_{n}^{c} + i {\bar{H}}_{n}^{s} = \frac{2}{N J} \sum_{j = 1}^{N J} S_{j} \exp (i n θ_{j}),

where

{\bar{H}}_{n}^{c}

and

{\bar{H}}_{n}^{s}

denote real functions, and the over-bar denotes the arithmetic mean of

N

independent pairs of repeated observations. Using the orthogonality of the system of trigonometric functions, the sample mean values of the FCs can be obtained as

{\bar{I}}^{'}_{0} = {\bar{H}}_{0}^{c} / (2 T_{i}),

{\bar{A}}^{'}_{n} = C_{n}^{c} {\bar{H}}_{n}^{c} - C_{n}^{s} {\bar{H}}_{n}^{s}, (n \geq 1),

{\bar{B}}^{'}_{n} = C_{n}^{c} {\bar{H}}_{n}^{s} + C_{n}^{s} {\bar{H}}_{n}^{c}, (n \geq 1),

where

C_{n}^{c} = \cos χ_{n} / (T_{i} \sin c ξ_{n})

,

C_{n}^{s} = \sin χ_{n} / (T_{i} \sin c ξ_{n})

, and

sinc x = \sin x / x

. The two data acquisition methods of the least-squares estimation (Eq. (12)) and the discrete Fourier transform on the exposures (Eqs. (14)-(16)) can gives the same values of

\bar{X^{'}}

.

To obtain the theoretical expression of the covariance between the FC means, we will adopt the latter data acquisition method. We assume that the random noise in Eqs. (10) and (11) originates from photon noise. Photon noise follows the Poisson distribution, wherein the population variance of the number of the photons detected by a PDE is equal to the arithmetic mean of the same [46]. Therefore, under the photon-noise-limited condition, the population variance of the randomly-varying exposure is given as

σ^{2} (S_{j}) ≅ η S_{j},

where the scaling factor

η = ε_{PH} / (μ_{QE} A_{PDE})

has the same dimension as

S_{j}

, and

ε_{PH}

denotes the average photon energy of the quasi-monochromatic photons incident on the PDE. Since the exposures measured in one integration time are not correlated with the other exposures, the estimate of the covariance associated with

S_{j}

and

S_{k}

is given by

s (S_{j}, S_{k}) = σ^{2} (S_{j}) δ_{j k},

where

δ_{j k}

denotes the Kronecker delta. Therefore, using the relation of Eq. (18), we obtain the estimate covariance associated with the measured FC means of Eqs. (14)-(16) as

s ({\bar{I}}^{'}_{0}, {\bar{I}}^{'}_{0}) = η {\bar{I}}^{'}_{0} / (N J T_{i}),

s ({\bar{A}}^{'}_{n}, {\bar{A}}^{'}_{n}) = q_{n, n} [2 {\bar{I}}^{'}_{0} + \sin c (ξ_{2 n}) {\bar{A}}^{'}_{2 n}],

s ({\bar{B^{'}}}_{n}, {\bar{B^{'}}}_{n}) = q_{n, n} [2 {\bar{I^{'}}}_{0} - \sin c (ξ_{2 n}) {\bar{A}}^{'}_{2 n}],

s ({\bar{I^{'}}}_{0}, {\bar{A^{'}}}_{n}) = η {\bar{A}}^{'}_{n} / (N J T_{i}),

s ({\bar{I}}^{'}_{0}, {\bar{B^{'}}}_{n}) = η {\bar{B^{'}}}_{n} / (N J T_{i}),

s ({\bar{A^{'}}}_{n}, {\bar{A}}^{'}_{m}) = q_{n, m} [\sin c (ξ_{n + m}) {\bar{A}}^{'}_{n + m} + \sin c (ξ_{n - m}) {\bar{A}}^{'}_{n - m}], (n \neq m),

s ({\bar{B^{'}}}_{n}, {\bar{B}}^{'}_{m}) = q_{n, m} [\sin c (ξ_{n - m}) {\bar{A}}^{'}_{n - m} - \sin c (ξ_{n + m}) {\bar{A}}^{'}_{n + m}], (n \neq m),

s ({\bar{A^{'}}}_{n}, {\bar{B}}^{'}_{m}) = {\begin{matrix} q_{n, n} \sin c (ξ_{n}) \cos (ξ_{n}) {\bar{B}}^{'}_{2 n}, (n = m), \\ q_{n, m} [\sin c (ξ_{n + m}) {\bar{B}}^{'}_{n + m} - \sin c (ξ_{n - m}) {\bar{B}}^{'}_{n - m}], (n \neq m), \end{matrix}

wherein

q_{n, m} = η / [N J T_{i} \sin c (ξ_{n}) \sin c (ξ_{m})]

and the symmetrical relations of

{\bar{A}}^{'}_{n - m} = {\bar{A}}^{'}_{m - n}

and

{\bar{B^{'}}}_{n - m} = - {\bar{B^{'}}}_{m - n}

, obtained by Eqs. (15)-(16), should be used for the special cases of

n - m < 0

. In the limit of

T_{i} / T ≪ 1

, the autocorrelations of Eqs. (19)-(21) converge to the earlier results [27] for the non-integrating PDEs. For the FCs of

X^{'}

, the estimated standard uncertainties

u ({x^{'}}_{j})

and the estimated correlation coefficients

r ({x^{'}}_{j}, {x^{'}}_{k})

are given as

u ({x^{'}}_{j}) = \sqrt{s ({\bar{x^{'}}}_{j}, {\bar{x^{'}}}_{j})},

r ({x^{'}}_{j}, {x^{'}}_{k}) = \frac{s ({\bar{x}}^{'}_{j}, {\bar{x}}^{'}_{k})}{u ({x^{'}}_{j}) u ({x^{'}}_{k})} .

For proper measurements, the azimuthal angle positions of the characteristic axes of the rotatable elements should be corrected for the real origin using the calibration procedures [5–9]. If the azimuthal angle of the constantly rotating element is represented as $ω t = θ_{r} + θ_{r 0, n}$ with respect to the real origin, in which $θ_{r 0, n}$ denotes the $n$ th order offset angle, the relationship between the original (unprimed) and experimental (primed) FCs can be obtained from the equality relation between Eqs. (2) and (10) as

I_{0} = {I^{'}}_{0},

A_{n} = {A^{'}}_{n} \cos (n θ_{r 0, n}) + {B^{'}}_{n} \sin (n θ_{r 0, n}),

B_{n} = - {A^{'}}_{n} \sin (n θ_{r 0, n}) + {B^{'}}_{n} \cos (n θ_{r 0, n}),

from which the observed values of the original FCs can be obtained. If the uncertainty of

θ_{r 0, n}

due to random noises in

ω

can be ignored for state-of-the-art RE-SEs, we obtain

s ({\bar{x}}_{j}, {\bar{x}}_{k}) ≃ s ({\bar{x}}^{'}_{j}, {\bar{x}}^{'}_{k})

. Therefore, if the value of the scaling factor

η

for each PDE in a RE-SE used is known, the estimated covariance in Eq. (8) and (9) can be determined quantitatively from the theoretical FC functions of Eqs. (3)-(5) through Eqs. (19)-(26).

4. Application

To examine the relationship of the reliability of the derived stochastic model functions and the estimated covariance between the FC means, we adopt a bare Si wafer as a sample and a homemade multichannel rotating-polarizer SE based on a three-polarizer design [45]. In our trials, each spectrum of the original FCs for the sample was measured at multiple azimuthal angles of the fixed analyzer from 0° to 358° in steps of 2° for the incidence angle of $ϕ =$ 70°; the measured data are shown in Fig. 1.

Fig. 1 Experimental (symbols) and theoretical (line) data of the Fourier coefficients for a bare c-Si wafer obtained at multiple azimuthal angles of the fixed analyzer from 0° to 358° in steps of 2° using a home-made multichannel rotating-polarizer spectroscopic ellipsometer based on a three-polarizer design.

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The observed values of the sample standard uncertainties $u (x_{j})$ and the sample correlation coefficients $r (x_{j}, x_{k})$ for the original FCs in Fig. 1 can be given as [44]

u (x_{j}) = {[\frac{1}{N (N - 1)} \sum_{l = 1}^{N} {(x_{j, l} - {\bar{x}}_{j})}^{2}]}^{1 / 2},

r (x_{j}, x_{k}) = \frac{\sum_{l = 1}^{N} (x_{j, l} - {\bar{x}}_{j}) (x_{k, l} - {\bar{x}}_{k})}{\sqrt{\sum_{l = 1}^{N} {(x_{j, l} - {\bar{x}}_{j})}^{2}} \sqrt{\sum_{l = 1}^{N} {(x_{k, l} - {\bar{x}}_{k})}^{2}}},

of which the experimental data are shown as circles in Figs. 2 and 3, respectively. In the study, using the relation

η = (N J T_{i}) u^{2} ({I^{'}}_{0}) / {\bar{I^{'}}}_{0}

of Eq. (19) and the corresponding experimental data, the mean of the data set of

η

, which was calculated at each scanning angle of the fixed analyzer, was obtained as

{\bar{η}}_{A} = (3.99 \pm 0.78) \times 10^{4}

count for

λ =

550 nm. Since

P =

18.23°,

κ_{P} = κ (1 + l_{1} \cos 2 P + l_{2} \sin 2 P)

in the third case of Eq. (6) is constant. In Eq. (3), the values of

V^{(SP)}

can be calculated by using the known optical properties of the sample, such as the refractive index of c-Si (4.077 - i0.025) and the refractive index (1.465) and film thickness (1.97 nm) of the native oxide, and the values of

i_{0}

can be determined by the known

P

and

A

values. Thus, the unknown parameters in Eq. (6) were determined as

κ_{P} = (9.86 \pm 0.12) \times 10^{7}

count/ms,

d_{12} = (2.46 \pm 0.14) \times 10^{- 2}

, and

d_{13} = (1.13 \pm 0.11) \times 10^{- 2}

by using the least-squares fitting for the function

I_{0} (κ_{P}, d_{12}, d_{13}; A)

of Eq. (3) and the corresponding experimental values. Substituting the calculated FCs into Eqs. (27) and (28), we obtained theoretically the estimated standard uncertainty and estimated correlation coefficient values, which also closely agreed with the corresponding experimental data, as shown in Figs. 2 and 3. We remark here that

N_{SP}

and

C_{SP}

can be derived as the functions of the five FCs {

I_{0}

,

A_{2}

,

B_{2}

,

A_{4}

,

B_{4}

} using the universal data-reduction method, and subsequently, their partial derivative functions can be obtained from the derivatives of the sample parameter functions with respect to the entries {

M_{11}

,

M_{12} (or M_{21})

,

M_{33}

} of the sample Mueller matrix. Consequently, by substituting the partial derivative, sensitivity coefficient, and covariance functions into the type A evaluation term in Eq. (9), we obtained the theoretical values of the combined standard uncertainties,

u_{c} (N_{SP})

and

u_{c} (C_{SP})

, of

N_{SP}

and

C_{SP}

for the repeated observations shown in Fig. 4. These analytic expressions exhibit excellent agreement with their corresponding experimental data (Fig. 4). The azimuthal angle dependency for

u_{c} (N_{SP})

and

u_{c} (C_{SP})

, as shown in Fig. 4, might be changed depending on both the wavelength and the type of sample parameters, i.e., their minimum positions are different in general. Therefore, to determine the optimized conditions for spectroscopic data, we define an uncertainty test function as follows:

Ζ (P, A, ϕ) = \sqrt{\frac{1}{N_{q} N_{λ}} \sum_{j = 1}^{N_{q}} \sum_{k = 1}^{N_{λ}} u_{c}^{2} (q_{j} (λ_{k}))},

where

N_{q}

is the number of types of sample parameters and

N_{λ}

denotes the number of measuring wavelengths. The analytic expression of

Z

can be obtained as a function of the azimuthal angles of the fixed elements and the angle of incidence, and subsequently used to find the optimized conditions in which the value of

Z

is minimum. In conclusion, we believe our approach can significantly aid in evaluating SE uncertainty in all types of multichannel RE-SEs.

Fig. 2 Experimental (symbols) and theoretical (line) data of the standard uncertainties for the Fourier coefficients as shown in Fig. 1.

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Fig. 3 Experimental (symbols) and theoretical (line) data of the correlation coefficients for the Fourier coefficients as shown in Fig. 1.

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Fig. 4 Experimental (symbols) and theoretical (line) data of the combined standard uncertainties for the ellipsometric sample parameters $N_{SP}$ and $C_{SP}$ for the wavelengths of 373 nm and 550 nm.

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5. Conclusion

In this work, for the first time, we obtained universal analytic expressions of the measuring uncertainty for multichannel RE-SEs. In the limit of $T_{i} / T ≪ 1$ , the analytic expressions converge to the measuring uncertainties of the RE-SEs adopting the non-integrating PDEs. A general theory of data-reduction procedures was introduced to obtain universal expressions of the theoretical combined standard uncertainties of the ellipsometric sample parameters. This required the determination of the analytic functions for the estimated covariance between the FC means, which has been undetermined thus far. Using a new closed-form Fourier analysis, we derived the analytic functions for estimated covariance between the FC means extracted from the detected radiant flux waveform. Since the uncertainties of the ellipsometric sample parameters are strongly related with not only the type of the sample parameters but also the wavelength, we introduced the uncertainty test function of Eq. (34) to find the optimal measurement conditions of an RE-SE for a given sample. The analytic expressions can be used for instrumentation, simulation, standardization, laboratory accreditation, and metrology. This approach will be also helpful for evaluating and expressing the measuring uncertainties for other types of SEs, such as photoelastic modulator SEs [47,48].

Acknowledgments

This work was supported by the Korea Research Institute of Standards and Science under the project “Development of core measurement technologies for the next generation of nano devices” (Grant No. 15011043).

References and links

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Universal evaluations and expressions of measuring uncertainty for rotating-element spectroscopic ellipsometers

Abstract

1. Introduction

2. Universal expressions of measuring uncertainty

3. Modeling of covariance between Fourier coefficients

4. Application

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (34)

Optics Express