CN103426031B - A kind of optimization method of ellipsometer system parameter - Google Patents

Abstract

The invention discloses a kind of ellipsometer system parameter optimization method, comprise the system model first setting up ellipsometer to be optimized, and obtain the matrix W comprising ellipsometer system parameter; Then the conditional number of ellipsometer system matrix W under different ellipsometer system parameter is calculated; Again calculate the difference of the maxima and minima of matrix W conditional number under each ellipsometer system parameter, and by the arrangement of difference descending order; Last putting in order according to difference, optimize each ellipsometer system parameter, after repeating n suboptimization, choose the systematic parameter that arrangement is the most front, difference when trying to achieve minimal condition numerical value and the (n-1)th suboptimization after its n suboptimization between matrix W minimal condition numerical value, when difference size meets the demands, then think that the ellipsometer system parameter after the n-th suboptimization is the optimal system parameter of current ellipsometer.The inventive method process is clear and definite, simple, and the selected standard flexibility and changeability of optimized parameter, goes for the optimization of the ellipsometer of existing different types of structure.

Description

Optimization method of ellipsometer system parameters

Technical Field

The invention belongs to the technical field of ellipsometers, and particularly relates to an ellipsometer system parameter optimization method.

Background

The ellipsometer is a measuring instrument most widely used in the field of ellipsometry, and is a general optical measuring instrument for acquiring information of a sample to be measured by using polarization characteristics of light.

In recent years, in order to adapt to different measurement conditions and the needs of user groups, ellipsometers of various configuration types have been developed, including a rotating polarizer type, a rotating analyzer type, a single rotation compensator type, a double rotation compensator type, and the like. When an ellipsometer is used to measure a sample to be measured, the measurement result often deviates from its true value to some extent. These deviations are caused by many reasons, including ellipsometer random noise, ellipsometer systematic errors, environmental random noise, and measurement artifacts. Even under the same deviation, the ellipsometer has different measurement results and sensitivity to the deviation according to different configured system parameters. Therefore, in order to reduce the influence of the deviation on the measurement result of the ellipsometer, a certain method must be used to perform optimization analysis on the system parameters of the ellipsometer, so as to obtain the optimal system parameters of the ellipsometer, thereby reducing the influence of the deviation on the measurement result of the ellipsometer as much as possible. During the ellipsometer design process, an optimal system parameter configuration for the current ellipsometer system configuration must be found.

Currently, for parameter optimization of an ellipsometer system, a main concern is to optimize the phase retardation of an ellipsometer compensator, for example, an optimal phase retardation of 127 ° is proposed to reduce the influence of the deviation on the ellipsometer measurement result. However, the ellipsometer has many parameters, such as the azimuth angles of the polarizer and the analyzer, the number of sampling points, the azimuth angle of the compensator, etc., which are also important influencing factors for the relationship between the deviation and the measurement result of the ellipsometer, and must be comprehensively considered for optimization.

Disclosure of Invention

The invention aims to provide an ellipsometer system parameter optimization method, which is simple and feasible and can be applied to ellipsometer measurement systems with various structural types.

In order to achieve the above object, the present invention provides an ellipsometer system parameter optimization method for determining an optimal system parameter of an ellipsometer, which is characterized in that the method includes:

(1) establishing a system model of the ellipsometer to be optimized, and arranging the system model of the ellipsometer into a form of a vector dot product and a vector dot product to obtain a system matrix W containing system parameters of the ellipsometer;

(2) calculating the condition number of the system matrix W under each ellipsometer system parameter;

(3) calculating the difference value between the maximum condition number and the minimum condition number of the system matrix W under each system parameter, and sequencing the corresponding system parameters according to the sequence of the difference values from large to small;

(4) and respectively and sequentially carrying out iterative optimization on each ellipsometer system parameter according to the sequence until the difference value between the minimum condition value of the most-front system parameter and the minimum condition value corresponding to the previous optimization is smaller than a threshold value, and finishing the optimization, wherein each corresponding system parameter is the optimized ellipsometer system parameter.

As a further preferred embodiment of the present invention, the specific process of iteratively optimizing the ellipsometer system parameters is as follows:

(1) for any ellipsometer system parameter, taking the current values of the rest system parameters as fixed values, and obtaining the relation between the system matrix W and the ellipsometer system parameter, so as to obtain the minimum value of the condition number of the system matrix W and the system parameter value corresponding to the minimum value, and take the minimum value and the system parameter value as the updated system parameter after one iteration;

in the same way, the updated values of the rest system parameters after one iteration are sequentially obtained;

(2) and repeating the process for multiple times, wherein each time the process is repeated, a group of updated ellipsometer system parameters are obtained, so that multiple iterative updates of each system parameter are completed until an iteration termination condition is met, and the latest group of ellipsometer system parameters can be obtained.

In the invention, the system model is a linear equation shown in formula (1):

$I_q=A_q^T{\mathrm{MS}}_q=\underset{j=0}{\overset{3}{\mathrm{\Σ}}}\underset{k=0}{\overset{3}{\mathrm{\Σ}}}{a}_{q,j}{m}_{j,k}{s}_{q,k}=\underset{j=0}{\overset{3}{\mathrm{\Σ}}}\underset{k=0}{\overset{3}{\mathrm{\Σ}}}{w}_{q,j,k}{m}_{j,k}---\left(1\right)$

wherein:

Α_q＝(a_q,0a_q,1a_q,2a_q,3)(2)

$M=\left(\begin{array}{cccc}{m}_{\mathrm{0,0}}& m_{\mathrm{0,1}}& m_{\mathrm{0,2}}& m_{\mathrm{0,3}}\\ m_{\mathrm{1,0}}& m_{\mathrm{1,1}}& m_{\mathrm{1,2}}& m_{\mathrm{1,3}}\\ m_{\mathrm{2,0}}& m_{\mathrm{2,1}}& m_{\mathrm{2,2}}& m_{\mathrm{2,3}}\\ m_{\mathrm{3,0}}& m_{\mathrm{3,1}}& m_{\mathrm{3,2}}& m_{\mathrm{3,3}}\end{array}\right)---\left(3\right)$

$S_q=\left(\begin{array}{c}{S}_{q,0}\\ S_{q,1}\\ S_{q,2}\\ S_{q,3}\end{array}\right)---\left(4\right)$

w_q,j,k＝a_q,js_q,k(5)

wherein, I_qIs the light intensity signal of the detector, A_qTo analyze the arm vector, S_qFor the deflection arm vector, subscript Q =0,1, …, Q-1 indicates the qth measurement component of the instrument, Q is the number of measurements, and M is the Mueller matrix of the sample to be measured, where the elements are represented by M_j,kIs shown as a_q,j,s_q,kRespectively correspond to A_qAnd S_qElement in vector, w_q,j,kRepresenting the measurement component I of the q-th order_qElement m to be measured for j row and k column_j,kI.e. the transfer characteristic of the ellipsometer measurement system, the subscripts j =0,1,2,3 and k =0,1,2,3 denote the fourth element in the vector, respectively. The system model of the ellipsometer is shown as formula (1).

The form of changing the sample matrix M to be tested of 4 × 4 in the formula (1) into the column vector of 16 × 1After finishing, the method can obtain:

$I_q=W_q\·\stackrel{\→}{M}=\left(\begin{array}{cccc}{a}_{q,0}{s}_{q,0}& a_{q,0}{s}_{q,1}& \·\·\·& a_{q,3}{s}_{q,3}\end{array}\right)\·\left(\begin{array}{c}{m}_{\mathrm{0,0}}\\ m_{\mathrm{0,1}}\\ \·\\ \·\\ \·\\ m_{\mathrm{3,3}}\end{array}\right)---\left(6\right)$

and all of the Q measurements may be represented as a Q × 16 ellipsometer system matrix W, where W is_qRepresenting the Q-th row of the matrix W, the Q measurements of the detector can be represented as a measurement vector I, where I is_qIs the q-th line of vector I. Equation (6) can be expressed in the form of a vector dot product:

$I=W(l,p,m)\·\stackrel{\→}{M}$

$=\left(\begin{array}{c}{I}_0\\ I_1\\ \·\\ \·\\ \·\\ I_{Q-1}\end{array}\right)=\left(\begin{array}{cccc}{w}_{\mathrm{0,0,0}}& w_{\mathrm{0,0,1}}& \·\·\·& w_{\mathrm{0,3,3}}\\ w_{\mathrm{1,0,0}}& w_{\mathrm{1,0,1}}& \·\·\·& w_{\mathrm{1,3,3}}\\ \·& \·& & \·\\ \·& \·& & \·\\ \·& \·& & \·\\ w_{Q-\mathrm{1,0,0}}& w_{Q-\mathrm{1,0,1}}& \·\·\·& w_{Q-\mathrm{1,3,3}}\end{array}\right)\·\left(\begin{array}{c}{m}_{\mathrm{0,0}}\\ m_{\mathrm{0,1}}\\ \·\\ \·\\ \·\\ m_{\mathrm{3,3}}\end{array}\right)---\left(7\right)$

the ellipsometer system matrix W in the formula (7) includes system parameters to be optimized for the ellipsometer, and the system matrix W includes different parameters for ellipsometers with different structure types, and for convenience of the following description, only three variables l, p, and m are used to represent the three parameters of the system matrix W, and actually the types and the number of the parameters vary according to different kinds of the ellipsometers.

In the invention, the condition number of the ellipsometer system matrix W under different ellipsometer system parameters is calculated, the condition number of the ellipsometer system matrix W is used as an evaluation standard for ellipsometer system parameter optimization, and the smaller the condition number is, the smaller the deviation in the measurement vector I is to the solution vector IThe smaller the effect of (c).

The condition number is a measure to mathematically evaluate the sensitivity of the solution of the linear system of equations Ax = b to errors or uncertainties in b. Then for equation (8), if the ellipsometer system matrix W condition number is large, a small deviation in the measurement vector I will cause a solution vectorLarge deviation and poor numerical stability, and if the condition number of the measurement matrix W of the ellipsometer system is small, the solution vector caused by the small deviation in the measurement vector IThe deviation of (2) is small and the numerical stability is good.

Because each ellipsometer system parameter has an influence on the W condition number, only one parameter is needed to be used as a variable quantity in the W condition number calculation process, and the corresponding relation between the condition number of the matrix W and the ellipsometer system parameter can be obtained by fixing the rest parameters.

The condition number of the ellipsometer system matrix W is shown in formula (9)

κ_p(W)＝||W||_p·||W^-1||_p(8)

${\left|\right|W\left|\right|}_p=\underset{x\∈D\left(W\right)}{\sup }\frac{{\left|\right|W\·x\left|\right|}_p}{{\left|\right|x\left|\right|}_p},{\left({\left|\right|x\left|\right|}_p\right)}^p=\underset{i}{\mathrm{\Σ}}{x}_i^p---\left(9\right)$

W^-1＝(W^T·W)^-1·W^T(10)

Wherein, κ_p(W) is the condition number of the ellipsometer system matrix W, | | W | | caltrop_pIs the p-norm of the ellipsometer system matrix W, equation (9) is the calculation method of the p-norm of the matrix W, D (W) is the space domain of the matrix W, x is an arbitrary vector in the space domain of the matrix W, x_iDenotes the ith element, W, in the vector x^-1The superscript T represents the matrix transpose, which is the generalized inverse of the matrix W.

In the invention, the difference value between the maximum value and the minimum value of the ellipsometer system matrix W condition number under each ellipsometer system parameter is calculated, the larger the difference value is, the larger the influence of the ellipsometer system parameter on the matrix W condition number is, the ellipsometer system parameter is used as a priority optimization object, and each ellipsometer system parameter is sequentially optimized according to the size sequence of the difference value. If the condition number difference of the matrix W corresponding to l in the three parameters of l, m and p is the maximum, and the condition difference of the matrix W corresponding to p is the minimum, the method is arranged as follows: l, m, p.

In the invention, according to the arrangement sequence of the parameters of the ellipsometer system in the third step, the parameters are optimized in sequence. In the optimization process, only one parameter is used as a variable quantity each time, and the numerical values of the rest fixed parameters all use the current optimal value as an initial value.

Step 1, respectively obtaining the values of ellipsometer system parameters l, m and p at the minimum condition number of an ellipsometer system matrix W and the corresponding minimum condition values: l₁，κ_p(W(l₁))；p₁，κ_p(W(p₁))；m₁，κ_p(W(m₁))。

Step 2, with p₁，m₁As an initial value, the relation between the matrix W condition number and the ellipsometer system parameter value l can be obtained, and the value of the matrix W condition number at the minimum position and the corresponding minimum condition value l are obtained₂，κ_p(W(l₂)). Then use₂，m₁The ellipsometer system parameter p is optimized as an initial value, and the corresponding p can be obtained₂，κ_p(W(p₂)). Also will l₂，p₂As initial value, corresponding m is obtained₂，κ_p(W(m₂)). The parameter value l of the ellipsometer system after the second optimization can be obtained₂，p₂，m₂。

And 3, repeating the optimization process of the step 2 to obtain a new group of ellipsometer system parameters, and obtaining a group of ellipsometer system parameter values and corresponding condition values of the matrix W after nth optimization: l_n，κ_p(W(l_n))；p_n，κ_p(W(p_n))；m_n，κ_p(W(m_n))。

Now choose the ellipsometer system parameter l with the largest influence on the condition number of the matrix W, let Δ κ_p(W(l))=κ_p(W(l_n))-κ_p(W(l_n-1) When Δ κ)_p(W(l))<The optimization is stopped as soon as possible. The ellipsometer system parameter value obtained after the nth optimization is the optimal system parameter of the current ellipsometer, wherein different values can be selected according to the optimization requirements as the judgment condition for stopping optimization.

Compared with a general instrument optimization method, the ellipsometer system parameter optimization method provided by the invention obtains the matrix W containing the ellipsometer system parameters by establishing an ellipsometer system model and expressing the ellipsometer system model in the form of vector dot product. And calculating the condition number of the matrix W relative to each parameter, and taking the condition number as the evaluation standard of the optimized configuration of the ellipsometer system. And calculating the difference value between the maximum value and the minimum value of the matrix W condition number under each ellipsometer system parameter again, and finally optimizing each ellipsometer system parameter from large to small according to the sequence of the difference values, wherein the optimal system parameter and the repeated calculation times are determined by the design requirements. The method can optimize all system parameters of the ellipsometer, obtain the optimal configuration of the ellipsometer system through repeated calculation for many times, and the optimization calculation times are variable according to different precision requirements, so that the method can be suitable for optimizing the ellipsometers with different structural types.

Drawings

FIG. 1 is a flow chart of a method for ellipsometer system parameter optimization provided in the present invention.

Fig. 2 is a schematic structural diagram of a dual-rotation compensator type muller matrix ellipsometer system.

Fig. 3 is a graph of a simulation result of a measurement of a mueller matrix ellipsometer of a dual rotation compensator type.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings. The description herein is intended to be illustrative only and not limiting with respect to the specific embodiments described.

The ellipsometer system parameter optimization method provided by the invention can be suitable for ellipsometers with different structure types, such as: rotary polarizer type, rotary analyzer type, single rotary compensator type, and dual rotary compensator type. The dual-rotation compensator type muller matrix ellipsometer system is the most complex in structure, and the system parameters to be optimized are the most. The principle and operation of the method of the present invention will be described in detail below with reference to the accompanying drawings by taking a dual-rotation compensator type muller matrix ellipsometer system as an example.

The implementation flow of the ellipsometer system parameter optimization method provided in this embodiment is shown in fig. 1.

(1) And establishing a system model of the ellipsometer to be optimized, and arranging the system model of the ellipsometer into a form of a vector dot product and a vector dot product to obtain a matrix W containing the system parameters of the ellipsometer.

For a dual-rotation compensator type muller matrix ellipsometer system structure, the system structure is shown in fig. 2. The dual-rotation compensator type muller matrix ellipsometer system mainly comprises a light source 1, a polarizer 2, a polarizing arm rotation compensator 3, a polarization analyzing arm rotation compensator 5, a polarization analyzer 6 and a detector 7. The polarizer 2 and the polarizing arm rotating compensator 3 form a polarizing arm 8 of the double-rotating-compensator type muller matrix ellipsometer measuring system, and the polarization analyzing arm rotating compensator 5 and the polarization analyzer form an polarization analyzing arm 9 of the double-rotating-compensator type muller matrix ellipsometer measuring system. The light beam emitted by the light source 1 is unpolarized light, is changed into linearly polarized light through the polarizer 2, is changed into elliptically polarized light through the modulation of the polarizing arm rotating compensator 3, then the polarized light is acted with the sample 4 to be detected, the polarization state of the polarized light is changed, the polarization detecting arm rotating compensator 5 demodulates the polarized light, and finally the light intensity signal of the emergent polarized light is detected through the polarization detector 6 and the detector 7.

The detector 7 detects to obtain a light intensity signal containing information of the sample 4 to be detected, then Fourier analysis is carried out on the light intensity signal detected by the detector 7 to obtain Fourier coefficients containing the Mueller matrix information of the sample 4 to be detected, and finally the Mueller matrix information of the sample 4 to be detected can be solved by the Fourier coefficients.

Wherein the polarizer 2 has a bias angle P relative to the incident surface, and the polarization arm rotation compensator 3 has a phase retardation₁And an initial phase angle C with respect to the incident surface_s1Amount of phase delay of the resolver arm rotation compensator 4₂And an initial phase angle C with respect to the incident surface_s2The analyzer 3 is relative to the incident surface angle a, the number of sampling points N in a single period in two measurements, and the like, which are all ellipsometer system parameters to be optimized in the optimization of the double-rotation compensator type ellipsometer.

And the polarizing arm rotating compensator 3 and the polarization analyzing arm rotating compensator 5 of the double-rotating compensator type muller matrix ellipsometer measuring system continuously rotate in a certain proportion, wherein the proportion is 5: 3, the light intensity signal detected by the detector 7 in this case can be written as an expression shown in equation (11) (R.W.Collins et al., J.Opt.Soc.Am.A, Vol.16, pp.1997-2006,1999),

$I\left(t\right)=I_0[1+\underset{n=1}{\overset{16}{\mathrm{\&Sigma;}}}({\mathrm{\&alpha;}}_{2n}\cos 2\mathrm{n\&omega;t}+{\mathrm{\&beta;}}_{2n}\sin 2\mathrm{n\&omega;t})]---\left(11\right)$

wherein I (t) represents the emergent light intensity of the ellipsometer measuring system, { I₀,I₀(α_2n,β_2n) Expressing the Fourier coefficients of each order of the light intensity signal when the light intensity signal is spread into a Fourier expansion form, wherein the Fourier coefficients contain the Mueller matrix information of the sample 4 to be tested, I₀Is a direct current component, (α)_2n,β_2n) To be a direct current component I₀Normalized Fourier coefficients of various orders, n represents the order of the Fourier coefficients, omega is the fundamental frequency of rotation of the two rotation compensators, and t is the rotation time of the rotation compensators.

Hadamard analysis of formula (11) gave the form shown in formula (12) (R.W.Collins et al., J.Opt.Soc.Am.A., Vol.16, pp.1997-2006,1999),

$I_q={\&Integral;}_{(q-1)\mathrm{\&pi;}/\mathrm{N\&omega;}}^{\mathrm{q\&pi;}/\mathrm{N\&omega;}}{I}_0[1+\underset{n=1}{\overset{16}{\mathrm{\&Sigma;}}}({\mathrm{\&alpha;}}_{2n}\cos 2\mathrm{n\&omega;t}+{\mathrm{\&beta;}}_{2n}\sin 2\mathrm{n\&omega;t})]\mathrm{dt}$

(12)

$=\frac{\mathrm{\&pi;}{I}_0}{\mathrm{N\&omega;}}+\underset{n=1}{\overset{16}{\mathrm{\&Sigma;}}}\frac{I_0}{\mathrm{n\&omega;}}\left(\sin \frac{\mathrm{n\&pi;}}{N}\right)[{\mathrm{\&alpha;}}_{2n}\cos \frac{(2j-1)\mathrm{n\&pi;}}{N}+{\mathrm{\&beta;}}_{2n}\sin \frac{(2j-1)\mathrm{n\&pi;}}{N}]$

in equation (12), N represents the total number of integration points in which the number of sampling points in a single period is a Hadamard component, N should be greater than the total number of fourier coefficients, I_qIndicating that the detector intensity in the qth measurement is the qth Hadamard integral component.

The relation between the Fourier coefficient and the Mueller matrix of the sample to be detected is shown as a formula (13),

${\mathrm{\&alpha;}}_{2n}=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}\underset{j}{\overset{4}{\mathrm{\&Sigma;}}}{d}_{{\mathrm{\&alpha;}}_{2n},m_{\mathrm{ij}}}{m}_{\mathrm{ij}},---\left(13a\right)$

${\mathrm{\&beta;}}_{2n}=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}\underset{j}{\overset{4}{\mathrm{\&Sigma;}}}{d}_{{\mathrm{\&beta;}}_{2n},m_{\mathrm{ij}}}{m}_{\mathrm{ij}},---\left(13b\right)$

in the formula (13), i represents a row of the mueller matrix of the sample to be measured, j represents a column of the mueller matrix of the sample to be measured, and m_ijThe element of the ith row and the jth column in the Mueller matrix of the sample to be tested is shown,andrespectively represent α_2nAnd β_2nAbout m_ijWhen the dual-rotation compensator type Mueller matrix ellipsometer measuring system is calibrated, the coefficientAndare all known quantities.

Thus, fourier coefficients of respective orders can be obtained by equation (12), and then a mueller matrix of the sample can be obtained by equation (13). Therefore, equations (11) to (13) are the system model of the dual-rotation compensator type muller matrix ellipsometer, and the transmission characteristics of the dual-rotation compensator type muller matrix ellipsometer system expressed by equations (11) to (13) conform to the linear relationship described in equation (1). Finally, after the formula (13) is substituted for the formula (12), the formula is expressed in a vector dot product form:

$I_q=W_q\&CenterDot;\stackrel{\&RightArrow;}{M}=W_q(P,A,{\mathrm{\&delta;}}_1,{\mathrm{\&delta;}}_2,N)\&CenterDot;\left(\begin{array}{c}{m}_{\mathrm{0,0}}\\ m_{\mathrm{0,1}}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ m_{\mathrm{3,3}}\end{array}\right)---\left(14a\right)$

$I=\left(\begin{array}{c}{I}_0\\ I_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ I_N\end{array}\right)=W\&CenterDot;\stackrel{\&RightArrow;}{M}=W(P,A,{\mathrm{\&delta;}}_1,{\mathrm{\&delta;}}_2,N)\&CenterDot;\left(\begin{array}{c}{m}_{\mathrm{0,0}}\\ m_{\mathrm{0,1}}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ m_{\mathrm{3,3}}\end{array}\right)---\left(14b\right)$

the formula (14) is a system model representing the dual-rotation compensator type muller matrix ellipsometer in the form of vector dot product, and obtains a matrix W containing parameters of the system to be optimized of the dual-rotation compensator type ellipsometer, wherein P, a,₁,₂and N is an ellipsometer system parameter to be optimized included in the system matrix W.

(2) And (3) solving the condition number of the matrix W of the dual-rotation compensator type ellipsometer system in the step (1).

The method for solving the p-norm of the ellipsometer system matrix W is defined in equations (9) to (11), and it is preferable to use the 2-norm of the ellipsometer system matrix W to find the condition number in this embodiment, as shown in equation (15):

κ₂(W)＝||W||₂·||W^-1||₂(15)

wherein, κ₂(W) is the corresponding condition number under 2 norms of the ellipsometer system matrix W, | W | | count₂Is the 2 norm of the matrix W.

Because each ellipsometer system parameter has an influence on the W condition number, only one parameter is needed to be used as a variable quantity in the W condition number calculation process, and the corresponding relation between the condition number of the matrix W and the ellipsometer system parameter can be obtained by fixing the rest parameters. The parameters P of the dual-rotation compensator ellipsometer system are optimized in turn,₁,₂a, N. Firstly, taking the bias angle P of the polarizer 2 relative to the incident surface as a variable, taking one of the other 4 parameters to be in accordance with an actual arbitrary fixed value, namely obtaining the relation that the value of the condition number of the matrix W changes along with the value of the P value, and obtaining the P value at the minimum position of the condition number of the matrix W at the moment and the corresponding condition value: p₁，κ₂(W(P₁)). The same method is adopted to obtain the relation between other parameters and the condition number of the matrix W, and the parameter value and the corresponding condition value at the minimum position of the condition number of the matrix W are obtained:_1,1，κ₂(W(_1,1))，_2,1，κ₂(W(_2,1))，A₁，κ₂(W(A₁))。

(3) and (3) calculating the difference between the maximum value and the minimum value of the matrix W condition number of each parameter by using the relation between the parameters of the ellipsometer system of each double-rotation compensator obtained in the step (2) and the matrix W condition number, and arranging the parameters from large to small according to the difference. A larger difference indicates a larger influence of the parameter on the condition number of the matrix W, and priority needs to be given in optimization. The system parameters of the double-rotation compensator type ellipsometer are arranged from large to small according to the difference value of matrix W condition numbers corresponding to the parameters:₁,₂,P,A,N。

(4) and (4) sequentially optimizing each parameter according to the arrangement sequence of the parameters of the double-rotation compensator type ellipsometer system in the step (3).

Step 1, firstly, optimizing the phase delay amount of the deflection arm rotation compensator 3₁Will be_2,1，P₁，A₁，N₁The system parameters are used as initial values to obtain the condition number and₁obtaining the minimum value of the condition number of the matrix W and the corresponding system parameter value:_1,2，κ₂(W(_1,2))。

will be provided with_1,2，P₁，A₁，N₁The system parameters are used as initial values to obtain the condition number of the matrix W and the phase delay of the analyzer arm rotation compensator 5₂The minimum value of the condition number of the matrix W and the corresponding phase delay amount are obtained according to the relation between the matrix W and the condition number:_2,2，κ₂(W(_2,2))。

will be provided with_1,2，_2,2,A₁，N₁And (3) taking the system parameters as initial values to obtain the relation between the matrix W condition number and the bias angle P of the polarizer 2 relative to the incident plane, and obtaining the minimum value of the matrix W condition number and the corresponding system parameter values: p₂，κ₂(W(P₂))。

Will be provided with_1,2，_2,2，P₁，N₁And (3) taking the system parameters as initial values to obtain the relation between the matrix W condition number and the offset angle A of the analyzer 6 relative to the incident surface, and obtaining the minimum value of the matrix W condition number and the corresponding system parameter values: a. the₂，κ₂(W(A₂))。

Will be provided with_1,2，_2,2，P₂，A₂Taking the system parameters as initial values, obtaining the relation between the matrix W condition number and the single-period sampling point number N, and obtaining the minimum value of the matrix W condition number and the corresponding system parameter values: n is a radical of₂，κ₂(W(N₂))。

Thus obtaining a group of optimized parameters of the dual-rotation compensator type Mueller matrix ellipsometer system:_1,2，_2,2，P₂，A₂，N₂。

and 2, repeating the optimization process of the step 1 for n times to obtain the n-th group of optimized parameters of the double-rotation compensator type ellipsometer system and the minimum condition number of the matrix W corresponding to each parameter:_1,n，κ₂(W(_1,n))，_2,n，κ₂(W(_2,n))，P_n，κ₂(W(P_n))，A_n，κ₂(W(A_n))，N_n，κ₂(W(N_n)). Selecting ellipsometer system parameters having the greatest influence on matrix W condition number₁The system parameter, Δ κ (W: (W) (W))₁))=κ₂(W(_1,n))-κ₂(W(_1,n-1) When Δ κ (W: (W) ()₁))<The optimization is stopped, wherein the termination conditions for the optimization process are determined according to the actual requirements.

Thus, the parameter values of the optimized double-rotation compensator type muller matrix ellipsometer system in the nth group can be considered as follows:_1,n，2,_n，P_n，A_n，N_ni.e. the optimal system parameters meeting the current optimization requirements.

Fig. 3 is a simulation result diagram of simulation measurement of a dual-rotation compensator type muller matrix ellipsometer. In simulation, a sample to be tested is ideal air, and the Mueller matrix of the sample is as follows:

$M=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)---\left(16\right)$

in the simulation, the ideal light intensity signal I is measured_qAdding a mean value of 0 and a standard deviation of 0.01I_qAfter one gaussian noise is generated, the mueller matrix of the ideal air is solved through equations (12) to (14). FIG. 3 shows the results of the solution under two different ellipsometer system parameters, wherein m is the value₀₀The constant value of 1 for the normalized element is not shown. The system parameter values of the unoptimized dual-rotation compensator type muller matrix ellipsometer are as follows:₁=90°，₂=90 °, P =10 °, a =10 °, N = 50. The optimized ellipsometer system parameter values are as follows:₁=124°，₂=124 °, P =45 °, a =45 °, N =70, and it can be seen from the figure that, for the same measurement noise, the simulation result under the optimized ellipsometer system parameter configuration condition fluctuates less than the unoptimized simulation result and is closer to the mueller matrix value of the ideal air. The optimized system parameters of the dual-rotation compensator type Mueller matrix ellipsometer obtained by the simulation are not the optimal system parameters of the structure type, and are the results after two times of optimization, and different optimization results can be obtained in the actual optimization process according to different optimization standards.

The optimization process of the ellipsometer system parameters of other different structure types is similar to the implementation process of the optimization of the dual-rotation compensator type muller matrix ellipsometer, and only a specific system model needs to be changed into a system model suitable for the ellipsometer of a specific structure type.

The above description is an embodiment of the present invention, but the present invention should not be limited to the disclosure of the embodiment and the drawings. Therefore, it is intended that all equivalents and modifications which do not depart from the spirit of the invention disclosed herein are deemed to be within the scope of the invention.

Claims (4)

1. A method for optimizing ellipsometer system parameters for determining optimal system parameters for an ellipsometer, the method comprising:

(1) establishing a system model of the ellipsometer, and arranging the ellipsometer system model into a vector and vector dot product form, so as to obtain a system matrix W containing ellipsometer system parameters;

(2) calculating the condition number of the system matrix W under each ellipsometer system parameter;

(3) calculating the difference value between the maximum condition number and the minimum condition number of the system matrix W under each system parameter, and sequencing the corresponding system parameters according to the sequence of the difference values from large to small;

(4) and respectively and sequentially performing loop iteration optimization on each system parameter of the ellipsometer according to the sequence until the difference value between the minimum condition value of the most-front system parameter and the minimum condition value of the system parameter optimized at the previous time is less than the threshold value, thereby completing the optimization, and at the moment, each corresponding system parameter is the optimized ellipsometer system parameter.

2. The ellipsometer system parameter optimizing method according to claim 1, wherein the specific process of performing the iterative optimization on the ellipsometer system parameter is as follows:

(1) for any ellipsometer system parameter, taking the current values of the rest system parameters as fixed values, obtaining the relation between the system matrix W and the ellipsometer system parameter, and further obtaining the minimum value of the condition number of the system matrix W and the system parameter value corresponding to the minimum value, so as to serve as the updated system parameter after one iteration;

in the same way, the updated values of the rest system parameters after one iteration are sequentially obtained;

(2) repeating the above process for multiple times, wherein each time the process is repeated, a group of updated ellipsometer system parameters is obtained, so that multiple iterative updates of each system parameter are performed, and the latest group of ellipsometer system parameters, namely the optimized ellipsometer system parameters, can be obtained until the iteration termination condition is met.

3. The method of claim 1 or 2, wherein the ellipsometer system matrix W has a condition number of:

κ_p(W)＝||W||_p·||W^-1||_p

wherein, $\left|\right|W|{|}_p=\underset{x\&Element;D\left(W\right)}{sup}\frac{\left|\right|W\&CenterDot;x|{|}_p}{\left|\right|x|{|}_p},{\left(\right|\left|x\right|{|}_p)}^p=\underset{i}{\&Sigma;}{x}_i^p,$ W^-1＝(W^T·W)^-1·W^T，κ_p(W) is the condition number of the ellipsometer system matrix W, | | W | | caltrop_pIs the p-norm of the ellipsometer system matrix W, D (W) is the spatial domain of the matrix W, x is an arbitrary vector in the spatial domain of the matrix W, x_iDenotes the ith element, W, in the vector x^-1The superscript T represents the matrix transpose, which is the generalized inverse of the matrix W.

4. The ellipsometer system parameter optimization method according to claim 1 or 2, wherein the system model is:

$I_q=A_q^T{\mathrm{MS}}_q=\underset{j=0}{\overset{3}{\&Sigma;}}\underset{k=0}{\overset{3}{\&Sigma;}}{a}_{q,j}{m}_{j,k}{s}_{q,k}=\underset{j=0}{\overset{3}{\&Sigma;}}\underset{k=0}{\overset{3}{\&Sigma;}}{w}_{q,j,k}{m}_{j,k}$

wherein, I_qIs the light intensity signal of the detector, A_qTo analyze the arm vector, S_qFor a polarizing arm vector, subscript Q is 0,1, …, Q-1 represents the qth measurement component of the instrument, Q is the measurement times, M is the Mueller matrix of the sample to be measured, wherein the elements are represented by M_j,kIs shown as a_q,j,s_q,kRespectively correspond to A_qAnd S_qElement in vector, w_q,j,kRepresenting the measurement component I of the q-th order_qElement m to be measured for j row and k column_j,kThe index j is 0,1,2,3 and k is 0,1,2,3 respectively represent the number of elements in the vector;

any element W in the system matrix W_qObtained by the following formula:

$I_q=W_q\&CenterDot;\stackrel{\&RightArrow;}{M}=\left(\begin{array}{cccc}{a}_{q,0}{s}_{q,0}& a_{q,0}{s}_{q,1}& ...& a_{q,3}{s}_{q,3}\end{array}\right)\&CenterDot;\left(\begin{array}{c}{m}_{0,0}\\ m_{0,1}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ m_{3,3}\end{array}\right).$