CN112378860A - Calibration method for system parameters of rotary device type Mueller matrix ellipsometer - Google Patents

Calibration method for system parameters of rotary device type Mueller matrix ellipsometer Download PDF

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CN112378860A
CN112378860A CN202011146847.2A CN202011146847A CN112378860A CN 112378860 A CN112378860 A CN 112378860A CN 202011146847 A CN202011146847 A CN 202011146847A CN 112378860 A CN112378860 A CN 112378860A
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陈修国
陈超
盛胜
周军宏
刘世元
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Huazhong University of Science and Technology
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Abstract

The invention belongs to the technical field related to the calibration of a precision optical measuring instrument system, and discloses a calibration method of a rotating device type Mueller matrix ellipsometer system parameter, which comprises the following steps: respectively replacing an imaging lens and an objective lens of the rotary device type Mueller matrix ellipsometer with an 1/4 standard wave plate and a reflector; rotating the azimuth angle of the 1/4 standard wave plate for multiple times, and respectively acquiring light intensity information under multiple azimuth angles from a detector of a rotary device type Mueller matrix ellipsometer; carrying out Fourier analysis on the light intensity information to obtain a Fourier coefficient of the light intensity information; establishing a relation model of Fourier coefficients of the light intensity information and parameters of a system to be calibrated; and continuously adjusting the value of the system parameter to be calibrated in the relation model until the error between the Fourier coefficient in the relation model and the Fourier coefficient is within a preset range, wherein the value corresponding to the system parameter to be calibrated is the calibration value. The method can realize the calibration of a plurality of calibration values through two-step calibration, and is simple and convenient.

Description

Calibration method for system parameters of rotary device type Mueller matrix ellipsometer
Technical Field
The invention belongs to the technical field related to calibration of a precision optical measuring instrument system, and particularly relates to a calibration method for system parameters of a rotary device type Mueller matrix ellipsometer.
Background
The rotating device type ellipsometer is a common ellipsometer and has the advantages of simple modulation, high measurement precision, nondestructive measurement and the like. The rotating device type high-resolution imaging muller matrix ellipsometer is based on a dual-rotation compensator type muller matrix ellipsometer, combines a microscopic imaging technology, utilizes a muller matrix containing more polarization information, can completely represent polarization characteristics of a sample, such as depolarization and the like, has high spatial resolution, can realize distribution measurement of a microcell material, and has greater advantages than a traditional rotating device type ellipsometer. In order to achieve higher resolution, the rotating device type high resolution imaging mueller matrix ellipsometer generally adopts a vertical objective lens type configuration scheme, in which a beam splitting device and an objective lens are used to form an optical path multiplexing system.
In order to ensure the accuracy of the measurement, before the material is measured and characterized by using a rotating device type high-resolution imaging muller matrix ellipsometer, system parameters of the instrument need to be calibrated, and the parameters to be calibrated include: 1, initial azimuth angles of a polarizer, an analyzer, a first rotating compensator and a second rotating compensator; 2, phase delay amount of the first and second rotation compensators; 3, residual polarization effect of the beam splitting device, and a beam splitting film system in the beam splitting device generally influences the amplitude ratio and the phase difference of polarized light; 4, the polarization aberration of the objective lens, generally, for the objective lens with Numerical Aperture (NA) larger than 0.6, the effect of the polarization aberration must be considered; and 5, the relationship between the incident angle of the rotary device type high-resolution imaging Mueller matrix ellipsometer and the electric rotation angle of the plane mirror.
The existing calibration method rarely considers the polarization aberration of the objective lens during calibration, or uses an off-line calibration mode to calibrate the beam splitting device and the objective lens, the calibration process is complicated, the calibration error is large, the calibration speed is slow, the universality is poor, and the precision measurement requirement of the rotary device type high-resolution imaging muller matrix ellipsometer cannot be met.
Disclosure of Invention
Aiming at the defects or the improvement requirements in the prior art, the invention provides a method for calibrating system parameters of a rotary device type muller matrix ellipsometer, which comprises the steps of firstly, respectively replacing an imaging lens and an objective lens of the rotary device type muller matrix ellipsometer with an 1/4 standard wave plate and a reflector, obtaining a Fourier coefficient of light intensity information at the moment, calibrating the Fourier coefficient in a relation model by using the Fourier coefficient, and obtaining the azimuth angles of a corresponding polarizer, an analyzer, a first rotary compensator and a second rotary compensator, the phase delay of the first rotary compensator and the second rotary compensator and the residual polarization effect of a beam splitter as the relation model is a function of the Fourier coefficient of the light intensity information and the system parameters to be calibrated; secondly, after the parameters of the system to be calibrated are calibrated, the 1/4 standard wave plate and the reflector are respectively replaced by the imaging lens and the objective lens, an isotropic uniform film is arranged on the sample stage, the objective lens and the film are taken as a whole to be a sample to be measured, a Mueller matrix of the sample to be measured is obtained, a calculation model of the Mueller matrix of the sample to be measured is established, and key parameters for solving the relation between the polarization aberration calibration value of the objective lens and the incident angle of the rotating device type Mueller matrix ellipsometer and the rotating angle of the reflector are obtained according to the Mueller matrix and the calculation model.
To achieve the above object, according to an aspect of the present invention, there is provided a method for calibrating a parameter of a rotating device type mueller matrix ellipsometer, the method including: s1, replacing the imaging lens and the objective lens of the rotary device type Mueller matrix ellipsometer with a 1/4 standard wave plate and a reflector respectively; s2, rotating the azimuth angle of the 1/4 standard wave plate for multiple times, and respectively acquiring light intensity information under the multiple azimuth angles from a detector of the rotary device type Mueller matrix ellipsometer; s3, carrying out Fourier analysis on the light intensity information to obtain Fourier coefficients of the light intensity information; s4, establishing a relation model of Fourier coefficients of light intensity information and system parameters to be calibrated, wherein the system parameters to be calibrated comprise azimuth angles of a polarizer, an analyzer, a first rotating compensator and a second rotating compensator, phase delay amounts of the first rotating compensator and the second rotating compensator and residual polarization effects of a beam splitting device; and S5, continuously adjusting the value of the system parameter to be calibrated in the relational model until the error between the Fourier coefficient in the relational model and the Fourier coefficient in the step S3 is within a preset range, and the value corresponding to the system parameter to be calibrated is the calibration value.
Preferably, the step S4 specifically includes: s41, obtaining the polarization state of the received light of the detector, and simplifying the polarization state to obtain the light intensity information of the received light of the detector; s42, obtaining the relation model of the Fourier coefficient and the system parameter to be calibrated according to the light intensity information in the step S41.
Preferably, the polarization state expression of the received light of the detector in step S41 is:
Sout＝MaR(Ap)R(-C2)Mc(δ2)R(C2)Mbt*R(Cs)Mc(δ)R(-Cs)MsR(-Cs)Mc(δ)R(Cs)Mbt*R(-C1)Mc(δ1)R(C1)R(-Pp)MpR(Pp)Sin
wherein S isoutIs the Stokes vector of the received light of the detector, SinStokes vector of the light emitted by the light source, Ma、Mc、MpAnd MsMueller matrices, delta, of analyzers, compensators, polarizers, and said mirrors, respectively1、δ2And δ is the phase retardation of the first rotation compensator, the second rotation compensator and the 1/4 standard waveplate, R (A)p)、R(C1)、R(C2)、R(Pp) And R (C)s) Respectively is the inspectionPolarizer, first rotation compensator, second rotation compensator, polarizer and rotation matrix of 1/4 standard wave plate Ap、C1、C2、PpAnd CsActual azimuth angles of the analyzer, the first rotation compensator, the second rotation compensator, the polarizer and the 1/4 standard wave plate respectively, wherein C1＝Cs1-5wt，C2＝Cs2-3wt，Cs1And Cs2Initial azimuth angles of the first rotary compensator and the second rotary compensator, w is a rotation fundamental frequency of the servo motor, MbrAnd MbtRespectively, the mueller matrices of the non-polarizing beam splitting device in reflection and in transmission, wherein,
Figure BDA0002739943730000031
Figure BDA0002739943730000041
Figure BDA0002739943730000042
in the formula, ΨrAnd ΔrAmplitude ratio and phase difference, psi, of orthogonally polarized light, respectively, upon reflection by the non-polarizing beam splitting meanstAnd ΔtThe amplitude ratio and the phase difference of the orthogonal polarized light when transmitted by the non-polarization beam splitting device are respectively.
Preferably, the expression of the light intensity information i (t) received by the detector after the polarization state is simplified in step S41 is as follows:
Figure BDA0002739943730000043
wherein, I0As a function of the spectral response, α0Is a direct Fourier coefficient, alpha2nAnd beta2nNamely fourier in the relational modelCoefficient, M11Is the mueller matrix element (1, 1).
Preferably, in step S5:
the calculation formula of the error between the fourier coefficient in the relational model and the fourier coefficient in step S3 is:
Figure BDA0002739943730000044
wherein, MFCiThe Fourier coefficient, Fc, obtained in step S3 at the i azimuth angle of the 1/4 standard waveplateiAnd (3) taking Fourier coefficients of the 1/4 standard wave plate in the relation model at the ith azimuth angle.
Preferably, the step S2 includes collecting light intensity information over a plurality of periods and averaging the light intensity information.
Preferably, the method further comprises: s6, the 1/4 standard wave plate and the reflector are respectively replaced by the imaging lens and the objective lens, and the objective lens and the film are taken as a whole to be a sample to be measured by arranging an isotropic uniform film on the sample stage; s7, adjusting the angle of the reflector to change the incident angle of the sample to be detected, and obtaining second light intensity information corresponding to the incident angle by the detector to further obtain a Mueller matrix of the sample to be detected; s8, establishing a computational model of the Mueller matrix of the sample to be measured; s9, continuously adjusting parameters in the calculation model until the error between the Mueller matrix of the sample to be measured calculated by the calculation model and the Mueller matrix obtained in the step S7 is within a certain range; and S10, substituting the parameters determined in the step S9 into the polarized aberration calculation formula of the objective lens to obtain the polarized aberration calibration value of the objective lens.
Preferably, the calculation model of the mueller matrix of the sample to be measured is:
Figure BDA0002739943730000051
wherein the content of the first and second substances,
Figure BDA0002739943730000052
is the angle of incidence
Figure BDA0002739943730000053
Mueller matrix of the said film, MOB(rho, theta) is the Mueller matrix of the objective lens under the polar angle theta and the polar diameter rho,
Figure BDA0002739943730000054
therein, ΨbrIs the amplitude ratio angle, Δ, of the objective lensbrIs the phase difference angle of the objective lens.
Preferably, the calculation formula of the polarization aberration of the objective lens is as follows:
Figure BDA0002739943730000055
Figure BDA0002739943730000056
wherein k is the number of sampling points, ZlIs the coefficient of the Zernike polynomial of the first term of the corresponding parameter, ∈ is the fitting error of the corresponding parameter, fl(ρk，θk) Is Zernike polynomial of the I term, and L is the number of the maximum term of the Zernike polynomial.
Preferably, the step S10 further includes obtaining a relationship between an incident angle and a mirror rotation angle of the rotating device type muller matrix ellipsometer according to the parameter.
In general, compared with the prior art, the calibration method for the parameters of the rotating device type muller matrix ellipsometer provided by the invention has the following beneficial effects:
1. according to the method, firstly, an imaging lens and an objective lens of a rotary device type muller matrix ellipsometer are respectively replaced by an 1/4 standard wave plate and a reflector, and a relation model of a Fourier coefficient of light intensity information and a system parameter to be calibrated is established to directly obtain azimuth angles of a polarizer, an analyzer, a first rotary compensator and a second rotary compensator, phase delay amounts of the first rotary compensator and the second rotary compensator and a calibration value of a residual polarization effect of a beam splitting device;
2. setting an isotropic uniform film on a sample stage, taking the film and an objective lens as a sample to be measured, establishing a calculation model of a Mueller matrix of the sample to be measured, and obtaining a polarization aberration calibration value of the objective lens and key parameters of a relation between an incident angle of a rotating device type Mueller matrix ellipsometer and a reflector rotation angle according to the calculation model;
3. the method for solving the parameters to be calibrated in the relation model and the calculation model by adopting the nonlinear regression fitting has the advantages of high calibration precision, good robustness, high measurement speed and the like compared with a numerical solving method;
4. the parameter calibration method of the rotary device type high-resolution imaging Mueller matrix ellipsometer system only needs two-step calibration, all parameters to be calibrated can be calibrated, calibration complexity is greatly reduced, calibration efficiency is improved, and error accumulation among calibration steps is reduced.
Drawings
Fig. 1 schematically illustrates an optical path diagram of a rotating device-type mueller matrix ellipsometer system according to an embodiment of the present disclosure;
fig. 2 schematically illustrates a step diagram of a first calibration method of a rotary device-type mueller matrix ellipsometer system according to an embodiment of the present disclosure;
FIG. 3 schematically illustrates an optical path diagram of a rotating device-type Mueller matrix ellipsometer system during a first calibration step according to an embodiment of the present disclosure;
FIG. 4 schematically illustrates a step diagram of a second calibration method for a rotary device type Muller matrix ellipsometer system according to an embodiment of the present disclosure;
FIG. 5 schematically illustrates an optical path diagram for a rotating device-type Mueller matrix ellipsometer system at varying angles of incidence during a second calibration step according to an embodiment of the present disclosure;
FIG. 6 is a schematic diagram illustrating data fitting results during a first calibration step according to an embodiment of the disclosure;
FIG. 7 is a diagram schematically illustrating a result of fitting data during a second calibration step according to an embodiment of the present disclosure.
The same reference numbers will be used throughout the drawings to refer to the same or like elements or structures, wherein:
101-light source, 102-collimator, 103-plane mirror, 401-polarizer, 402-first compensator, 105-non-polarization beam splitting device, 106-lens, 107-objective lens, 108-sample stage, 901-second compensator, 902-analyzer, 801-1/4 standard wave plate, 802-plane mirror and 110-detector.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
Fig. 1 shows a schematic diagram of an optical path of a rotating device type mueller matrix ellipsometer system to be calibrated according to the present invention. The rotating device type muller matrix ellipsometer system comprises a light source 101, wherein the light source emits light beams to a collimator 102 to be collimated, the collimated light beams are sent to a plane reflector 103, the plane reflector 103 sends reflected light to a polarizer 401 and a first compensator 402, the first compensator 402 transmits the light beams to a non-polarization beam splitting device 105, the non-polarization beam splitting device 105 sends partial light beams to a lens 106 and an objective lens 107 which are arranged below the non-polarization beam splitting device, the objective lens 107 irradiates the light beams to a sample of a sample stage 108, the sample on the sample stage 108 reflects the light beams back to the objective lens 107 and the lens 106, and the light beams are transmitted to a second compensator 901, an analyzer 902 and a detector 110 which are arranged above the sample stage through the non-polarization beam splitting device 105. The polarizer 401 and the first compensator 402 form a polarizing arm, the second compensator 901 and the analyzer 902 form an analyzing arm, the first compensator 402 and the second compensator 901 are controlled by a servo motor and are provided with high-precision encoders, synchronous acquisition of a control system is realized based on STM32, and the objective lens 107 and the lens 106 form an imaging system. The plane mirror 103 is used for controlling the illumination incident angle of the sample, the positions of the objective lens 107 and the lens 106 are in a conjugate relation, and the deflection of the incident beam is changed by adjusting the rotation angle of the plane mirror 103. Thereby changing the angle of light exiting the objective lens 107. The sample stage 108 is composed of an electric rotary stage and an electric positioning stage, the electric rotary stage can change the azimuth angle of the measured sample, and the electric displacement stage can adjust the focal length of the sample and move the measurement area.
The accuracy of the calibration of the rotary device-type mueller matrix ellipsometer system directly affects the measurement accuracy of the instrument, and therefore, errors in the instrument must be analyzed and reasonably compensated before measurement.
The present application primarily considers the following calibration parameters: (1) azimuth angle errors of the polarizer 401, the analyzer 902, the first compensator 402 and the second compensator 901 cannot accurately obtain azimuth angle information when the device is installed; (2) the phase delay amounts of the first compensator 401 and the second compensator 402, which are functions of the wavelength and are affected by the measurement environment; (3) due to the residual polarization effect of the non-polarization beam splitter 105, the ideal non-polarization beam splitter 105 does not affect the polarization device of the light beam, but the polarization state of the incident light beam is affected due to the complex film system of the actual beam splitter; (4) the polarization aberration of the objective lens 107 is larger as the NA of the objective lens 107 is increased, and the included angle of the light refraction on the objective lens 107 is larger, so that the fresnel formula shows that the polarization state of the light is greatly influenced, the birefringence of the coating material is smaller, the stress birefringence effect also influences the polarization state of the light, and the influence of the objective lens 107 on the polarization state of the light is the polarization aberration, so that the polarization aberration of the objective lens 107 needs to be considered in order to improve the measurement accuracy of the instrument; (5) the incidence angle of the rotary device type muller matrix is related to the rotation angle of the electric rotating platform of the plane reflector 103, the rotary device type muller matrix ellipsometer drives the plane reflector by using the electric rotating platform to realize different illumination incidence angles, and the accuracy of the illumination incidence angles has great influence on the measurement accuracy of the high-resolution imaging muller matrix ellipsometer.
The method realizes the calibration of the instrument in two steps, and comprises the following steps: (1) azimuth angles of the polarizer 401, the analyzer 902, the first compensator 402, and the second compensator 901; (2) the phase delay amounts of the first compensator 401 and the second compensator 402; and (3) the residual polarization effect of the non-polarizing beam splitting device 105. And a second step of calibration: (4) the polarization aberration of the objective lens 107 and (5) the incident angle of the rotating device type mueller matrix, and the rotation angle of the electric rotating stage of the plane mirror 103.
First, a first calibration is performed, and as shown in fig. 2, the method specifically includes the following steps S1 to S5.
S1, replacing the imaging lens and the objective lens of the rotary device type Mueller matrix ellipsometer with a 1/4 standard wave plate and a reflector respectively;
to realize the calibration, the optical path configuration of the rotating device-type mueller matrix ellipsometer system needs to be changed, the imaging part, that is, the objective lens 107 and the lens 106, is detached and replaced with the standard wave plate 801 and the plane mirror 802 of the calibration sample 1/4, as shown in fig. 3, the 1/4 standard wave plate 801 is placed on the mirror 802 to ensure the parallelism therebetween, and at this time, the system can be regarded as a common mueller matrix ellipsometer without imaging function. The plane mirror 103 is adjusted so that the incident light is perpendicularly incident on the calibration sample.
S2, rotating the azimuth angle of the 1/4 standard wave plate for multiple times, and respectively acquiring light intensity information under the multiple azimuth angles from a detector of the rotary device type Mueller matrix ellipsometer;
the azimuth angle of the calibration sample piece is changed at equal intervals through the electric rotating platform, namely, the azimuth angle of the 1/4 standard wave plate is rotated for multiple times to measure the calibration sample piece by using an instrument to be calibrated, so that light intensity information under a plurality of groups of azimuth angles of the calibration sample piece is obtained, the light intensity information under each azimuth angle is collected at the detector 110, preferably, the light intensity information in a plurality of periods is collected, the light intensity information is averaged to reduce the influence of the stability of the light source, and therefore the measurement accuracy is improved. The calibration azimuth angle is larger than two groups, and the final fitting result tends to be stable along with the increase of the number of the measurement groups.
S3, carrying out Fourier analysis on the light intensity information to obtain Fourier coefficients of the light intensity information;
and Fourier analysis is carried out on the light intensity information to obtain a Fourier coefficient required by calibration.
S4, establishing a relation model of Fourier coefficients of light intensity information and system parameters to be calibrated, wherein the system parameters to be calibrated comprise azimuth angles of a polarizer, an analyzer, a first rotating compensator and a second rotating compensator, phase delay amounts of the first rotating compensator and the second rotating compensator and residual polarization effects of a beam splitting device;
in the relation model, the input is a parameter to be calibrated, and the output is a Fourier coefficient of a measurement signal. The method comprises the following steps:
s41, obtaining the polarization state of the received light of the detector, and simplifying the polarization state to obtain the light intensity information of the received light of the detector;
the polarization state expression of the received light of the detector is as follows:
Figure BDA0002739943730000101
wherein S isoutIs the Stokes vector of the received light of the detector, SinStokes vector of the light emitted by the light source, Ma、Mc、MpAnd MsMueller matrices, delta, of analyzers, compensators, polarizers, and said mirrors, respectively1、δ2And δ is the phase retardation of the first rotation compensator, the second rotation compensator and the 1/4 standard waveplate, R (A)p)、R(C1)、R(C2)、R(Pp) And R (C)s) The rotation matrixes of the analyzer, the first rotation compensator, the second rotation compensator, the polarizer and the 1/4 standard wave plate are respectively，Ap、C1、C2、PpAnd CsActual azimuth angles of the analyzer, the first rotation compensator, the second rotation compensator, the polarizer and the 1/4 standard wave plate respectively, wherein C1＝Cs1-5wt，C2＝Cs2-3wt，Cs1And Cs2Initial azimuth angles of the first rotary compensator and the second rotary compensator, w is a rotation fundamental frequency of the servo motor, MbrAnd MbtRespectively, the mueller matrices of the non-polarizing beam splitting device in reflection and in transmission, wherein,
Figure BDA0002739943730000111
Figure BDA0002739943730000112
Figure BDA0002739943730000113
in the formula, ΨrAnd ΔrAmplitude ratio and phase difference, psi, of orthogonally polarized light, respectively, upon reflection by the non-polarizing beam splitting meanstAnd ΔtThe amplitude ratio and the phase difference of the orthogonal polarized light when transmitted by the non-polarization beam splitting device are respectively.
The expression of the light intensity information i (t) received by the detector after the polarization state is simplified is as follows:
Figure BDA0002739943730000114
wherein, I0As a function of the spectral response, α0Is a direct Fourier coefficient, alpha2nAnd beta2nI.e. the Fourier coefficients, M, in the relational model11Is the mueller matrix element (1, 1).
S42, obtaining the relation model of the Fourier coefficient and the system parameter to be calibrated according to the light intensity information in the step S41.
From the analytical development, α1，α3，α5，…，α29，α31And beta1，β3，β5，…，β29，β31All are 0, and the remaining fourier coefficients are expressed as:
α2＝-0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s2·sinδ1·cos2(Pp-Ap) (6a)
β2＝-0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s2·sinδ1·sin2(Pp-Ap) (6b)
α4＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·sinδ1·sinδ2·cos2(Pp-Ap) (6c)
β4＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·sinδ1·sinδ2·sin2(Pp-Ap) (6d)
α6＝sin2Ψr·sin2Ψt·sin(Δr+Δt)·c1·sinδ2·sin2Pp·sin2Ap (6e)
β6＝-sin2Ψr·sin2Ψt·sin(Δr+Δt)·c1·sinδ2·sin2Pp·cos2Ap (6f)
Figure BDA0002739943730000121
Figure BDA0002739943730000122
α10＝sin2Ψr·sin2Ψt·sin(Δr+Δt)·c2·sinδ1·sin2Pp·sin2Ap (6i)
β10＝-sin2Ψr·sin2Ψt·sin(Δr+Δt)·c2·sinδ1·cos2Pp·sin2Ap (6j)
Figure BDA0002739943730000123
Figure BDA0002739943730000124
α14＝-0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s1·sinδ2·cos2(Pp-Ap) (6m)
β14＝-0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s1·sinδ2·sin2(Pp-Ap) (6n)
Figure BDA0002739943730000125
Figure BDA0002739943730000126
Figure BDA0002739943730000131
Figure BDA0002739943730000132
α22＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s2·sinδ1·cos2(P+Ap) (6s)
β22＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s2·sinδ1·sin2(Pp+Ap) (6t)
α26＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s1·sinδ2·cos2(Pp+Ap) (6u)
β26＝0.5·sin2Ψr·sin2Ψt·sin(Δr+Δt)·s1·sinδ2·sin2(Pp+Ap) (6v)
Figure BDA0002739943730000133
Figure BDA0002739943730000134
wherein, cj＝cos2(δj/2)，sj＝sin2(δj/2)。
And S5, continuously adjusting the value of the system parameter to be calibrated in the relational model until the error between the Fourier coefficient in the relational model and the Fourier coefficient in the step S3 is within a preset range, and the value corresponding to the system parameter to be calibrated is the calibration value.
As can be seen from the above equations (6a) - (6x), the input variable in the equation with the output fourier coefficient is the parameter variable to be calibrated, i.e. the fourier coefficient Fc at the i-th azimuth angle of the calibration sampleiIs the parameter to be calibrated (A)p，Pp，Cs1，Cs2，Cs，δ1，δ2，δ，Ψr，Ψt，Δr，Δt) As a function of (c). Fourier coefficient MFc obtained in connection with step S3iAnd obtaining the error calculation formula of the two as follows:
Figure BDA0002739943730000135
wherein, MFCiThe Fourier coefficient, Fc, obtained in step S3 at the i azimuth angle of the 1/4 standard waveplateiAnd (3) taking Fourier coefficients of the 1/4 standard wave plate in the relation model at the ith azimuth angle. The corresponding parameters to be calibrated can be obtained by solving according to a nonlinear regression method, and the fitting result is shown in fig. 6, wherein the abscissa is 24 fourier coefficients, and the ordinate is the numerical value of the fourier coefficient.
To this end, the first calibration step is completed, after the calibration of the corresponding device is performed by using the data after the first calibration step, the calibration sample is removed, and the imaging part (the lens 106 and the objective lens 107) is installed again, the objective lens 107 is a core component in the high-resolution imaging system, as the NA of the objective lens 107 increases, the included angle of the light refraction on the NA increases, which may have a large influence on the polarization state of the light, and in addition, the birefringence effect and stress of the coating material may also have an influence on the polarization state of the light, and this influence on the polarization state of the light by the objective lens becomes polarization aberration, especially the polarization aberration of the high-NA objective lens with the NA greater than 0.6 is more obvious. In order to improve the measurement accuracy of the instrument, the polarization aberration of the objective lens 107 must be taken into account; another purpose of the second calibration step is to calibrate the relationship between the measured incident angle and the angle of rotation of the mirror electric rotating table. The specific steps of the second calibration are as follows, and as shown in FIG. 4, the method includes steps S6-S10.
S6, the 1/4 standard wave plate and the reflector are respectively replaced by the imaging lens and the objective lens, and the objective lens and the film are taken as a whole to be a sample to be measured by arranging an isotropic uniform film on the sample stage;
the objective lens 107 is adjusted so that the light coincides with the axis of the objective lens 107 such that the incident angle of the light emitted from the objective lens 107 becomes 0 °. An isotropic, homogeneous film is placed on the sample stage with the objective lens 107 and isotropic sample as a whole.
S7, adjusting the angle of the reflector to change the incident angle of the sample to be detected, and obtaining second light intensity information corresponding to the incident angle by the detector to further obtain a Mueller matrix of the sample to be detected;
a measurement is taken using the instrument already calibrated in the first step and light intensity information at an angle of incidence of 0 deg. is obtained at the detector 110. As shown in fig. 5, the rotation angle of the plane mirror 103 is adjusted, so as to change the incident angle of the sample to be measured, the detector obtains second light intensity information corresponding to the incident angle, and further obtains a mueller matrix of the sample to be measured, where the mueller matrix is a mueller matrix required by calibration in the subsequent step.
S8, establishing a computational model of the Mueller matrix of the sample to be measured;
the calculation model of the mueller matrix of the sample to be measured is established based on the Zernike polynomial. The Zernike polynomials are a set of functions defined on a unit circle, with completeness and orthogonality, since the form of the Zernike polynomials, which are usually used to describe the wavefront aberrations, is almost identical to the aberrations of the optical system, and any image in the unit circle can be expanded by the Zernike polynomials, which are defined as follows:
Figure BDA0002739943730000151
where N is the order of the polynomial, m is the angular frequency of the sinusoidal component, (p, θ) are the polar diameter and polar angle, respectively, in the corresponding polar coordinates, N is the normalization factor,
Figure BDA0002739943730000152
is a radial polynomial.
The computational model of the mueller matrix of the sample to be measured in the application is as follows:
Figure BDA0002739943730000153
wherein the content of the first and second substances,
Figure BDA0002739943730000154
is the angle of incidence
Figure BDA0002739943730000155
The Mueller matrix of the film can be calculated from the film transmission matrix, MOB(ρ, θ) is the mueller matrix of the objective lens at the polar angle θ and the polar diameter ρ. The original calibration mode utilizes the characteristics of a spherical mirror, takes an objective lens as a sample, measures the polarization characteristics of the objective lens, uses a Mueller matrix to represent the polarization characteristics of the objective lens in research, and can see that the polarization characteristics of the objective lens have no obvious anisotropy from the Mueller matrix elements, so that the polarization aberration of the objective lens can be represented by a polarization parameter amplitude ratio angle and a phase difference angle. Thus, the mueller matrix for each point on the objective can be expressed as follows:
Figure BDA0002739943730000156
therein, ΨbrIs the amplitude ratio angle, Δ, of the objective lensbrIs the phase difference angle of the objective lens.
The calculation formula of the polarization aberration of the objective lens is as follows:
Figure BDA0002739943730000161
Figure BDA0002739943730000162
wherein k is the number of sampling points, ZlIs the coefficient of the Zernike polynomial of the first term of the corresponding parameter, ∈ is the fitting error of the corresponding parameter, fl(ρk，θk) Is Zernike polynomial of the I term, and L is the number of the maximum term of the Zernike polynomial.
By performing Zernike decomposition on equations (11) and (12), it can be found from the decomposition results of the previous 49-term Zernike coefficients that only the values of several terms of coefficients are relatively large, and it can be observed that the Zernike coefficients are relatively large and are Zernike moments of the same type, selecting a multi-pattern corresponding to the Zernike coefficients with coefficient values greater than 1% to describe the polarization aberration of the objective lens, and using the decomposed data as the initial values of the subsequent fitting.
The relationship between the incident angle of the rotary device type muller matrix ellipsometer and the rotation angle of the reflector can be obtained according to Abbe sine theorem, and the sine value of the deflection angle alpha of the reflector and the illumination incident angle
Figure BDA0002739943730000163
Sine values are in a direct proportional relationship, but due to instrument optical path defects and hardware noise, the actual relationship can be expressed as:
Figure BDA0002739943730000164
from the above analysis, the above equations (11) to (13) contain 12 unknowns, i.e., 8 Zernike coefficients, slope K, intercept D, and fitting thickness D of the standard film sample.
S9, continuously adjusting parameters in the calculation model until the error between the Mueller matrix of the sample to be measured calculated by the calculation model and the Mueller matrix obtained in the step S7 is within a certain range;
the following nonlinear equation (14) is solved to obtain the above 12 unknown parameters by using a nonlinear regression fitting method, where the expression of the nonlinear equation (14) is:
Figure BDA0002739943730000165
wherein M issimu(ρ, θ) is a sample mueller matrix obtained by calculating the model, and the mueller matrix obtained in the above step S7.
And S10, substituting the parameters determined in the step S9 into the polarized aberration calculation formula of the objective lens to obtain the polarized aberration calibration value of the objective lens.
The 12 unknowns can be solved by the formula (14), and the polarization aberration of the objective lens can be obtained by substituting the unknowns into the formula (11) and the formula (12); and substituting the solved K and D into the formula (13) to obtain the relation between the incident angle and the reflector rotation angle of the rotary device type Mueller matrix ellipsometer.
Fig. 7 shows the measurement and fitting results of the calibration mueller matrix, wherein the abscissa is the incident angle and the ordinate is the value of the mueller matrix. It can be seen from the figure that the measured data indicate that the polarization effect of the objective lens is isotropic, i.e. the mueller matrix elements in the off-diagonal part are close to 0. And as the incidence angle increases, the change amplitude of the Mueller matrix increases, because the light deflection increases near the edge of the objective lens, and the influence on the polarized light increases.
In summary, the calibration parameters of a plurality of devices can be obtained through two-step calibration, so that the calibration complexity is greatly reduced, the calibration efficiency is improved, and the error accumulation among the calibration steps is reduced; and the method of nonlinear regression fitting is combined to solve the relation model and the parameters to be calibrated in the calculation model, and compared with a numerical solving method, the method has the advantages of high calibration precision, good robustness, high measurement speed and the like.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (10)

1. A calibration method for system parameters of a rotary device type Mueller matrix ellipsometer is characterized by comprising the following steps:
s1, replacing an imaging lens and an objective lens of the rotary device type Mueller matrix ellipsometer with a 1/4 standard wave plate and a reflector respectively;
s2, rotating the azimuth angle of the 1/4 standard wave plate for multiple times, and respectively acquiring light intensity information under the multiple azimuth angles from a detector of the rotary device type Mueller matrix ellipsometer;
s3, carrying out Fourier analysis on the light intensity information to obtain Fourier coefficients of the light intensity information;
s4, establishing a relation model of Fourier coefficients of light intensity information and system parameters to be calibrated, wherein the system parameters to be calibrated comprise azimuth angles of a polarizer, an analyzer, a first rotating compensator and a second rotating compensator, phase delay amounts of the first rotating compensator and the second rotating compensator and residual polarization effects of a beam splitting device;
and S5, continuously adjusting the value of the system parameter to be calibrated in the relational model until the error between the Fourier coefficient in the relational model and the Fourier coefficient in the step S3 is within a preset range, and the value corresponding to the system parameter to be calibrated is the calibration value.
2. The calibration method according to claim 1, wherein step S4 specifically comprises:
s41, obtaining the polarization state of the received light of the detector, and simplifying the polarization state to obtain the light intensity information of the received light of the detector;
s42, obtaining the relation model of the Fourier coefficient and the system parameter to be calibrated according to the light intensity information in the step S41.
3. The calibration method according to claim 2, wherein the polarization state expression of the received light of the detector in step S41 is:
Sout＝MaR(Ap)R(-C2)Mc(δ2)R(C2)Mbt*R(Cs)Mc(δ)R(-Cs)MsR(-Cs)Mc(δ)R(Cs)Mbr*R(-C1)Mc(δ1)R(C1)R(-Pp)MpR(Pp)Sin
wherein S isoutIs the Stokes vector of the received light of the detector, SinStokes vector of the light emitted by the light source, Ma、Mc、MpAnd MsMueller matrices, delta, of analyzers, compensators, polarizers, and said mirrors, respectively1、δ2And delta is the first rotation compensator,Phase retardation of the 1/4 standard wave plate, R (A)p)、R(C1)、R(C2)、R(Pp) And R (C)s) Rotation matrices of the analyzer, the first rotation compensator, the second rotation compensator, the polarizer and the 1/4 standard wave plate, Ap、C1、C2、PpAnd CsActual azimuth angles of the analyzer, the first rotation compensator, the second rotation compensator, the polarizer and the 1/4 standard wave plate respectively, wherein C1＝Cs1-5wt，C2＝Cs2-3wt，Cs1And Cs2Initial azimuth angles of the first rotary compensator and the second rotary compensator, w is a rotation fundamental frequency of the servo motor, MbrAnd MbtRespectively, the mueller matrices of the non-polarizing beam splitting device in reflection and in transmission, wherein,
Figure FDA0002739943720000021
Figure FDA0002739943720000022
Figure FDA0002739943720000023
in the formula, ΨrAnd ΔrAmplitude ratio and phase difference, psi, of orthogonally polarized light, respectively, upon reflection by the non-polarizing beam splitting meanstAnd ΔtThe amplitude ratio and the phase difference of the orthogonal polarized light when transmitted by the non-polarization beam splitting device are respectively.
4. The calibration method according to claim 3, wherein the light intensity information I (t) received by the detector after the polarization state is simplified in step S41 is expressed as:
Figure FDA0002739943720000031
wherein, I0As a function of the spectral response, α0Is a direct Fourier coefficient, alpha2nAnd beta2nI.e. the Fourier coefficients, M, in the relational model11Is the mueller matrix element (1, 1).
5. Calibration method according to claim 1 or 4, wherein in step S5:
the calculation formula of the error between the fourier coefficient in the relational model and the fourier coefficient in step S3 is:
Figure FDA0002739943720000032
wherein, MFCiThe Fourier coefficient, Fc, obtained in step S3 at the i azimuth angle of the 1/4 standard waveplateiAnd (3) taking Fourier coefficients of the 1/4 standard wave plate in the relation model at the ith azimuth angle.
6. The calibration method according to claim 1, wherein step S2 comprises collecting light intensity information over a plurality of periods and averaging the light intensity information.
7. The calibration method according to claim 1, wherein the method further comprises:
s6, the 1/4 standard wave plate and the reflector are respectively replaced by the imaging lens and the objective lens, and the objective lens and the film are taken as a whole to be a sample to be measured by arranging an isotropic uniform film on a sample stage;
s7, adjusting the angle of the reflector to change the incident angle of the sample to be detected, and obtaining second light intensity information corresponding to the incident angle by the detector to further obtain a Mueller matrix of the sample to be detected;
s8, establishing a computational model of the Mueller matrix of the sample to be measured;
s9, continuously adjusting parameters in the calculation model until the error between the Mueller matrix of the sample to be measured calculated by the calculation model and the Mueller matrix obtained in the step S7 is within a certain range;
and S10, substituting the parameters determined in the step S9 into the polarized aberration calculation formula of the objective lens to obtain the polarized aberration calibration value of the objective lens.
8. The calibration method according to claim 7, wherein the computational model of the mueller matrix of the sample to be measured is:
Figure FDA0002739943720000041
wherein the content of the first and second substances,
Figure FDA0002739943720000042
is the angle of incidence
Figure FDA0002739943720000043
Mueller matrix of the said film, MOB(rho, theta) is the Mueller matrix of the objective lens under the polar angle theta and the polar diameter rho,
Figure FDA0002739943720000044
therein, ΨbrIs the amplitude ratio angle, Δ, of the objective lensbrIs the phase difference angle of the objective lens.
9. The calibration method according to claim 8, wherein the calculation formula of the polarization aberration of the objective lens is:
Figure FDA0002739943720000045
Figure FDA0002739943720000046
wherein k is the number of sampling points, ZlThe coefficients of the Zernike polynomials of the first term for the corresponding parameters, e the fitting error of the corresponding parameters, fl(ρk，θk) Is Zernike polynomial of the I term, and L is the number of the maximum term of the Zernike polynomial.
10. The calibration method according to claim 9, wherein the step S10 further comprises obtaining a relationship between an incident angle and a mirror rotation angle of the rotating device type muller matrix ellipsometer according to the parameter.
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