Rear focal plane ellipsometry method based on Mueller matrix modeling solution

Abstract

The invention belongs to the related field of optical measurement technology, and discloses a rear focal plane ellipsometry method based on Mueller matrix modeling solution, which comprises the following steps of: S1, establishing a system theoretical Mueller matrix response model based on a theoretical Mueller matrix of an optical element of a rear focal plane ellipsometry system and a beam vector propagation path; s2, calibrating an optical element influencing the polarization state of a measurement light beam in the measurement system, and establishing an actual Mueller matrix response model of the rear focal plane ellipsometry measurement system; and S3, collecting a sample measurement signal, and calculating Mueller response matrix elements of the sample for the actual Mueller matrix response model by using a matrix least square method or fitting matching. According to the invention, Mueller matrix optical modeling is applied to the rear focal plane ellipsometry system, analysis and extraction of polarization response characteristics of the sample are realized, the measurement precision and data reliability of the system are improved, and the problem that the polarization state change of the system is difficult to accurately characterize in the traditional ellipsometry process is solved.

Description

Technical Field

[0001] The present invention belongs to the field related to optical measurement technologies, and more specifically, relates to a rear focal plane ellipsometry measurement method based on Mueller matrix modeling and solution. Background Art

[0002] Modern optical measurement technologies play a crucial role in fields such as semiconductor manufacturing, nanomaterial research, and surface analysis. Among them, ellipsometry measurement technology is widely used in the measurement of film thickness, material optical constants, and surface characteristics due to its non-contact and high-precision characteristics. Rear focal plane ellipsometry can accurately analyze the polarization characteristics of a sample in a single shot by collecting the pattern on the rear focal plane of the objective lens after reflection from the sample, and has become a research hotspot in recent years.

[0003] Mueller matrix optical modeling is one of the cores in rear focal plane ellipsometry. The Mueller matrix comprehensively describes the change in the polarization state before and after the interaction between light and matter, providing a theoretical basis for in-depth analysis of the polarization response in an optical system. However, in a rear focal plane ellipsometry system, due to the complex optical path and multiple optical elements involved, such as the complex inner layer structure and material medium in the objective lens system and the large incident angle, the modulation effect on the polarization state of light waves is complex, and the influence of polarization aberration is inevitable. Traditional modeling methods based on the Jones matrix cannot accurately represent the polarization response of components, such as changes in anisotropic birefringence, depolarization, etc. This leads to a deviation between system modeling and actual measurement, thus affecting the measurement accuracy, especially in the optical characterization of high-resolution and complex samples.

[0004] Therefore, in a rear focal plane ellipsometry system, it is urgent to develop a Mueller matrix optical modeling method that can comprehensively consider the system polarization aberration to improve the measurement accuracy of the rear focal plane ellipsometry system. Summary of the Invention

[0005] Aiming at the defects and deficiencies of the existing technologies described in the background, the present invention proposes a rear focal plane ellipsometry measurement method based on Mueller matrix modeling and solution, which is used to solve the problem that traditional modeling methods cannot accurately characterize the influence of key components on the polarization state in a measurement system, and accurately characterize the change in the polarization state in a rear focal plane ellipsometry system through Mueller matrix optical modeling.

[0006] To achieve the above object, the present invention proposes a rear focal plane ellipsometry measurement method based on Mueller matrix modeling and solution, which includes the following: S1. Based on the theoretical Mueller matrix of the optical components in the rear focal plane ellipsometry system and the beam vector propagation path, establish a system theoretical Mueller matrix response model; S2. Calibrate the optical components that affect the polarization state of the measurement beam in the measurement system, and establish an actual Mueller matrix response model of the rear focal plane ellipsometry system; S3. Collect the sample measurement signals, and use the matrix least squares method or fitting matching for the actual Mueller matrix response model to calculate the Mueller response matrix elements of the sample.

[0007] Further preferably, S1 includes: determining the optical elements in the post-focal plane ellipsometer measurement system, and establishing a system theoretical Mueller matrix response model according to the propagation path of the measurement beam in the system and the standard Mueller matrices of each element: wherein, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model predicted value matrix, is the incident Stokes vector of the measurement light.

[0008] Further preferably, the optical elements that need to be calibrated in step S2 include but are not limited to the polarization generator, the polarization analyzer, the beam splitter, and the objective lens.

[0009] Further preferably, the actual Mueller matrix response model of the system after calibration in step S2 is: wherein, is the actual Mueller matrix of the polarization generator after calibration, is the actual Mueller matrix of the polarization analyzer after calibration, is the beam splitter transmission Mueller matrix, is the beam splitter reflection Mueller matrix, is the objective lens polarization aberration, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model predicted value matrix, is the incident Stokes vector of the measurement light.

[0010] Further preferably, the polarization generator includes an optical component selected from the following structures: at least one linear polarizer; or a combination of a linear polarizer and a compensator; or a combination of optical elements that can equivalently achieve a predetermined polarization control function.

[0011] Further preferably, the polarization analyzer includes an optical component selected from the following structures: at least one linear polarizer; or a combination of a linear polarizer and a compensator; or a combination of optical elements that can equivalently achieve a polarization analysis function.

[0012] Further preferably, the polarization aberration of the objective lens is the polarization aberration at the Fourier plane pupil, which varies with the pupil position, i.e., the incidence angle and azimuth angle. To simplify the establishment of the Mueller matrix model, it can be appropriately simplified , is set to have rotational symmetry.

[0013] Further preferably, step S3 includes: using matrix least squares method or fitting matching on the actual Mueller matrix response model in S2 to calculate the Mueller matrix of the sample , where the actual Mueller matrix response model is simplified to: where, is the model prediction value matrix, is the Mueller response of the sample outgoing light propagating to the detector, is the Mueller response matrix of the incident light propagating to the sample, is the Mueller matrix of the sample.

[0014] Further preferably, matrix least squares calculation is realized by matrix vectorization of the back focal plane data: where, is the Mueller response of the sample outgoing light propagating to the detector, is the Mueller response matrix of the incident light propagating to the sample, is the Mueller matrix of the sample, and Vec represents the vectorization operation.

[0015] Generally speaking, the present invention proposes a back focal plane ellipsometry method based on Mueller matrix modeling and solution, which is used to solve the problem that the traditional modeling method cannot accurately characterize the influence of key components in the measurement system on the polarization state. Through Mueller matrix optical modeling, the accurate characterization of the polarization state change in the back focal plane ellipsometry system can be realized, and it can be used to analyze geometric and physical properties such as the thickness, surface roughness, and refractive index of the sample thin film. This method has the following beneficial effects: 1. The present invention uses the Mueller matrix to establish an optical model considering the influence of optical components on the polarization state, and accurately describes the change of the polarization state in the back focal plane ellipsometry system; 2. The present invention utilizes the advantage that the back focal plane ellipsometry system can realize the accurate analysis of the polarization characteristics of the sample in a single shot, and combines the Mueller matrix optical modeling method of the invention to quickly and accurately calculate parameters such as the thickness and material optical constants of the sample. Brief Description of the Drawings

[0016] To more clearly illustrate the technical solutions in the embodiments of the present application, the following will briefly introduce the accompanying drawings required for the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other accompanying drawings can be obtained based on these drawings.

[0017] Figure 1 FIG. 4 is a schematic flow chart of a post-focal plane ellipsometry method based on Mueller matrix modeling and solution provided according to a preferred embodiment of the present invention.

[0018] Figure 2 FIG. 8 is a schematic diagram of a post-focal plane ellipsometry system device constructed according to a preferred embodiment of the present invention.

[0019] Figure 3 FIG. 12 is a schematic diagram of a polarization element parameter calibration configuration constructed according to a preferred embodiment of the present invention.

[0020] Figure 4 FIG. 16 is a schematic diagram of a beam splitter Mueller matrix calibration configuration constructed according to a preferred embodiment of the present invention.

[0021] Figure 5 FIG. 20 is a schematic diagram of an objective lens polarization aberration calibration configuration constructed according to a preferred embodiment of the present invention.

[0022] Figure 6 FIG. 24 is a schematic diagram of a sample post-focal plane image collected according to a preferred embodiment of the present invention.

[0023] Figure 7 FIG. 28 is a schematic diagram of some Mueller matrix elements at different incident angles obtained according to a preferred embodiment of the present invention.

[0024] Figure 8 FIG. 32 is an error graph of film thickness measurement under different calibration steps obtained according to a preferred embodiment of the present invention.

[0025] In all the accompanying drawings, the same reference numerals are used to represent the same elements or structures, where: 1 - polarization generator, 2 - non-polarizing beam splitter, 3 - first lens, 4 - rear focal plane, 5 - objective lens, 6 - sample, 7 - polarization analyzer, 8 - second lens, 9 - CMOS camera, 10 - motion controller. Detailed Embodiments

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are part of the embodiments of the present invention, rather than all of them. Here, they are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0027] Figure 1 As shown in the schematic flow chart of a rear focal plane ellipsometry method based on Mueller matrix modeling and solution provided by the embodiments of the present invention, the specific steps are as follows: S1. Based on the theoretical Mueller matrix of the optical elements in the rear focal plane ellipsometry system and the beam vector propagation path, establish a system theoretical Mueller matrix response model; As Figure 2 As shown in the schematic diagram of the rear focal plane ellipsometry system device constructed according to the embodiments of the present invention, the rear focal plane ellipsometry system includes a polarization generator 1, a non-polarizing beam splitter 2, a first lens 3, a rear focal plane 4, an objective lens 5, a sample 6, a polarization analyzer 7, a second lens 8, a CMOS camera 9, and a motion controller 10.

[0028] Determine the optical elements in the rear focal plane ellipsometry system. According to the propagation direction of the measurement light beam and the position of each optical element, successively combine the standard Mueller matrices of each optical element to form a system theoretical Mueller matrix response model: Among them, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model prediction value matrix, The incident Stokes vector of the measurement light beam. This formula can be appropriately simplified or transformed. For example, use to replace Or in a system with a small numerical aperture, ignore .

[0029] S2. Calibrate the optical elements in the measurement system that affect the polarization state of the measurement light beam, and establish an actual Mueller matrix response model of the rear focal plane ellipsometry system; As Figures 3 - 5 As shown in the schematic diagram of the calibration configuration of the key optical elements constructed according to the embodiments of the present invention, Figure 3The azimuth angles of the linear polarizers in the polarization generator and the polarization analyzer, as well as the azimuth angle and retardation amount of the retarder, are calibrated. The Mueller matrix model of this calibration configuration is as follows: Wherein, is the azimuth angle of the linear polarizer in the polarization generator, is the azimuth angle of the linear polarizer in the polarization analyzer, is the azimuth angle of the retarder in the polarization generator, is the retardation amount of the retarder in the polarization generator, is the azimuth angle of the retarder in the polarization analyzer, is the retardation amount of the retarder in the polarization analyzer, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer, is the Mueller matrix of the sample. For the solution of such problems, the method of data fitting can be adopted.

[0030] Figure 4 For calibrating the reflection and transmission Mueller matrices of the beam splitter, the Mueller matrix model of this calibration configuration can be expressed as: Wherein, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer. Under the two calibration configurations of reflection and refraction of the beam splitter, they respectively correspond to the matrices related to the reflection and transmission of the beam splitter under their respective configurations. Such problems can be solved by the matrix least squares method.

[0031] Figure 5 For the configuration of obtaining the objective lens polarization aberration, combined with the calibration of the first two steps, the actual Mueller matrix response model of the corrected system is: Wherein is the actual Mueller matrix of the polarization generator after calibration, is the actual Mueller matrix of the polarization analyzer, is the transmission Mueller matrix of the beam splitter, is the reflection Mueller matrix of the beam splitter, is the objective lens polarization aberration, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model prediction value matrix, is the incident Stokes vector of the measurement light. By measuring the standard sample with a dual-rotating retarder to obtain the measurement value data, and then can be solved; thus, the calibration of the back focal plane ellipsometer system constructed in the embodiment of the present invention has been completed.

[0032] S3. Collect the sample measurement signals, and use matrix least squares or fitting matching for the actual Mueller matrix response model to calculate the Mueller matrix of the sample ; The actual Mueller matrix response model can be simplified as: Wherein, is the Mueller response of the sample outgoing light propagating to the detector, The light source is the Mueller response matrix of the incident light propagating to the sample, is the Mueller matrix of the sample. As Figure 6 is a schematic diagram of the sample back focal plane image collected according to the embodiment of the present invention. Matrix least squares calculation is realized by vectorizing the back focal plane data: Wherein, is the Mueller response of the sample outgoing light propagating to the detector, The light source is the Mueller response matrix of the incident light propagating to the sample, Vec represents the vectorization operation, and the Mueller matrix of the sample can be quickly calculated by using the above least squares for the back focal plane annular signal .

[0033] For an optically isotropic planar thin film sample, is: Wherein, N, C, S is the Mueller matrix element of the optically isotropic planar thin film sample. As Figure 7 is obtained at different incident angles according to the embodiment of the present invention N, C, S Schematic diagram. The solid line in the figure represents the measured value obtained in the embodiment of the present invention, and the dotted line is the theoretical value calculated according to the thin film optical model. The measured value is close to the theoretical value, verifying the accuracy of the back focal plane ellipsometry measurement method based on Mueller matrix modeling and solution proposed by the present invention

[0034] Figure 8 is the error graph of thin film thickness measurement under different calibration steps. The ordinate is the difference between the average value of several thickness measurement results and the nominal thickness. Among them, calibration step 1 is the calibration of polarization element parameters, calibration step 2 is the calibration of the Mueller matrix of the beam splitter, and calibration step 3 is the calibration of the objective lens polarization aberration; from Figure 8 it can be seen that the calibration steps can greatly improve the accuracy of thin film thickness measurement, verifying the excellent effect of the back focal plane ellipsometry measurement method based on Mueller matrix modeling and solution proposed by the present invention in solving the thin film thickness

[0035] The following are the Mueller matrices appearing in the above text, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer, is the Mueller matrix representing the rotation operation, represents the rotation angle, , , , , , , , is the Mueller matrix for performing the subscript angle rotation operation, is the Mueller matrix of the linear polarizer in the polarization generator, is the Mueller matrix of the linear polarizer in the polarization analyzer, is the Mueller matrix of the retarder, represents the retardation, and is the Mueller matrix of the retarder with the retardation being the subscript angle, is the Mueller matrix for coordinate transformation: In summary, a rear focal plane ellipsometry method based on Mueller matrix modeling and solution proposed in this embodiment can be used to solve the problem that the traditional modeling method cannot accurately characterize the influence of key components in the measurement system on the polarization state, and realizes the accurate characterization of the polarization state change in the rear focal plane ellipsometry system through Mueller matrix optical modeling.

[0036] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution, characterized in that: It includes the following steps: S1. Based on the theoretical Mueller matrix of the optical elements of the back focal plane ellipsometer system and the beam vector propagation path, establish a system theoretical Mueller matrix response model; S2. Calibrate the optical elements in the measurement system that affect the polarization state of the measurement beam, and establish an actual Mueller matrix response model of the back focal plane ellipsometer system; S3. Collect the sample measurement signal, and use matrix least squares or fitting matching on the actual Mueller matrix response model to calculate the Mueller response matrix elements of the sample.

2. The post-focal-plane ellipsometry measurement method based on Mueller matrix modeling and solution according to claim 1, characterized in that: S1 includes: Determine the optical elements in the back focal plane ellipsometer system, and establish a system theoretical Mueller matrix response model according to the propagation path of the measurement beam in the system and the standard Mueller matrix of each element: Among them, is the Mueller matrix of the polarization generator, is the Mueller matrix of the polarization analyzer, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model predicted value matrix, is the incident Stokes vector of the measured light.

3. A method for measuring the back focal plane ellipsometry based on Mueller matrix modeling and solving as claimed in claim 1, characterized in that: The optical elements that need to be calibrated in step S2 include but are not limited to a polarization generator, a polarization analyzer, a beam splitter, and an objective lens.

4. The post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution according to claim 3, characterized in that: The actual Mueller matrix response model of the system after calibration in step S2 is: Among them, is the actual Mueller matrix of the polarized generator after calibration, is the actual Mueller matrix of the polarization analyzer after calibration, is the Mueller matrix of the beam splitter transmission, is the Mueller matrix of the beam splitter reflection, is the polarization aberration of the objective lens, is the Mueller matrix of the sample, is the transformation matrix of the unified coordinate system, is the model prediction value matrix, is the incident Stokes vector of the measured light ray.

5. A post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution, characterized in that: The polarization generator includes an optical component selected from the following structures: at least one linear polarizer; or a combination of a linear polarizer and a compensator; or a combination of optical elements that can equivalently achieve a predetermined polarization control function.

6. The post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution according to claim 3, characterized in that: The polarization analyzer includes an optical component selected from the following structures: at least one linear polarizer; or a combination of a linear polarizer and a compensator; or a combination of optical elements that can equivalently achieve a polarization analysis function.

7. A method for measuring the post-focal plane ellipsometry based on Mueller matrix modeling and solution, as claimed in claim 3, wherein: Polarization Aberration of the Objective Lens It is the polarization aberration at the Fourier plane pupil and varies with the incident angle and azimuth angle.

8. The post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution according to claim 1, characterized in that: Step S3 includes: calculating the Mueller matrix of the sample by using matrix least squares method or fitting matching for the actual Mueller matrix response model in S2 , where the actual Mueller matrix response model is simplified to: Among them, is the matrix of model prediction values, is the Mueller response of the sample outgoing light propagating to the detector, is the Mueller response matrix of the incident light propagating to the sample, is the Mueller matrix of the sample.

9. A post-focal plane ellipsometry measurement method based on Mueller matrix modeling and solution, characterized in that: Vectorize the back focal plane data to implement matrix least squares calculation: wherein, is the Mueller response of the sample output light propagating to the detector, is the Mueller response matrix of the incident light propagating to the sample, is the Mueller matrix of the sample, and Vec represents the vectorization operation.

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