Measurement technologies can be broadly categorized into model-free and model-based techniques. Model-free metrology, such as conventional optical microscopy, can directly derive geometric dimensions of a specimen from acquired images through edge detection algorithms. While this "what-you-see-is-what-you-get" approach offers advantages of simplicity and intuitiveness, it often exhibits certain limitations in practical applications. For example, due to the limitation of optical diffraction limit, it is generally difficult for traditional optical microscopy to measure structural features below 200 nm. Compared with model-free techniques, the model-based metrology is often based on the interaction between a certain physical field (such as electromagnetic field, acoustic field, temperature field, etc.) with the object to be measured, and then uses instruments to collect the state changes of the physical field before and after the interaction, and then extracts relevant information of the object through inverse problem solving. Although the model-based measurement process is not as simple and intuitive as those model-free techniques, it can break through the limitations of model-freetechniques with certain prior knowledge.

The theoretical basis of model-based metrology is the interaction mechanism between the physical field and the object to be measured, which is often extremely complex and needs to be described by partial differential equations such as Maxwell's equations. The success of this measurement technique mainly depends on two aspects, namely, the forward modeling and the inverse problem solving. Since model-based metrology usually involves a large number of complex scientific calculations, especially numerical calculation problems, we clearly proposed to call it "computationalmetrology" for the first time, and defined it as "a measurement technique that models the complex measurement process and collects observation data under certain conditions through instruments, and then obtain the parameters to be measured by solving inverse problems".

Fig. 1 Fundamentals and basic elements of computationalmetrology

ⅠScientific problems and methods

In view of the basic characteristics of computational metrology, we summarize a number of scientific problems that need to be solved in computational metrology, including the measurability problem, the error analysis and uncertainty estimation, and the contradiction between measurement speed and measurement precision. In order to address the above scientific problems, with the help of modern mathematical theory and computational methods as well as the latest advances in the fields of artificial intelligence and machine learning, the research focuses on the fast and accurate numerical calculation method of the forward model, the robust and accurate inverse parameter extraction, the measurement configuration optimization, and the uncertainty evaluation of the measurement results. Through the above research, we attempt to break through the limitation of traditional model-free metrology and explore the sensitive mechanism of computational metrology and their enhancement methods. We hope to provide new principles and approaches for in-line, non-destructive and accurate measurements in advanced manufacturing processes, such as IC manufacturing and atomic-scale manufacturing.

ⅡDevelopment and application of the instruments

(1) Development andapplication ofadvanced Muellermatrixellipsometry

The ellipsometer, an optical instrument that utilizes polarization properties of light, has been widely employed for the characterization of thin-film thicknesses and optical constants, as well as for the measurement of 3D profiles of nanostructures.Compared with conventional ellipsometers, which can only obtain two measured parameters—the amplitude ratio and the phase difference—each measurement of a Mueller matrix ellipsometer (MME)obtains a complete 4×4 Mueller matrix containing 16measurement parameters, providing abundant information about the sample’s anisotropy, depolarization, and other optical properties. In view of this, for different application requirements, new principles ofMueller matrix measurement are studied, and advanced Mueller matrix ellipsometers such asbroadband MME, high-resolution imagingMME, and high-speedMEEare developed.The developed instruments are then applied for the characterization ofnew materials, new processes and new devices in the fields ofIC,next-generation display, photovoltaics, AR/VR, etc.

Fig.2 Broadband Mueller matrix ellipsometer: (a) Optical layout; (b) Prototype

Fig.3 High-resolution imaging Mueller matrix ellipsometer: (a) Optical layout; (b) Principle ofthe backfocal-plane scanning with a high-NA objective; (c)Prototype

Fig.4 High-speed Mueller matrix ellipsometer:(a) Optical layout; (b) Prototype

(2) Development andapplication ofsmall-angle X-rayscatterometry

Small-angle X-ray scattering (SAXS) refers to the scattering phenomenon in which X-rays, upon interacting with a sample, are deflected at small angles near the incident beam due to nanoscale variations in the sample’s electron density. Based on this phenomenon, SAXS is widely used in polymers, biomacromolecules, nanomaterials and other fields to analyze sub-micro-scale information such as particle size, shape, distribution, orientation and specific surface area. In recent years, it has also been applied to the measurement of3D profiles of nanostructures inICmanufacturing.In the research, for different application requirements, the principle ofSAXSmeasurement is studied, aSAXSinstrument is developed, and the combination of SAXStechnique withthecomputer tomography (CT) and ghost imagingtechniquesis explored.The developed instruments are then applied forthe characterization ofnew polymers, nanomaterials, nanostructures, etc.

Fig.5 Optical layout and prototype of the developed lab-SAXSinstrument

(3) Development and application of ultrafast photoacoustic measuring instrument

The photoacoustic effect refers to the phenomenon that the absorbing medium generates sound waves under periodic light conditions. Ultrafast photoacoustic measurement technology is a new type of ultrasonic nondestructive detection technology. It uses ultrashort pulse laser to excite high-frequency ultrasound through solid photoacoustic effect, and uses laser beam to detect the propagation of ultrasound, combining the hightransverse resolution of optical measurements and the penetration of acoustic measurement.At the same time, it has the advantages of high longitudinal resolution, noncontact and fast measurement, and has important application prospects in the measurement of micro-nano scale non-transparent films and structures in IC manufacturing. Focusing on the ultrafast photoacoustic measurement technology, the theoretical model of photoacoustic measurement of micro-nano film structure, the signal processing method of photoacoustic measurement and the extraction method of parameters to be measured of micro-nano film structure are studied, and the ultrafast photoacoustic measuring instrument is developed. Applications are then carried out for single-layer films, multilayer films and patterned film structures.

Fig.6 The basic principle of ultrafast photoacoustic measurement

Fig.7 Some typical nanostructures measured by the developed instruments

Fig.8 Thelayer-dependent optical constants of 2D MoS2 materials revealed using the developed instruments

(4) Development and application of polarization angle-resolved scattering instrument

The polarization angle-resolved scattering (PARS) instrument is an instrument for measuring the profiles of nanostructures by using the image of the scattering pupil. The PARS adopts an objectivelenswith a high NA in the optical path, which can simultaneously collectdiffraction light within a certain angle range. At the same time, the light incident on the objective lens converges after passing through the objective lens, which can form different incident and azimuthal angles, and form a one-to-one correspondence with each point on the rear focal plane of the objective lens, so as to realize the measurement dataat multiple incident angles andfullazimuthal angles. Since there are no moving parts in the optical path, the measurement speed of PARS is only restricted by the response time ofthe detector, which is very suitable for inline measurement in IC manufacturing. For different measurement objects, the PARS has two working modes. For traditional IC overlay targets with periods of 600-800 nm, the PARS can collect multiple diffraction orders of light spots at the same time, and then use the approximate linear relationship between the high-order (generally±1storders) diffraction intensity difference and the overlay error to realize the overlay error measurement. For overlay targets with periods oftens of nanometers, thezero-orderdiffraction light of the target to be tested is collected by the high NA objective lensapproachingto realize the collection of key size and overlay error information in single measurement, and introduce polarization modulation and orthogonal polarization demodulation units and the cross-polarization frequency domain image asymmetric information separation technology to realize the separation of key size and overlay error information, and combine the intelligent parameter extraction method based on machine learning to obtain the key size and overlay error of the device to be tested.

Fig.9 Prototype of the polarization angle-resolved scatterometer

Representative Papers：

1. S. Liu, "Computational metrology: Problems and solution methods," J. Mech. Eng. 50, 1-10 (2014).

2. S. Liu, X. Chen, and C. Zhang, "Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology," Thin Solid Films 584, 176-185 (2015).

3. X. Chen, H. Gu, J. Liu, C. Chen, and S. Liu, "Advanced Mueller matrix ellipsometry: Instrumentation and emerging applications," Sci. China Tech. Sci. 65, 2007-2030 (2022).

4. X. Chen, C. Wang, T. Yang, J. Liu, C. Luo, and S. Liu, "Inline Optical Measurement and Inspection for IC Manufacturing: State-of-the-Art, Challenges, and Perspectives," Laser Optoelectron. Prog. 59, 0922025 (2022).

5. C. Chen, X. Chen, C. Wang, S. Sheng, L. Song, H. Gu, and S. Liu, "Imaging Mueller matrix ellipsometry with sub-micron resolution based on back focal plane scanning," Opt. Express 29, 32712-32727 (2021).

6. S. Zhang, H. Jiang, H. Gu, X. Chen, and S. Liu, "High-speed Mueller matrix ellipsometer with microsecond temporal resolution," Opt. Express 28, 10873-10887 (2020).

7. B. Song, H. Gu, M. Fang, X. Chen, H. Jiang, R. Wang, T. Zhai, Y. Ho, and S. Liu, "Layer-dependent dielectric function of wafer-scale 2D MoS2," Adv. Opt. Mater. 7, 1801250 (2019).

8. S. Liu, X. Chen, and S. Liu, "Physics-enhanced learning for automated determination of material optical constants," Laser Photon. Rev. 19, 2500809 (2025).

9. T. Yang, X. Chen, S. Liu, J. Zhang, and S. Liu, "Bootstrap Method for Uncertainty Evaluation in Critical Dimension Small-Angle X-ray Scattering," IEEE Trans. Instrum. Meas. 73, 1008010 (2024).

10. J. Zhang, X. Chen, T. Yang, and S. Liu, "X-ray-based overlay metrology using reciprocal space slicing analysis," Opt. Lett. 48, 6380-6383 (2023).

11. Z. Wang, J. Min, Y. Sun, X. Wang, X. Chen, Z. Tang, and S. Liu, "Temperature dependence of femtosecond photoacoustic process in high-precision characterization for metal nanofilms," Photoacoustics 41, 100678 (2025).

12. J. Min, X. Chen, Z. Wang, J. Hu, Y. Sun, Z. Tang, and S. Liu, "Deep learning-based identification of characteristic regions for picosecond ultrasonics metrology," Measurement 218, 113205 (2023).

13. J. Zhang, J. Liu, J. Zhu, H. Jiang, and S. Liu, "In-situ calibration of objective lens of angle-resolved scatterometer for nanostructure metrology," Appl. Opt. 62, 3829-3838 (2023).