n advanced manufacturing and precision metrology, accurate measurement of the surface topography of each surface within transparent parallel optical components is crucial for controlling device performance. Multi-surface wavelength-tuned interferometry often suffers from spectral crowding and severe crosstalk due to the superposition of harmonic signals. Conventional FFT-based analysis is prone to spectral leakage under conditions of low signal-to-noise ratio (SNR) and non-integer period sampling, making it difficult to ensure measurement accuracy. To address these challenges, we propose a Hilbert-subspace Joint Parameter Estimation for Multi-surface Phase-shifting Interferometry (HJPE-MPI) algorithm. This algorithm utilizes the Hilbert transform to construct analytic signals, which are then used to form a Hankel matrix for subspace decomposition, enabling high-resolution frequency estimation of sampled interference signals. The estimated frequency offsets are subsequently used to correct the amplitude and phase of the harmonic components corresponding to each surface, thereby enabling joint and precise parameter localization. Simulation and experimental results demonstrate that the proposed method significantly outperforms conventional algorithms in the estimation accuracy of frequency, phase, and amplitude for multi-harmonic interference signals, and exhibits excellent repeatability and stability in measurements of a 30 mm-thick transparent plate.

Multi-surface wavelength-tuning interferometry is an essential technique for acquiring the nanoscale three-dimensional topography of optical parallel surfaces. Nevertheless, constrained by the superposition of multi-frequency signals, the finite sampling length, and the inherent discrete frequency resolution, the demodulation process is prone to spectral aliasing, leakage, and the picket-fence effect, which severely degrades the estimation accuracy of harmonic parameters. In this paper, we propose a complex-ratio correction method based on multi-snapshot spectral estimation algorithm (MSSE-mCRC). It utilizes a modified Singular Value Decomposition (SVD) to estimate the source number of superimposed interference harmonics, thereby assessing the sampling performance. By constructing a covariance matrix via multiple snapshots and employing eigendecomposition to separate the signal and noise subspaces, high-resolution spectral estimation is achieved. Subsequently, the Hanning-windowed Complex-Ratio Correction technique is applied to correct the amplitudes and phases. Simulation results demonstrate that the proposed algorithm achieves frequency and amplitude estimation errors on the order of 10-4 and 10-2, respectively, outperforming traditional FFT-based methods. Furthermore, in the repeated experiments performed on two different optical components, the maximum RMS, PVQ, and PV values of the measured surface profiles are 0.0111μm, 0.0454μm, and 0.0594μm for Experiment 1, and 0.0030μm, 0.0137μm, and 0.0308μm for Experiment 2, respectively. These results successfully validate the repeatability and stability of the proposed algorithm.

Interferometry enables sub-nanometer accuracy in optical surface metrology. The wavelength-tuning phase-shifting interferometry facilitates simultaneous multi-surface measurements by leveraging the distinct frequencies induced by optical path differences. However, acquired interferograms contain superimposed interference patterns forming three harmonic signals, which cannot be processed by conventional signal separation methods due to single-channel constraints. Additionally, wavelength-tuning errors cause spectral leakage that degrades parameter reconstruction. To address issues presented, we propose a multi-surface wavelength-tuning interferometry algorithm using virtual interference channels for blind source separation and refined spectral analysis (MWI-VBSA). This method employs dual-criterion separation with orthogonalization to extract signal components, followed by spectral refinement for harmonic frequency and efficient phase demodulation functions. Comprehensive validation demonstrates significant improvements over FFT and multi-surface advanced iterative algorithm: simulations under varying SNR (25 similar to 50 dB) show reduced frequency errors and low surface reconstruction errors, while diverse sampling lengths/intervals confirm robust harmonic separation. Experimental measurements on 50 mm transparent plates achieve surface retrieval errors below 3.32 x 10(-3) lambda(0) for RMS, verifying consistent accuracy. Furthermore, the algorithm's robustness was decisively confirmed through a four-surface interferometry experiment, in which it successfully separated six superimposed harmonic signals. This outcome provides evidence of the algorithm's exceptional performance and reliability when processing complex signal superposition.

To achieve high-precision measurement, robust anti-interference capability, and good interchangeability, a six-axis force sensor based on a six-bar closed-loop parallel mechanism is proposed. The sensor employs a modular split-body design and incorporates a rigid-flexible coordinated decoupling unit within the measuring branch. By introducing a steel ball and a spring-guided mechanism, the load transmission path is optimized, and the coupling among six-dimensional channels is effectively suppressed, thereby realizing mechanical decoupling. A static model is developed based on the structural equivalence principle, and finite element simulation and static calibration are conducted to verify the effectiveness of the structural design and modeling method, as well as their engineering feasibility. The static calibration results indicate that the linearity, repeatability, hysteresis, and coupling error of the sensor are within 1.38 % FS, 0.43 % FS, 1.13 % FS, and 1.73 % FS, respectively, demonstrating high measurement accuracy and promising potential for engineering applications.

Efficient and precise measurements of optics with parallel surfaces are crucial for ensuring component performance. However, current interferometry faces challenges from an uncertain number of interference sources and adverse effects on parameter demodulation due to sampling errors and noise. To address these issues, we first use the minimum description length (MDL) criterion on a Hankel matrix of intensity signals to estimate the number of effective surfaces by minimizing an information-theoretic cost function. Then, variational mode decomposition (VMD) iteratively extracts the superimposed signal into interference modal components (IMCs) with distinct frequencies. Frequency estimation employs a zero-padded Fourier transform with a Blackman window, followed by phase and surface reconstruction. In error analysis, under signal-to-noise ratio (SNR) = 30-50 dB and up to 20% phase-shifting errors, frequency solution errors remain below 0.7%. The Monte Carlo trials show low mean and standard deviation (STD) in solution errors. Repeated measurements on a transparent plate yield stable profiles with minimal peak-to-valley (PV), root mean square (RMS), and PVQ errors.

Integrated sensing and communications (ISAC) is an essential 6G capability for joint data transmission and environmental sensing. To support 6G scenarios with stringent ISAC performance requirements, existing massive-MIMO-based systems are expected to scale toward ultra-massive MIMO. However, this scaling incurs prohibitive cost and power consumption when realized using widely adopted phased arrays with complex phase shifters and feeding networks. Recently, holographic integrated sensing and communications (HISAC) has emerged as a promising paradigm to address this issue. It employs reconfigurable holographic surfaces (RHSs), a type of leaky-wave antenna, as a cost- and energy-efficient implementation of ultra-massive MIMO-based ISAC, and offers enhanced flexibility for ISAC beam synthesis through holographic beamforming. In this paper, we provide a comprehensive tutorial on HISAC, focusing on how RHS-enabled holographic beamforming can be exploited to jointly support communication and sensing under practical hardware constraints. We first introduce the fundamentals of RHSs and discuss the unique leakage power constraint of holographic beamforming. We then present a general optimization framework for HISAC and show how HISAC enhances joint communication and sensing, sensing-assisted communication, and communication-assisted sensing. We further present HISAC system implementations and experimental results. Finally, we outline promising research directions for HISAC, highlighting the potential of HISAC in advancing efficient, flexible, and high-performance ISAC networks.

The fringe analysis method, using phase modulation with wavelength tuning, is widely used in profiling the surfaces and thickness variation in optical flat interferometry. However, the measurement accuracy may deteriorate owing to the phase demodulation errors resulting from the offsets in frequency estimation due to the unavoidable spectrum leakage. To address this challenge, a dual-window all-phase harmonic demodulation on wavelength tuning interferometry (Dwa-WTI) is developed to improve the measurement accuracy of surface profiles and thickness variation of the optical flat. The Dwa-WTI compensates for the effect of the sampling sequence on the spectrum by synthesizing the new interferograms, which effectively suppresses the undesirable error of phase demodulation. With the developed Dwa-WTI, the initial phase of the wavefront distribution at the spectrum peak of the synthesized sequence is independent of the frequency offset. The reliability of the Dwa-WTI has been validated by numerical simulations and experiments. In experiments, the optical flats with thicknesses of 5, 10, and 30 mm are measured. The difference of peak-to-valley (PV) and root-mean-square (rms) for the profiles of the thickness variation, front and rear surfaces are compared in repeated experiments. Meanwhile, the measurements under different spatial carriers are compared to assess the stability of the Dwa-WTI and the relative standard deviation (RSD) for the root-mean-square of the theoretical error (RMSTE) is about 0.085.

The non-contact phase-shifting interferometry technique enables the demodulation of optoelectronic signals to extract profile information of the measured object, achieving nanoscale surface reconstruction. However, challenges in multi-surface interferometry include: (1) the need for a disturbance quantification model to correct parameter estimations; (2) difficulty in simultaneously solving frequency and amplitude under disturbances, as methods like FFT and least-squares iterations struggle to balance both; and (3) efficient and accurate resolution of harmonic parameters for real-valued interference signals. To address these issues, and achieve high-precision and simultaneous measurements of multi-surface optical elements, the EQA-CHFA method is proposed. The proposed algorithm constructs a fuzzy entropy model to evaluate interference signal disturbances, employs a root-matrix spectral estimation method to determine harmonic frequencies, and uses window functions to correct harmonic amplitudes. Initial phases of all surfaces are then demodulated via weighted phase demodulation functions. Simulations and error analyses demonstrate the algorithm's robustness against additive Gaussian noise and phase-shifting errors, outperforming FFT and least-squares methods. Experiments on a 30 mm-thick optical element further validate the accuracy and practical applicability of EQA-CHFA for multi-surface measurements.

To efficiently and accurately realize four-surface measurements with flexible cavity lengths and sampling frequencies, a wavelength-tuning phase-shifting matching algorithm based on harmonic selection modes is developed. The developed MPSA-AHR method utilizes pre-iterative wavefront reconstruction errors to quantitatively analyze the multi-harmonic reconstruction performance and to obtain efficient sample combinations. Combined with the densified power spectral density method, the harmonic frequencies can be obtained with high accuracy, enabling the simultaneous measurement of front/rear surfaces, thickness variation, and inhomogeneous distribution of refractive index for the tested transparent plates. The proposed method outperforms the existing methods and can realize multi-surface measurements with fewer sampling frames under the designed harmonic selecting mode, which is verified by sufficient simulations and error analysis under several measurement conditions. Repeatability measurements of a transparent plate with an average thickness of 50 mm using a Fizeau wavelength-tuning phase-shifting interferometer also verify the practical validity of our method.

To obtain the distribution of front/rear surfaces, thickness variation, and refractive index inhomogeneity of transparent parallel optics at non-rigorous cavity lengths and sampling frequencies, a combined phase-shifting method that integrates anti-under-sampling and anti-overlapping based on the laser tuning performance has been developed. Then six independent and easily interpretable spectra can be obtained, whereby efficient and accurate phase demodulation can be realized. Further, a phase difference-based harmonic frequency and amplitude solution algorithm is introduced to obtain accurate frequency characteristics. The proposed eFFT-MAA method is robust to spectral leakage, and its effectiveness is verified by error analysis for different combinations of cavity lengths and surface shapes. Repeated and comparative measurements in the experiment on a transparent plate with an average thickness of 43 mm also verify that it is effective.

It is important to evaluate the reliability of the key silicon-based structures such as Through silicon vias (TSV) and Micro-electromechanical Systems (MEMS). The thermo-mechanical stress of TSV and the bonding stress of MEMS are quantitatively determined in this paper based on real-time phase shifting using a polarization camera. The polarizated images based on finite element simulation are reconstructed by stress-optic law and Mueller matrix multiplication for experimental verification. A economical infrared polariscope without rotation of optical elements is developed to provide a rapid measurement of stress in silicon-based structures. The error correction algorithm for the low extinction ratio in infrared polariscope is used for measurement of stress. Experimental results indicate that the infrared photoelastic system enables measurement of stress in TSV and MEMS.

Multiple-surface interferometry with nanoscale accuracy is important in the precise manufacturing of optically transparent parallel plates. To measure the surface profile and thickness variation of the plates simultaneously, the frequencies of the interferometric signal must be estimated from overlaid interferograms. Traditional algorithms typically suffer from issues such as spectrum leakage, reliance on initial iterative values, and the need for prior knowledge. In this study, the time-domain estimation algorithm for multiple-surface interferometry (MSI-TDe) is introduced based on a difference model to improve the accuracy of frequency estimation. The MSI-TDe algorithm is based on a normal equation that is insensitive to environmental noise. Using the algorithm, the frequencies of an interferometric signal can be estimated without prior knowledge and employed for wavefront reconstruction in multi-surface interferometry. Numerical simulation results indicate that the MSI-TDe algorithm has better frequency estimation performance than the discrete Fourier transform (DFT) algorithm. The relative error of the frequency estimation is on the order of 10–4. Three-surface interferometry was first performed. The root-mean square repeatability standard deviations of 0.07, 0.12 and 0.11 nm for the thickness variation, front surface profile, and rear surface profile, respectively, indicate the stability of the MSI-TDe algorithm. Four-surface interferometry with six frequency components was then performed. The adaptability of the MSI-TDe algorithm is validated by the measurement results.

The core issue of multi-surface wavelength-shifting interferometry at free cavity lengths is mainly centered on adaptively matching the tuning step. To avoid harmonic aliasing and undersampling, the discrete-time Fourier transform method with global spectrum densification is used to comprehensively analyze the spectral characteristics under different sampling cases and cavity lengths, then a composite multi-surface wavelength-shifting interferometry algorithm is proposed accordingly. The proposed method includes a preprocessing algorithm and three spectral criteria to identify and avoid undesirable sampling problems for in-situ measurements. The algorithm performance is quantitatively evaluated from several aspects of the simulation analysis. A flat plate with an average thickness of 20 mm was repeatedly measured using a Fizeau wavelength-shifting interferometer in the experiments, and the PVQ and RMS errors at different measurement distances were less than 5 mm and 1 mm, respectively.

We present a barycentric Lagrange interpolation collocation method (BLICM) and investigate its capability in avoiding volumetric locking in nearly incompressible and incompressible plane elasticity. Specifically, the governing equations of plane elasticity are expressed as mixed formulas combining displacement and pressure. The unknown functions of displacement and pressure are approximated using barycentric Lagrange interpolation. The governing equations and boundary conditions are discretized into algebraic equations in matrix form. The boundary conditions are imposed by additional method to form an over-constraint system of algebraic equations. A regular domain method is introduced to tackle irregularly shaped domains. Extensive numerical experiments were conducted to validate the proposed method and the results demonstrate its ability to avoid locking issues while achieving high precision in nearly incompressible and incompressible plane elasticity.

Consisting of the front and rear surfaces of the measured optics and two reference surfaces, the four-surface interferometry system can obtain parameters of the measured surface profiles and refractive index inhomogeneity by acquiring and decoding the fringe patterns of the measured plate and the rear reference plate respectively. However, the difficulty of such measurement comes from the complex superposition of at least six interference sub-signals, and the setting of the phase-shift interval determines whether the harmonics can be reconstructed or not. To improve the measurement accuracy, two kinds of convolutional fitting window functions are redesigned based on the Hanning window to enhance the function performance. Then, by analyzing the reconstruction errors of each wavefront under different combinations of sampling intervals and cavity lengths during four-surface measurements, an error quantization matrix obtained by triple iteration is constructed. By setting a threshold condition, the optimal phase-shifts under different cavity lengths can be matched adaptively. The multi-surface measurement accuracy and reliability of adaptive phase-shift matching are verified in the simulation. The interferogram acquisition and phase demodulation of two transparent flats were performed using a Fizeau wavelength-tuning interferometer, and the reconstruction errors of the surfaces in the repetitive measurements further verified the reliability of the proposed method.

Addressing the common limitations of current six-axis force sensors, including the degradation of strain gauge accuracy due to environmental influences and the necessity for a complete replacement when damaged, this paper proposes an innovative six-axis force sensor that integrates a hybrid serial-parallel structure featuring flexible hinges. The sensor's bracket is designed using a combination of serial-parallel flexible hinges, which confer load distribution capabilities, are lightweight, have high strength, and facilitate replacement. The static force model and the overall stiffness model of the sensor were developed based on the 3-universal-prismatic-universal parallel mechanism theory, providing a theoretical foundation for evaluating the sensor's performance. To validate the accuracy of the stiffness model, finite element software is utilized for simulation-based verification. Subsequently, a calibration experiment is performed on the sensor, yielding results that conclusively show an error margin of less than 5% between experimental and theoretical values, satisfying the design criteria for sensor measurement.

As a non-destructive detection technique for measuring parallel transparent optics, wavelength-shifting interferometry provides a separation basis for multiple measured surfaces by assigning different frequencies to the interference harmonics based on their optical-path-difference. However, the non-synchronous sampling and spectral aliasing can decrease the measurement accuracy. Therefore, an approximate entropy algorithm is utilized to determine the sampling performance. Then the frequency search accuracy is guaranteed by the zero-padding method, and the all-phase Fourier transform method is introduced to correct the harmonic frequencies and amplitudes, where the spectral aliasing problem is solved by the relative amplitude analysis. So efficient surface measurement combining the above advantages is realized, and the validity is verified by simulation analysis of different profile features and de-tuning errors. The effectiveness of the proposed method is fully confirmed by the repetitive measurements on two parallel plates in the experiments.

The difficulty in multi-surface measurements comes from that each measured surface contributes harmonics to the fringe patterns, and the wavelength-shifting technique is an efficient technique that allows the harmonics to have different frequencies. When under-sampling in multi-surface measurements occurs, harmonic frequencies can not be detected by the spectrum directly. As a typical case in under-sampling, the problem of harmonic frequencies folded in the obtained spectrum as false frequencies are addressed in this paper. This contribution demonstrates an efficient algorithm enabling the detection of real and accurate harmonic frequencies, whereby the wavefronts can also be correctly reconstructed. Then the frequency detection accuracy is improved by redividing a smaller spectral interval of the selected frequency bands. Comparative studies based on the Zernike polynomials verify the effectiveness of the proposed method. Experiments for a transparent flat were performed, and the maximum PVQ, PV, and maximum RMS errors for repeatable measurements verified the effectiveness of the proposed algorithm in practical application.