Phase Shift Error Research Articles

Rigorous simulation model of double-Ronchi shearing interferometry on lithographic tools.

Double-Ronchi shearing interferometry is widely used in wavefront aberration measurements for advanced lithography projection lens systems. A rigorous simulation model of double-Ronchi shearing interferometry on lithographic tools is the precondition for phase-shifting retrieval algorithm design and error analysis. This paper presents a rigorous simulation model of double-Ronchi shearing interferometry considering the vector nature of light. The model is accurate and can be used in the study of double-Ronchi shearing interferometry.

Comparing different configurations for rotary transformer of wound-rotor resolvers

Wound-Rotor resolvers are the oldest and the most widely used resolvers. They are two-phase synchronous generators with high frequency AC excitation. Rotary Transformers (RTs) are used to supply the rotating excitation winding of the resolver. Although RTs offer the benefits of contactless power transmission, harmonic contents of the secondary voltage, phase shift error between the primary and the secondary voltages, and high leakage flux are the main challenges in their usage along with resolvers. Therefore, in this paper, different configurations are examined for RT’s ferromagnetic core to overcome the mentioned problems. All the simulations are done using 3-D time variant finite element method (TVFEM) and the best configuration is chosen for experimental prototyping and measurements. Close agreement between the simulation and the experimental test results approves the performed analysis.

A Hybrid-Integrated Photonic Spiking Neural Network Framework Based on an MZI Array and VCSELs-SA

We propose an inferencing framework of a hybrid-integrated photonic spiking neural network (PSNN) to perform pattern recognition tasks, where the linear computation is realized based on a 4 × 4 silicon photonic Mach-Zehnder interferometer (MZI) array, and the nonlinear computation is performed by an InP-based spiking neuron array based on vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSELs-SA). With the modified Tempotron-like remote supervised method (ReSuMe) training algorithm, we realize two pattern recognition tasks, the recognition of numbers “0-3” and optical character recognition (OCR). The phase shifts in the MZI array are accurately configured to represent the weight matrix according to the decomposition procedure of a 4 × 4 triangular MZI mesh. Besides, the effects of the phase shift error and quantization precision of phase shifters (PSs) on the recognition performance are analyzed. For the OCR task, the 400 × 10 PSNN is realized by multiplexing the 4 × 4 MZI array based on the matrix blocking and the reconfigurability of the MZI array. This work provides a systematic computational model of the hybrid-integrated PSNN based on the silicon photonics and InP platforms, enabling the co-design and optimization of hardware architecture and algorithm, which contributes one step forward toward the construction of a hybrid-integrated PSNN hardware system.

Algorithm improvement for the surface morphology diagnostics based on the Gram-Schmidt orthonormalization and the least square ellipse fitting under the EAST-like vibrational environments

To realize in situ surface morphology diagnosis of Plasma-Facing Materials (PFMs) in the Experimental Advanced Superconducting Tokamak (EAST), a diagnostic system which is called DUT-SIEP (Speckle Interferometry Experiment Platform in Dalian University of Technology) is in development. Mimicking the actual vibrational conditions in EAST, Dual-wavelength Phase-Shifting Interferometry (DPSI) should be sensitive to vibrational environment and would reduce accuracy in the morphology measurements. Therefore, an approach combining the Gram–Schmidt orthonormalization and the Least-Square Ellipse Fitting (GS-LSEF) was proposed to replace the conventional phase extraction process to improve the measurement accuracy in the EAST-like vibration conditions. The new approach only requires two interferograms at each wavelength while the conventional method needs three at least for each wavelength. This significantly reduces the phase errors in the acquisition processes. The Gram–Schmidt orthonormalization is performed to correct the phase shift errors and determine the elliptic coefficients. In the Least-Square Ellipse Fitting, the elliptic coefficients are used to compensate the phase errors from the disturbed background intensity. The approach can significantly improve the measurement efficiency and avoid the infinite loops caused by the numeric divergent. The method was verified by laser ablation on tungsten (W) material for simulating erosion-processes in EAST. Results show that the proposed approach has significantly improved in accuracy by approximately twofold comparing with those by the convenient Four-step phase-shifting approaches due to reducing the noise superposition problems in the EAST-like vibrational conditions.

Joint least-squares algorithm correcting phase-shift errors and detector nonlinearity simultaneously in phase-shifting interferometry.

Phase-shifting interferometry may suffer from the errors caused by the miscalibration of the phase shifter and the nonlinearity of the detector simultaneously. These errors are not easy to eliminate because they are generally coupled with each other in interferograms. For solving this issue, we suggest a joint least-squares phase-shifting algorithm. It allows one to decouple these errors through an alternate least-squares fitting procedure, thus accurately estimating phases, phase shifts, and coefficients of the detector response simultaneously. The converging condition of this algorithm, associated with the uniqueness of the equation solution and anti-aliasing phase shifting, is discussed. Experimental results demonstrate that this proposed algorithm is helpful for improving phase-measuring accuracy in phase-shifting interferometry.

Estimation Method Based on Extended Kalman Filter for Uncertain Phase Shifts in Phase-Measuring Profilometry

Phase-measuring profilometry (PMP) is increasingly applied in high-accuracy three-dimensional shape measurement. However, various factors may result in the uncertainty of phase shift values in the PMP system, and phase errors induced by actual phase shift often bring about the reconstruction failure of a measured object. A quadratic phase estimation method using the extended Kalman filter is proposed to remove the phase error introduced by uncertain phase shift. After eliminating the background and fringe modulation, the state estimation is employed to evaluate the quadratic phase coefficients in a selected mask window, and the phase shifts of adjacent fringe patterns can be estimated to compute the unwrapping phase. This paper presents a novel method for improving the accuracy of the PMP system influenced by phase shift errors, and the proposed region-wise method significantly enhances the reconstruction quality and efficiency. Experimental results show that the proposed algorithm effectively evaluates the actual phase shift and directly compensates the phase error, and has the advantages of high speed, high accuracy, and robustness.

Large Interferometer For Exoplanets (LIFE)

Context. In the fourth paper in this series, we identified that a pentagonal arrangement of five telescopes, using a kernel-nulling beam combiner, shows notable advantages for some important performance metrics for a space-based mid-infrared nulling interferometer over several other considered configurations for the detection of Earth-like exoplanets around solar-type stars. Aims. We aim to produce a physical implementation of a kernel-nulling beam combiner for such a configuration, as well as a discussion of systematic and stochastic errors associated with the instrument. Methods. We developed a mathematical framework around a nulling beam combiner, and then used it along with a space interferometry simulator to identify the effects of systematic uncertainties. Results. We find that errors in the beam combiner optics, systematic phase errors and the root-mean-squared (RMS) fringe tracking errors result in instrument-limited performance at ~4–7 μm, and zodiacal light limited at ≳10 μm. Assuming a beam splitter reflectance error of |ΔR| = 5% and phase shift error of Δϕ = 3°, we find that the fringe tracking RMS error should be kept to less than 3 nm in order to be photon limited, and the systematic piston error be less than 0.5 nm to be appropriately sensitive to planets with a contrast of 1 × 10−7 over a 4–19 μm bandpass. We also identify that the beam combiner design, with the inclusion of a well-positioned shutter, provides an ability to produce robust kernel observables even if one or two collecting telescopes were to fail. The resulting four-telescope combiner, when put into an X-array formation, results in a transmission map with a relative signal-to-noise ratio equivalent to 80% of a fully functioning X-array combiner. Conclusions. The advantage in sensitivity and planet yield of the Kernel-5 nulling architecture, along with an inbuilt contingency option for a failed collector telescope, leads us to recommend this architecture be adopted for further study for the LIFE mission.

System design and error correction of simultaneous phase-shifting point-diffraction interferometer for dynamic wavefront detection with 400 mm aperture.

In this paper, a simultaneous phase-shifting point-diffraction interferometer (SPS-PDI) at632.8nm is designed with the assistance of an off-axis parabolic mirror (OAPM), through which the dynamic wavefront with 400mm aperture can be detected. In the system, a polarization point-diffraction plate (P-PDP) is developed to modulate the polarization states of the reference light and the test light through a simultaneous phase-shifting system based on a chessboard phase grating and a retarder array, and four phase-shifting interferograms can be acquired to realize dynamic detection. Furthermore, the circular carrier squeezing interferometry (CCSI) is proposed to suppress the phase errors generated by position mismatch, intensity distortion, and phase-shift error. The detection result of the SPS-PDI is consistent with the 4D PhaseCam6000 dynamic interferometer. The difference of the peak-to-valley (PV) and root-mean-square (RMS) values are only 0.04λ and 0.008λ. Additionally, the capacity to detect dynamic wavefront is good.

Model-independent nonlinearity rectification algorithm using a phase-probability-equalization-based look-up table.

In the fringe projection system, nonlinearities often result in severe artifacts, such as the gamma effect and the phase-shifting error. Most previous methods can only eliminate the nonlinearity of a particular model. Additionally, the problems of coupling nonlinearities are difficult to solve. Therefore, this paper proposes a model-independent nonlinear rectification algorithm. By applying phase probability equalization (PPE) on several complete periods of a flat area, we built a look-up table (LUT) between the phase error and the wrapped phase, and retrieved an accurate phase with the subtraction of a searched phase error. The simulation and experimental results show that, compared with the traditional full-field PPE algorithm, the proposed algorithm is more robust to the object height distributions and has better rectification on incomplete fringe periods. Besides, the proposed algorithm also has higher efficiency because of the characteristics of local processing and noniterative characteristics.

Joint Transmissive and Reflective RIS-Aided Secure MIMO Systems Design Under Spatially-Correlated Angular Uncertainty and Coupled PSEs

This paper investigates a joint transmissive and reflective reconfigurable intelligent surfaces (RIS) -aided secure multiple-input multiple-output (MIMO) system, where both a RIS-assisted transmitter and a RIS-based reflector are deployed to defend against the simultaneous jamming attack and wiretapping threat. Our design focuses on maximizing the sum rate under the unknown jammer’s beamforming, joint RISs’ coupled phase shift errors (PSEs), and spatially-correlated angular channel uncertainties. Besides, we take into account the various quality-of-service (QoS) requirement constraints for guaranteeing the secure performance. Since the problem is non-convex and mathematically intractable, a new optimization framework is established to facilitate the solution development to the formulated problem. Specifically, armed with the Akaike information criterion, a novel diagonalization method is first proposed to estimate the unknown jamming covariance matrix. Then, a series of fractional-eliminated rate expressions is derived that facilitates the application of the proposed Double Deterministic Transformation (DDT) to tackle the coupled stochastic PSEs. Besides, regardless of the spatial correlation matrix, a general discretization method is proposed to convert the e spatially-correlatd angular uncertainties into a worst-case robust one. Subsequently, building upon the above transformations which transform the original problem into tractable one, a two-layer iterative Lagrange multiplier algorithm capitalizing a low-complexity dual method is proposed to obtain the globally optimal solution of the digital precoder, where the multiple QoS constraints are handled without iteration. Meanwhile, we develop a novel polyblock-based multiple penalty method to obtain the globally optimal solutions to RISs’ phase shifts which can simultaneously satisfy the multiple QoS constraints. Moreover, to address the narrow feasibility region induced by the multiple QoS constraints, a heuristic initial optimization method is proposed, which strengthens the existing result. Finally, theoretical analysis and numerical results demonstrate the optimality and the excellent performance of our proposed optimization framework.

Phase Preselector Intended to Suppress Image Frequency in the Radio Receiver

Until now, to suppress the image channel, radio receivers use a frequency preselector consisting of oscillatory circuits of the input device and a radio frequency amplifier. But the frequency preselector does not suppress strongly enough the image frequency. Therefore, two frequency conversions are often used in railway radio communication systems. For example, for station radio communications at frequencies f = 151‒156 MHz, two frequency converters with fpr1 = 24 MHz and fpr2 =1,596 MHz are used, which greatly complicates the design of the receiver.The article proposes to use a phase preselector, which is the second parallel quadrature frequency converter, the signal of which, after passing additional elements, is summed with the signal of the first frequency converter with high suppression of the image frequency. With this method of organising communication, the bandpass phase shifter will be a problematic block in the phase preselector, since it must have a small error Δφ.A bandpass phase shifter made on RC circuits with a minimum phase shift error Δφ = 0,1˚ within a frequency band of 25 kHz is described. In case of increasing the frequency band, the phase shift error remains minimal, but it is necessary to use limiting amplifiers to equalise the amplitudes of the signals at its output. The use of a phase preselector instead of a frequency preselector will help to reduce the intermediate frequency, which will lead to high selectivity, both in the adjacent and image channels with single-sideband modulation, and will also simplify and reduce the cost of the signal receiver.

Phase-shifting digital holographic microscopy for microstructure measurement by sweeping the repetition rate of femtosecond laser

This paper proposes a phase-shifting digital holographic microscopy (PSDHM) for microstructure measurement by sweeping the repetition rate of femtosecond laser, and a multiple reflection arrangement between two quasi-parallel mirrors is constructed for optical multiplication. High precision phase-shifting can be achieved by sweeping the repetition rate of the femtosecond laser referenced to a Rb atomic clock without any mechanical sweeping. Optical multiplication can shorten the spatial distance of the optical delay line used for pulses alignment, make the PSDHM structure compact and stable, and avoid certain environmental disturbances. In the experiments, a ten-step phase-shifting test was first carried out for evaluating the phase-shifting accuracy, and the phase-shifting error was calculated to be in the range of −1° to 0.25°. Then, a USAF 1951 resolution target and a microstructure standard target were measured using a four-step PSDHM, and the measurement results were compared with those from a stylus profiler and a white light interferometer, respectively. The lateral resolution of PSDHM was tested to be about 2.1 μm, and the maximum error of the longitudinal measurement was within 6 nm. Experiments verify that the PSDHM system has good performance in terms of phase-shifting accuracy, surface topography measurement and coherent noise suppression.

Wavelength-tuning phase-shifting interferometry of transparent plates using sub-signal frequency correction

The difficulty of measurements on transparent flats comes from that the measured surfaces contribute corresponding sub-signals to the captured interferograms. As the cornerstone of phase demodulation, the frequency solution of the inter-harmonics is significant. The frequency solution accuracy is not satisfactory in traditional algorithms since the unavoidable frequency leakage. To overcome the problem above, a Fourier transform-based frequency correction algorithm with a narrower spectrum interval (FT-FCA) is proposed. By the developed algorithm, reliable multi-surface measurements can be ensured by the obtained sub-signal frequencies and the designed sampling weight functions. In the comparative studies, the robustness of the proposed FT-FCA to the random wavelength-tuning error and phase-shifting error is verified. In the experiments, two transparent flats with thicknesses of 20 mm and 43 mm were measured using a Fizeau wavelength-shifting interferometer, and the wavefronts were reconstructed by the developed algorithm. The comparative experimental analysis verifies the effectiveness of the FT-FCA.

Analysis of the influence of simultaneous phase-shifting shearing interference polarizer

The aspheric surface testing system based on the shearing interference principle is used to better analyze how the installation and fabrication errors of linear polarizer affects the precision of the aspheric surface. The effects of the angle error of the linear polarizer and the phase-shifting error of the wave plate on the reconstruction of the aspheric surface are studied. Constructing the model of the Jones matrix with the error term system, the reconstruction surface and residual after the corresponding error system are obtained using the four-step phase-shifting and differential Zernike method. The polarizer array of an angle of θ was varied from ( θ − 1 deg ) to ( θ + 2 deg ) every 0.2 deg, and the changes of the fitting surface and the residual surface were simulated. The phase-shifting error of the simulated one-fourth wave plate was <λ / 300. Then, the change of the fitting surface and residual surface when these two errors exist at the same time were simulated. The simulation results show that the influence of the phase-shifting error of the wave plate on the fitting surface is much greater than the angle error of the polarizer array. When assembling the interference system, the phase-shifting error of the wave plate should be <λ / 300 when the light transmission angle error of the polarizer array is between −0.2 deg and 0.2 deg, which will ensure the testing precision of the aspheric surface testing system.

Coverage Probability of STAR-RIS-Assisted Massive MIMO Systems With Correlation and Phase Errors

In this letter, we investigate a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisting a massive multiple-input multiple-output (mMIMO) system. In particular, we derive a closed-form expression for the coverage probability of a STAR-RIS assisted mMIMO system while accounting for correlated fading and phase-shift errors. Notably, the phase configuration takes place at every several coherence intervals by optimizing the coverage probability since the latter depends on statistical channel state information (CSI) in terms of large-scale statistics. As a result, we achieve a reduced complexity and overhead for the optimization of passive beamforming, which are increased in the case of STAR-RIS networks with instantaneous CSI. Numerical results corroborate our analysis, shed light on interesting properties such as the impact of the number of RIS elements and the effect of phase errors, along with affirming the superiority of STAR-RIS against reflective-only RIS.

RIS Assisted Wireless Powered IoT Networks With Phase Shift Error and Transceiver Hardware Impairment

Considering a reconfigurable intelligent surface (RIS) aided wireless powered Internet of Things (WP IoT) network. To address the energy-limitation issue, IoT devices in such a network can be wirelessly powered by a power station (PS) first and then connect with an access point (AP) using their own harvested energy. The RIS helps enhance energy and information receptions in the downlink wireless energy transfer (WET) and uplink wireless information transfer (WIT), respectively. This work unveils the impact of phase shift error (PSE) and transceiver hardware impairment (THI) on the considered network. Our investigation starts with a scenario where only the impact of the PSE on system under study is considered, then moves toward a scenario with the compound effect of both PSE and THI. A maximization problem of the system sum throughput is formulated to evaluate the overall performance for these two scenarios, subject to the constraints of the adjustable RIS phase shifts, the statistical PSE and the transmission time scheduling. To handle the non-convexity of the formulated problem due to those coupled variables, we first adopt the Lagrange dual method and Karush-Kuhn-Tucker (KKT) conditions to derive the optimal time scheduling in closed-form. Next, we recast the stochastic PSE into the deterministic counterpart for its tractability. Then, we adopt a successive convex approximation (SCA) to iteratively derive the optimal WIT’s phase shifts, and element-wise block coordinate decent (EBCD) and complex circle manifold (CCM) methods to iteratively derive the optimal WET’s phase shifts. Finally, we complete our solution approach for the scenario with both PSE and THI. Simulation results highlight the performance of the proposed scheme and the benefits induced by the RIS in comparison to benchmark schemes.

Quasi-pixelwise motion compensation for 4-step phase-shifting profilometry based on a phase error estimation.

Phase-shifting profilometry (PSP) is widely used in 3D shape measurement due to its high accuracy. However, in dynamic scenarios, the motion of objects will introduce phase-shifting errors and result in measurement errors. In this paper, a novel compensation method based on 4-step phase-shifting profilometry is proposed to reduce motion-induced errors when objects undergo uniform or uniformly accelerated motion. We utilize the periodic characteristic of fringe patterns to estimate the phase errors from only four phase-shifting patterns and realize a pixel-wise error compensation. This method can also be applied to non-rigid deforming objects and help restore high-quality texture. Both simulation and experiments demonstrate that the proposed method can effectively improve the measurement accuracy and reduce surface ripples introduced by motion for a standard monocular structured-light system.

Accurate EOM-based phase-shifting digital holography with a monitoring interferometer.

Phase-shifting digital holography (PSDH) can effectively remove the zero-order term and twin image in on-axis holography, but the phase-shifting error deteriorates the quality of reconstructed object images. In this paper, accurate PSDH with an electro-optic modulator (EOM) is proposed. The EOM is used to generate the required phase shift of on-axis digital holography, and the required phase shift is precisely measured with orthogonal detection of a homodyne interferometer and controlled with proportional-integral-derivative feedback in real time. The merits of our method are that it can achieve fast and accurate phase shifting without mechanical motion or sacrificing the resolution and field of view. The optical configuration was designed, an experimental setup was constructed, and real-time phase shifting was realized. Experiments of the phase-shifting accuracy evaluation, suppression effectiveness of the zero-order and twin image terms, and the specimen measurement demonstrate that the proposed method has significant application for precision topography measurement.

Performance analysis of a UAV-based IRS-assisted hybrid RF/FSO link with pointing and phase shift errors

Intelligent reflecting surface (IRS)-assisted wireless communication has recently become a promising technology to relax the line-of-sight requirement and also to provide a communication link in uncovered and remote areas. In this work, an alternate implementation of unmanned aerial vehicle (UAV)-based IRS-assisted hybrid free space optics/radio frequency wireless communications is presented. Phase shift errors occur due to the presence of reflecting elements in the IRS and affect the reflected beam. We consider the effect of phase shift error along with atmospheric turbulence, which is modeled as gamma-gamma distribution. Furthermore, a statistical model is considered for pointing errors due to the position and orientation fluctuation of the mounted IRS on the UAV. We derive closed-form expressions of the average symbol error rate and spectral efficiency of the considered system, under the combined effects of atmospheric turbulence, pointing error, and phase shift error. In addition, we present asymptotic analysis of the average symbol error rate of the system in a high SNR region and evaluate the diversity order and coding gain. The derived results are further validated using Monte Carlo simulation.

Parameter mismatch phase extraction method for spatial phase-shifting interferograms

Mismatch of interferogram parameters, such as the contrast and the fringe frequency inconsistency, can lead to phase error in simultaneous phase-shifting interferometry. In the dynamic Fizeau interferometer (DFI) based on the lateral displacements of light sources, an extended light source is used to suppress coherent noise. Aiming at the contrast mismatch generated by the extended light source and the inherent fringe frequency mismatch of the DFI, We propose a phase-extraction method to solve these problems. Based on this dynamic interferogram model, a linear relationship is constructed between the tilt angle and interferogram contrast, carrier frequency error, phase-shift error. Use the two steps of initial estimation and fine iterations to extract the phase. The RMS of the residuals was better than 1.5 × 10–5λ for the simulation and 0.004 λ for the experimental calculation, which demonstrates that the ripple error was suppressed during phase extraction.