Improving the measurement accuracy of shadow moiré by three-step random phase-shifting algorithm

In this study, a simultaneous dual-wavelength phase-shifting interferometry system using two reference arms was built, and a simple phase retrieval method for the system with blind phase shifts was proposed using five phase-shifted interferograms. First, the unknown phase shifts were controlled independently to separate the information of single wavelength, which can be calculated directly by using the Euclidean matrix norm algorithm. Second, a wrapped phase of a single wavelength was obtained using a three-step phase-shifting algorithm. Finally, a continuous phase of a synthetic wavelength was obtained. The simulation and experimental results indicated the feasibility and greater accuracy of the proposed method. Moreover, the strict requirement of the phase shifter was eliminated because of the use of arbitrary and unknown phase shifts, thereby making experimental operation easier.

An algorithm is proposed for extraction of the unknown phase shift steps and correction of the phase shift error in partially coherent illuminated phase-shifting digital holography. The mean contrast of noise (MCN) of the reconstructed amplitude image is developed to evaluate the noise level resulting from phase shift error. Actual phase shift steps are first extracted by minimizing the MCN. Then the extracted actual phase shift steps are used to retrieve the original object wave without phase shift error. The validity of the proposed method is verified by simulations, and then implemented in partially coherent illuminated phase-shifting digital holography where a spatial light modulator with unknown phase modulation characteristic is used as the phase shifter. The results demonstrated that the actual phase shift steps can be extracted and the phase shift error is corrected effectively.

A direct phase shift extraction and wavefront reconstruction method in two-step generalized phase-shifting interferometry (GPSI) with arbitrary unknown phase shift is proposed. In this method the unknown phase shift α can be extracted by a determinate formula directly without iteration or additional judgment of its correct value from two or more phase shift solutions as necessary before. By an appropriate formula of GPSI the complex object field in the recording plane can be calculated and then the object wavefront in the original object plane obtained. This method is applicable for GPSI of any frame number K≥2 and for both the amplitude and phase objects. Computer simulations have shown that the phase shift extraction errors are below 0.01 rad in a wide range of 0.4 rad<α<2.6 rad and the computation time is greatly reduced by a factor of about 20 compared with the previous method. The effectiveness and accuracy of this method are also verified by optical experiments.

Projector nonlinearity is a common problem for digital structured light-based three-dimensional (3-D) shape measurement techniques. A temporal-spatial binary encoding method is presented for the purpose of eluding it. We build a 3-D shape measurement scheme by combining our proposed method with phase measurement profiling. A standard sinusoidal fringe pattern is divided into more than two binary fringe patterns using specially designed temporal and spatial binary encoding rule based on intensity hierarchic quantification, and then are in-focus projected onto the measured object at a time sequence to reconstruct a frame phase-shifting fringe image. On account of the projected binary fringe pattern strictly consisting of zeros and ones, the influence of the projector nonlinearity on the measurement result can be effectively ruled out and simultaneously enables high-quality sinusoidality. In-depth investigations by theoretical analysis and experiments are conducted to demonstrate the performance of this method.

Phase-shifting interferometry with a genetic algorithm is proposed. The correction of unknown phase-shifting error is an important task in general phase-shifting interferometry. Since phase-shifting errors generate twin image noise in a reconstructed image, we can reduce the phase-shifting errors indirectly by trying to eliminate the twin-image noise in the reconstructed image. By Zernike polynomial expansion, the reconstructed image is represented as the evenness and oddness, where the ratio of the evenness and oddness is a measure of the amount of the twin image noise. We employ the genetic algorithm for finding the fittest phase shifts of interferograms by reducing the evenness of the reconstructed image, which leads to reduction of phase-shifting errors. This phase-shifting interferometry with a genetic algorithm is confirmed experimentally.

A novel method is presented to extract phase distribution from phase-shifted interferograms with unknown tilt phase shifts. The proposed method can estimate the tilt phase shift between two temporal phase-shifted interferograms with high accuracy, by extending the regularized optical flow method with the spatial image processing and frequency estimation technology. With all the estimated tilt phase shifts, the phase component encoded in the interferograms can be extracted by the least-squares method. Both simulation and experimental results have fully proved the feasibility of the proposed method. Particularly, a flat-based diffractive optical element with quasi-continuous surface is tested by the proposed method with introduction of considerably large tilt phase shift amounts (i.e., the highest estimated tilt phase shift amount between two consecutive frame reaches 6.18λ). The phase extraction result is in good agreement with that of Zygo's MetroPro software under steady-state testing conditions, and the residual difference between them is discussed. In comparison with the previous methods, the proposed method not only has relatively little restrictions on the amounts or orientations of the tilt phase shifts, but also works well with interferograms including open and closed fringes in any combination.

The phase shift interferometer can be used to detect the wavefront phase of optical system and the piezoelectric transducer is widely used as a phase shifter in phase shift interferometer. However, the displacement of piezoelectric transducer presents hysteresis nonlinearity. Hence, when the equal-step phase shift algorithm (e.g. the Schwider-Hariharan phase-shifting algorithm which obtains five images and sets the phase-shifting between images to 90 degrees) is used in the phase shift interferometer to calculate the wavefront phase, the nonlinearity of piezoelectric transducer will introduce error to the 90 degrees phase-shifting between images. Then the calculated wavefront phase will contain the error. Hence, it is necessary to propose a method to diminish the phase error. The method by choosing an optimized phase-shifting algorithm has been used to mitigate the phase error. However, the method does not directly focus on piezoelectric transducer itself. In the paper, we propose a method by compensating for the hysteresis nonlinearity displacement of piezoelectric transducer to mitigate the phase error. We analyze the relationship between the nonlinearity displacement of piezoelectric transducer and phase shift error and establish the error models to analyze the relationship between the phase shift error and phase error. Furtherly, we propose the Proportion Integration Differentiation (PID) closed-loop feedback control method to compensate for the hysteresis nonlinearity displacement of piezoelectric transducer. The results indicate that, with compensated, the maximum tracking error is reduced from 18.89 % to 5.5 %. And in the ascending stage, the PV error and RMS error with compensated are less 12.35nm and 4.34nm than the PV error and RMS error without compensated. In the descending stage, the PV error and RMS error with compensated are less 19.15nm and 6.76nm than the PV error and RMS error without compensated. Hence, the proposed compensation method can improve the detection accuracy of the phase shift interferometer.&amp;#xD;

When phase-shifting (PS) and temporal-phase-unwrapping algorithms are employed for three-dimensional (3D) shape measurement, the measured object must be kept static during the projection and acquisition. If the measured object is moving, deviation among multiple fringe images will inevitably occur, which results in the 3D measurement error. In this paper, a novel 3D measurement technique for rigid moving objects obtained by using the PS algorithm and the three-pitch heterodyne-unwrapping (TPHU) algorithm is proposed, which consists of six steps. First, the pixel offset is estimated based on a centroid deviation calculation. Second, the phase offsets among multiple fringes are calculated. Third, a novel set of fringes for the projection is generated. Fourth, we offset the captured fringe images according to the pixel offsets to generate a new set of captured fringe images for the phase calculation. Fifth, the wrapped phase is calculated by the PS algorithm. Finally, the unwrapped phase is calculated by the TPHU algorithm. The proposed method can be applied to the 3D shape reconstruction of a rigid movement object with a determined movement trajectory and constant movement speed. This approach not only greatly improves the measurement efficiency but also inherits the high accuracy and robustness of the PS and TPHU algorithms.

A phase shift selection method is proposed to design algorithms immune against phase shift errors in two-step generalized phase-shifting interferometry. A general formula for wavefront reconstruction error is derived, and its specific expressions for two common errors are also given. Calculation results suggest that the proper range of phase shift for general application is about from 0.5 to 2.0 rad for both the fixed and linear phase shift errors. Computer simulations have demonstrated the effectiveness of this phase shift selection method by decreasing the wave reconstruction errors to one-fifth.

In this paper, a 90° phase-shift delay-locked loop (DLL) with closed-loop duty-cycle correction (DCC) is proposed. The proposed DLL increases the accuracies of the duty cycle and 90° phase shift by compensating PVT variation using a closed-loop DCC and a closed-loop 90° phase shift, and achieves a smaller area by reducing the delay-line length by 25% to 50% as compared to conventional DLLs with the DCC and/or 90° phase shift. We also propose a 90° phase detector consisting of a time-to-voltage converter and a sense amplifier to independently control and lock the DCC and 90° phase-shift loops. The proposed DLL is designed using the 0.13 μm process technology with a 1.2 V supply voltage. The operating frequency range is from 400 MHz to 800 MHz. Monte Carlo simulation results show that the output duty cycle error ranges from -0.21% to 0.22% for an input duty cycle range of 30% to 70%, and the 90° phase-shift error is less than 1.66°, which is lower than a 5.7ps error at 800 MHz. Finally, the power consumption of the proposed DLL is 2.8 mW at 800 MHz.

Phase-shifting interferometry has many advantages, and the phase shifting nature of the Liquid Crystal Point Diffraction Interferometer (LCPDI) promises to provide significant improvement over other current OMEGA wavefront sensors. However, while phase-shifting capabilities improve its accuracy as an interferometer, phase-shifting itself introduces errors. Phase-shifting algorithms are designed to eliminate certain types of phase-shift errors, and it is important to chose an algorithm that is best suited for use with the LCPDI. Using polarization microscopy, the authors have observed a correlation between LC alignment around the microsphere and fringe behavior. After designing a procedure to compare phase-shifting algorithms, they were able to predict the accuracy of two particular algorithms through computer modeling of device-specific phase shift-errors.

A new fringe analysis for shadow moire new fringe analysis for shadow moire using a SEM is proposed in order to perform a high resolution 3-D shape measurement. Two sheets of shadow moire fringe images are grabbed by shifting the grating using PZT. One sheet of deformed shadow of grating image which does not include any original grid image is reconstructed from two shadow moire fringe images by using the new method. In experiments, a sphere that is a bearing ball(diameter:700μm) was measured by the method. The standard deviation of the difference between the measured result and the ideal shape as the sphere was 95nm. From the results, it was confirmed that the proposed method had not only a high measuring accuracy but also a spatial high-resolution power(330nm). using a SEM is proposed in order to perform a high resolution 3-D shape measurement. Two sheets of shadow moire fringe images are grabbed by shifting the grating using PZT. One sheet of deformed shadow of grating image which does not include any original grid image is reconstructed from two shadow moire fringe images by using the new method. In experiments, a sphere that is a bearing ball(diameter:700μm) was measured by the method. The standard deviation of the difference between the measured result and the ideal shape as the sphere was 95nm. From the results, it was confirmed that the proposed method had not only a high measuring accuracy but also a spatial high-resolution power(330nm).

To apply phase-shifting interferometry (PSI) to in-situ measurement, we have proposed an algorithm to detect and suppress phase-shifting error and contrast fluctuation. The phase shift and contrast are analyzed in spatial-frequency domain. The strength of baseband and sideband implies the pattern contrast. The position and phase angle of the sideband indicates the tilt gradients and translational value of phase shift. Thus, the phase shift error and contrast fluctuation could be extracted. A contrast-compensated equation is established to calculate the wavefront phase. The proposed algorithm was applied to the interferograms subjecting to vibration and wavefront phase was calculated. The experimental results show that, under vibration of one micron amplitude and 60Hz frequency, the error of wavefront PV value is less than 0.01wave and the 2σ repeatability is less than 0.01wave. For no hardware is required, the proposed algorithm provides a cost-effective method for wavefront in-situ measurement with PSI.

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 shifts from scattering are used to analyze and engineer acoustic radiation forces. With the aid of phase shifts, analytical results for acoustic radiation forces can be simplified into compact and physically meaningful expressions, which can be used to develop a simplified procedure for the engineering of the radiation force. The desired radiation force can be fulfilled by a specific set of phase shifts up to a certain order, and then the required phase shifts can be fulfilled by engineering object and beam parameters. As an example, the phase shift approach is used to engineer the inner-to-outer radius ratio and the outer radius of a spherical shell to show how to use the phase shift method for the design of acoustic radiation forces. The example here is a force that is desired to pull particles against the propagation of a Bessel beam. A small paraxial parameter to pull a spherical shell is satisfied by in-phase scattering of monopole, dipole, quadrupole, octupole, and beyond. The example presented here is relatively simple yet reveals the advantages of the phase shift approach. The phase shift method can provide a simplified route for the design of acoustic tweezers using either traveling beams or standing waves.