Phase Step Errors Research Articles

Numerical four phase shifting microwave holography with phase step error analysis

Microwave holography utilizes an antenna to record the hologram. In a phase shifting microwave holographic method, the antenna has to perform the 2D scanning over an area corresponding to each phase shift in reference wave. Since this methodology requires phase shifting in each scan, this makes the process lengthy and laborious. In this work, a numerical 4-step phase shifting method is presented for generating the microwave hologram, in which a single (0° phase) hologram is recorded experimentally and the other three holograms are generated numerically by shifting the phase in reference wave. The amplitude and phase information of object is extracted using angular spectrum diffraction method. An error introduced by various components in experiments is calculated and corrected to enhance the accuracy of reconstructed phase values. The calculated phase step error for the experiment is 0.0501 rad. The quantitative and qualitative results along with the experimental phase shifting phase results are shown to envision the applications of numerical phase shifting microwave holography in imaging of metallic objects and highly scattered biological phantom.

Online Estimation of System Inertia in a Power Network Utilizing Synchrophasor Measurements

System inertia plays a vital role in controlling the angular stability of the system during a disturbance. Due to increased penetration of power electronic interfaced sources, such as Solar Photovoltaic (SPV) source, the overall system inertia reduces and varies depending on their operating conditions. In this paper, an approach for online inertia estimation in the power system network with SPV sources is proposed, using the synchronized measurements from Phasor Measurement Units (PMUs). An equivalent swing equation is used to emulate the network dynamics. A relationship between the inertia constant and the roots of this equation is determined. In order to numerically obtain the roots, the Estimation of Signal Parameter via Rotational Invariance Techniques (ESPRIT) method is first used to find the modes present in the frequency signal. A new formulation is proposed to extract an equivalent mode from all the obtained modes. Also, to avoid phase step error, Rate Of Change Of Frequency (ROCOF) is estimated from the equivalent mode of the frequency signal. Results obtained for the 39 bus New England system for various test cases, using Real-Time Digital Simulator (RTDS), prove the efficacy and superiority of the proposed approach over the existing approaches in the literature.

Multi-rate phase interpolator for high speed serial interfaces

A flexible and all digital CMOS phase interpolator is proposed in this paper suitable for high speed multi-Gigabit serial transceivers applications. The topology is constructed by a parallel combination of identical CMOS inverters grouped in two segments, a phase multiplexer and a capacitive tank. The proposed phase interpolator is quite simple and generic and can be easily fitted in a clock and data recovery system. The circuit is designed in a 65nm CMOS technology node under 1V supply voltage. The circuit robustness is verified through its application to the most updated M-PHY serial interface standard requirements supporting multiple data rates of 1.5/3/6Gbps. It is capable to deliver two orthogonal (I/Q) output phases with 5-bit phase resolution resulting in 32 equally spaced discrete steps of 11.25°. Post-layout simulation results confirm less than 1.7° worst case phase step error, settling time less than 2 clock cycles and power consumption 0.6mW at 6Gbps.

相移干涉测量中相移误差的自修正

The reasons for the phase shifting errors in a phase shifting interferometric measurement were analyzed,and a method to correct the phase shifting errors was researched based on the least square method.The calculation and correction for the relative phase step amount between arbitrary two interferograms were implemented.With the combination of optical,mechanical,electrical and calculating subsystems,more than three step interferometric images with arbitrary intervals were achieved.Then,the iterative least-square algorithm was used to calculate the relative phase step amount between arbitrary two interferograms.The phase shift error was feedbacked to the hardware system and the phase step interval was corrected in self-adaption until the phase step interval was equal to the characteristic value of the phase shift.A phase shift error correction system based on iterative least-square algorithm was constructed to achieve phase step error self-correction on a phase shifting interferometry.The real measurement of optical surface was conducted to verify the correction of algorithm and the feasibility of phase step error self-correction system.The results show that the relative phase step error is less than 5%,and the root mean square error(RMS) for measurement repeatability of the optical surface is less than λ/1 000 with the help of the proposed phase step self-correction process.

Invited Review Article: Measurement uncertainty of linear phase-stepping algorithms

Phase retrieval techniques are widely used in optics, imaging and electronics. Originating in signal theory, they were introduced to interferometry around 1970. Over the years, many robust phase-stepping techniques have been developed that minimize specific experimental influence quantities such as phase step errors or higher harmonic components of the signal. However, optimizing a technique for a specific influence quantity can compromise its performance with regard to others. We present a consistent quantitative analysis of phase measurement uncertainty for the generalized linear phase stepping algorithm with nominally equal phase stepping angles thereby reviewing and generalizing several results that have been reported in literature. All influence quantities are treated on equal footing, and correlations between them are described in a consistent way. For the special case of classical N-bucket algorithms, we present analytical formulae that describe the combined variance as a function of the phase angle values. For the general Arctan algorithms, we derive expressions for the measurement uncertainty averaged over the full 2π-range of phase angles. We also give an upper bound for the measurement uncertainty which can be expressed as being proportional to an algorithm specific factor. Tabular compilations help the reader to quickly assess the uncertainties that are involved with his or her technique.

Experimental requirement of the error-compensating five-frame interferogram-collecting sequence used in phase-shifting interferometry

Digital phase-shifting interferometry (PSI), a technique widely used in optical testing, requires interferograms collected at optical phase differences separated by a definite phase step. The five-frame interferogram-collecting sequence suggested by Hariharan et al. [Appl. Opt. 26, 2504 (1987)] is extremely effective in significantly reducing the errors in height profiles derived using PSI that are caused by phase-step errors. In this Letter, we report on a class of five-frame sequence that, owing to its mathematical equivalence with the one suggested by Hariharan et al. and its ease of execution, is more commonly used but is much less effective in reducing the height profile errors caused by phase-step errors.

High speed phase shifting interferometry using injection locking of the laser frequency to the resonant modes of a confocal Fabry-Perot cavity

We present a high speed phase shifting interferometer which utilizes the self injection locking of a frequency tunable laser diode. By using a confocal Fabry-Perot cavity made of ultra low expansion glass, and linearly modulating the laser diode current, the laser frequency could be injection locked to the resonant modes of the Fabry-Perot cavity consecutively. It provided equal phase steps to the interferograms which are ideal to be analyzed by the Carré algorithm. The phase step error was evaluated to be about 3 MHz which corresponds to 0.2 nm in length measurement. With this technique, profile measurements are insensitive to external vibration since four 640x480 pixels images can be acquired within 4 ms. Difference of two profile measurements, each made with and without vibration isolation, respectively, was evaluated to be 0.5 nm (rms).

Identification of nonlinear recording error in phase shifting interferometry

The phase shifting method for quantitative fringe pattern analysis provides high accuracy if stringent requirements on the component interferogram recording are met. In the paper the issue of detection and identification of error sources in the two-beam interferogram phase shifting experiment is discussed. The phase shift angle histogram and lattice-site representation are applied for that purpose. Special attention is paid to possible nonlinear recording of component interferograms in the presence of linear and nonlinear phase step errors. Four and five step phase shifting algorithms are considered. The superiority of the lattice-site representation is shown. In the case of phase steps equal to π/2, however, the lattice-site representation of shift angles for five frame algorithm does not allow to detect recording nonlinearity. The four frame counterpart shows to be very helpful in this respect. Its properties related to the fringe pattern profile under study, including a defocused Ronchi grating, are discussed.

High resolution, molecular-specific, reflectance imaging in optically dense tissue phantoms with structured-illumination

Structured-illumination microscopy delivers confocal-imaging capabilities and may be used for optical sectioning in bio-imaging applications. However, previous structured-illumination implementations are not capable of imaging molecular changes within highly scattering, biological samples in reflectance mode. Here, we present two advances which enable successful structured illumination reflectance microscopy to image molecular changes in epithelial tissue phantoms. First, we present the sine approximation algorithm to improve the ability to reconstruct the in-focus plane when the out-of-focus light is much greater in magnitude. We characterize the dependencies of this algorithm on phase step error, random noise and backscattered out-of-focus contributions. Second, we utilize a molecular-specific reflectance contrast agent based on gold nanoparticles to label disease-related biomarkers and increase the signal and signal-to-noise ratio (SNR) in structured illumination microscopy of biological tissue. Imaging results for multi-layer epithelial cell phantoms with optical properties characteristic of normal and cancerous tissue labeled with nanoparticles targeted against the epidermal growth factor receptor (EGFR) are presented. Structured illumination images reconstructed with the sine approximation algorithm compare favorably to those obtained with a standard confocal microscope; this new technique can be implemented in simple and small imaging platforms for future clinical studies.

Determination of phase-step errors of kinoform gratings from their diffraction efficiencies

Phase-step heights of a kinoform grating are determined from its diffraction efficiencies. The diffraction orders of the measured kinoform grating are compared with theoretical ones of the pure phase- stepped grating, and then, by a minimization procedure, the set of heights of the phase steps is found.

Derivation of retardation phase in computer-aided photoelasticity by using carrier fringe phase shifting

Previous phase-shifting schemes in computer-aided photoelasticity required the processing of six fringe patterns to derive the phase difference due to retardation. A technique in which a carrier fringe is used is demonstrated to reduce to four the number of fringe patterns required. The use of fewer fringe patterns lowers the computation time and the number of phase-step errors. The basis of the technique is outlined in detail, and experimental results are presented as well.

Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps

The authors describe an algorithm that extends the usefulness of phase stepping as a method for phase recovery from Fabry–Perot cavities. The use of arbitrary step sizes eliminates the need to operate with predetermined phase steps, thus minimising the effect of phase step errors. Furthermore, an iterative technique that accounts for multiple-reflection cavities is discussed. The algorithm is demonstrated on data obtained from an intrinsic fibre Fabry–Perot heat-flux sensor.