Two-step Phase-shifting Interferometry Research Articles

Dynamic phase measurement based on two-step phase-shifting interferometry with geometric phase grating

Abstract This paper proposes a novel dynamic 2-step phase-shifting interferometry method with a geometric phase grating and direct aberration-free object phase recovery algorithm. By exploring the amplitude-phase modulation property of the geometric phase grating, two-step phase-shifting interferograms with π / 2 phase shift can be obtained in a single shot through two kinds of spatial-multiplexing arrangements. Then the restriction on the background uniformity of the 2-step phase-shifting algorithms and the system aberration can be avoided simultaneously, based on the presented direct and fast object phase demodulation strategy with an object-free state recorded in advance, so an accurate object measurement result can be guaranteed. Accordingly, a detailed analysis of the theoretical model of error sources is conducted by taking the depolarization error of geometric phase grating and phase measurement model error into consideration, with a Least Squares Iteration optimization strategy established to further improve the measurement accuracy. Experiments on various types of phase objects, such as a photo-etched quartz substrate, the spatial light modulator, and a silicon chip, validate the effectiveness and accuracy of our method when compared with the reference phase obtained from 4-step temporal phase-shifting method. The dynamic phase measurement capability is demonstrated by exhibiting the evaporation process of alcohol droplet.

Two-frame advanced iterative self-tuning algorithm for accurate phase retrieval

Two-shot random phase-shifting interferometry has been actively investigated and improved owing to more demanding requirements in investigating dynamic phenomena or sensing some transient events. Given the background intensity and phase step knowledge, a reconstruction algorithm can be used to recover a complex-valued signal from intensity-only measurements. Most algorithms assume the background intensity and modulation amplitude in a frame of interferogram are constant variables; however, in realistic systems, they vary laterally and axially across the field of view. The model can be used but often appears less precise. In this work, we propose an approach that leverages the iterative method to cope with spatially varying background intensity and modulation amplitude. Our approach, termed the two-frame advanced iterative self-tuning algorithm, uses the spatial mean algorithm paired with an iterative procedure to search the phase step from two captured fringe patterns. It represents a novel approach to reliable and practical two-step phase-shifting interferometry without pre-filtering due to the cancelation of simultaneously obtaining the phase shifts and measured phase through iterative operation in the spatial–temporal domain. Experimental results obtained using the proposed method indicate that it is a simple and robust solution for phase extraction from a two-frame unknown phase shift fringe pattern with spatially varying background intensity and modulation amplitude.

Two-step phase shifting algorithm based on cost function minimization

A computationally efficient phase shift estimation algorithm is proposed for two-step phase-shifting interferometry based on a cost function minimization technique. The proposed method can accurately estimate the phase shift over a wide range resulting in reliable phase estimation. Due to the analytical solution offered by the proposed method, its computational efficiency is found to be higher than the existing state-of-art methods. While the performance of some methods is user-defined parameter-dependent, the proposed method is non-parametric and user-independent. The accuracy of phase shift and subsequent phase estimation using the proposed algorithm is found to be maximum around phase shift of π/2rad which further decreases for the phase shift values closer to 0rad and πrad. The experimental results demonstrate the practical applicability of the proposed method.

Two-step phase-shifting interferometry for self-interference digital holography

We propose a phase-shifting interferometry technique using only two in-line phase-shifted self-interference holograms. There is no requirement for additional recording or estimation in the measurement. The proposed technique adopts a mathematical model for self-interference digital holography. The effectiveness of the proposed technique is demonstrated by experiments on incoherent digital holographic microscopy and color-multiplexed fluorescence digital holography with computational coherent superposition. Two-color-multiplexed four-step phase-shifting incoherent digital holography is realized for the first time, to the best of our knowledge, using the proposed technique.

Two-step blind phase-shifting interferometry based on the extreme value of interference and least-squares iterative algorithm

To improve the measuring accuracy in two-step phase-shifting interferometry (PSI), a new approach combining the extreme value of interference (EVI) and the least-squares iterative algorithm (LSIA) is proposed to extract the phase from two-frame blind phase-shifting interferograms. This method first evaluates the phase shift between two interferograms by the EVI algorithm, and then constructs the fitted interferogram by the addition of two interferograms after filtering the corresponding background intensities, so the phase with high precision can be retrieved by combining two real interferograms and this fitted interferogram using the LSIA method. The proposed algorithm expands the flexibility of the LSIA method and has the high-precision performance compared with the existing algorithms in two-step PSI. Simulation and experiment are performed to verify the feasibility of the proposed algorithm.

Phase-shifting digital holography with a tri-focal Fibonacci telescope in a single exposure

In this work a tri-focal Fibonacci telescope based on a Greek-ladder photon sieve was constructed to realize three isometric images at the same time. Two of these interfere with the plane reference wave to form two frames of a phase-shifting hologram and the last unused one is referred to as the object intensity. Two-step phase-shifting interferometry can be used to reconstruct the test object in a single exposure. Simulations and experiments were carried out which verified the effectiveness of our proposed method. With advantage of real-time phase and a light-weight diffractive lens, the tri-focal sieve can be applied in bio-microscopy, surface testing and so on.

Two-step phase-shifting interferometry for illumination highly non-uniform by fitting 2D polynomials of degree K

This paper presents a robust and non-iterative algorithm for phase retrieval from two interferograms changed by an arbitrary unknown phase-step. First, the object phase is algebraically eliminated; second, the background and modulation light are approximated to 2D polynomials of degree K; third, an error function is algebraically derived; fourth, the least squares method is applied for estimating the phase-step, background, and modulation light, and finally the object phase is retrieved. With this approach, it is not necessary to do additional measurements. The important advantages of this method are its simple computer implementation and its capacity to support high spatial variations in the illumination. This approach is described theoretically, and verified numerically and experimentally.

Accurate reconstruction of electric field of ultrashort laser pulse with complete two-step phase-shifting

This paper presents a complete two-step phase-shifting (TSPS) spectral phase interferometry for direct electric-field reconstruction (SPIDER) to improve the reconstruction of ultrafast optical fields. Here, complete TSPS acts as a balanced detection that can not only remove the effect of the dc term of the interferogram, but also reduce measurement noises, and thereby improve the capability of SPIDER to measure the pulses with narrow spectra or complex spectral structures. Some prisms are chosen to replace some environment-sensitive optical components, especially reflective optics to improve operating stability and improve signal-to-noise ratio further. Our experiments show that the available shear can be decreased to 1.5% of the spectral width, which is only about $1/3$ compared with traditional SPIDER.

Fast phase-shifting technique for 3-D surface micro-topography measurement

This paper presents a two-step phase-shifting interferometry technique for fast inspection of uniformity of optoelectronic samples. The in-line configuration was applied to make full use of the space-bandwidth product (SBP) of the camera and hence can resolve finer spatial details of the sample. Two-step phase-shifting (TSPS) algorithm was used to reconstruct the object with only two in-line interferograms generated when a quarter wave plate (QWP) is placed in the light path of one arm of the Mach-Zehnder interferometer. Experimental results show that the extracted phases calculated by the TSPS technique match with the phases extracted by the three-step phase-shifting (THSPS) technique.

Multiple-image encryption by two-step phase-shifting interferometry and spatial multiplexing of smooth compressed signal**Project supported by the National Natural Science Foundation of China (Grant Nos. 61775121, 61307003, 61405122, and 11574311), Key R&D Program of Shandong Province, China (Grant No. 2018GGX101002), the Natural Science Foundation of Shandong Province, China (Grant No. R2016FM03), and the Fundamental Research Funds of Shandong University, China (Grant

A multiple-image encryption method based on two-step phase-shifting interferometry (PSI) and spatial multiplexing of a smooth compressed signal is proposed. In the encoding and encryption process, with the help of four index matrices to store original pixel positions, all the pixels of four secret images are firstly reordered in an ascending order; then, the four reordered images are transformed by five-order Haar wavelet transform and performed sparseness operation. After Arnold transform and pixels sampling operation, one combined image can be grouped with the aid of compressive sensing (CS) and spatial multiplexing techniques. Finally, putting the combined image at the input plane of the PSI encryption scheme, only two interferograms ciphertexts can be obtained. During the decoding and decryption, utilizing all the secret key groups and index matrices keys, all the original secret images can be successfully decrypted by a wave-front retrieval algorithm of two-step PSI, spatial de-multiplexing, inverse Arnold transform, inverse discrete wavelet transform, and pixels reordering operation.

Phase gradient algorithm based on co-axis two-step phase-shifting interferometry and its application

A phase gradient method based on co-axis two-step phase-shifting interferometry, is used to reveal the detailed information of a specimen. In this method, the phase gradient distribution can only be obtained by calculating both the first-order derivative and the radial Hilbert transformation of the intensity difference between two phase-shifted interferograms. The feasibility and accuracy of this method were fully verified by the simulation results for a polystyrene sphere and a red blood cell. The empirical results demonstrated that phase gradient is sensitive to changes in the refractive index and morphology. Because phase retrieval and tedious phase unwrapping are not required, the calculation speed is faster. In addition, co-axis interferometry has high spatial resolution.

Two-level image authentication by two-step phase-shifting interferometry and compressive sensing

A two-level image authentication method is proposed; the method is based on two-step phase-shifting interferometry, double random phase encoding, and compressive sensing (CS) theory, by which the certification image can be encoded into two interferograms. Through discrete wavelet transform (DWT), sparseness processing, Arnold transform, and data compression, two compressed signals can be generated and delivered to two different participants of the authentication system. Only the participant who possesses the first compressed signal attempts to pass the low-level authentication. The application of Orthogonal Match Pursuit CS algorithm reconstruction, inverse Arnold transform, inverse DWT, two-step phase-shifting wavefront reconstruction, and inverse Fresnel transform can result in the output of a remarkable peak in the central location of the nonlinear correlation coefficient distributions of the recovered image and the standard certification image. Then, the other participant, who possesses the second compressed signal, is authorized to carry out the high-level authentication. Therefore, both compressed signals are collected to reconstruct the original meaningful certification image with a high correlation coefficient. Theoretical analysis and numerical simulations verify the feasibility of the proposed method.

Statistical estimation method for phase shift in generalized two-step phase-shifting interferometry

Based on the statistical property of the phase and the spatial frequency relationship between the object and reference waves, a simple estimation method for phase shift extraction is proposed for generalized two-step phase-shifting interferometry. In this method, a phase shift constrained within 0–π is firstly worked out with statistics, and then it is corrected as its principal value within the range (0, 2π) except the singular value of π, according to the sign of tangent form of one half of the relative phase shift. Thus, this method provides a possible method to solve the uncertainty of the phase shift existing in some common algorithms. After the determination of the phase shift, a phase can be retrieved easily. The feasibility and the accuracy of the method were verified by numerical simulations and an optical experiment. In addition, this method is applicable to two or more frames off-axis generalized PSI, including slightly off-axis situation.

特殊点的两步广义相移干涉相移值提取算法

特殊点的两步广义相移干涉相移值提取算法

Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor

Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information from wavelength-multiplexed holograms is presented. Specific phase shifts for respective wavelengths are introduced to remove the crosstalk components and extract only the object wave at the desired wavelength from the holograms. Object waves at multiple wavelengths are selectively extracted by utilizing 2π ambiguity and the subtraction procedures based on phase-shifting interferometry. Two-step phase-shifting interferometry is merged into the procedures to decrease the number of the recorded holograms. The proposed technique is numerically investigated and experimentally demonstrated. The applicability to the case where the number of wavelengths is more than two and possible noise when using two-step phase-shifting interferometry are discussed.

A derivative method of phase retrieval based on two interferograms with an unknown phase shift

Phase retrieval is one of the important study fields in the optical metrology of phase objects. In this paper, a rapid derivative method of phase retrieval is proposed that can both extract phase and determine an unknown phase shift based on two-step phase-shifting interferometry (PSI). By numerically calculating the first-order derivatives of two PSI images with the random phase distribution, one can directly obtain the quantitative phase image, without the need for dealing with the background term and reference intensity. We give the method with the theory, and illustrate it using the simulations of a ball and a red blood cell (RBC) and an optical experiment of a polystyrene sphere. From which, the effectiveness and accuracy of this method is verified, but the error of the experimental result is larger than that of the simulated result.

Fast phase retrieval method based on twice derivatives in phase-shifting interferometry with a blind phase shift

A derivative method for phase retrieval in two-step phase-shifting interferometry (PSI) with a blind phase shift is proposed in this Letter. By numerically calculating the first-order and second-order derivatives of two PSI images, one can directly determine the phase shift and then obtain the quantitative phase image. We illustrate the method with both a theoretical analysis and some simulations of an average-sized red blood cell in different interferometry recording modes. From the results, the validity, accuracy, as well as efficiency of this method are verified. Moreover, this method can be applied to any quantitative phase imaging, including on-axis, off-axis, and slight off-axis interferometry.

Determination of Poisson’s Ratio by the Methods of Two-Step Phase-Shifting Interferometry

We propose procedures for the contactless determination of Poisson’s ratio by the methods of phase-shifting interferometry for beams with smooth and rough surfaces and a scheme of experimental installation used for their realization. We also present some results of verification of these procedures for steel and duralumin beams.

Non-iterative blind phase-shifting algorithm for two-step phase-shifting interferometry based on an analytical formula

An algorithm for retrieval of the complex amplitude of the recorded object in blind two-step phase-shifting interferometry is proposed, in which the complex amplitude of the object wave can be found out directly from the recorded intensity patterns without the process of extracting the phase-shifting value in advance. A simple analytical formula for calculating the complex amplitude (including both the amplitude and the phase) of the object wave based on the recorded intensities is derived. Both the theoretical analysis and experimental results demonstrated the feasibility of this kind of blind phase-shifting algorithm in small phase shift situations.

Optical Image Split-encryption Based on Object Plane for Completely Removing the Silhouette Problem

We propose a split-encryption scheme on converting original images to multiple ciphertexts. This conversion introduces one random phase-only function (POF) to influence phase distribution of the preliminary ciphertexts. In the encryption process, the original image is mathematically split into two POFs. Then, they are modulated on a spatial light modulator one after another. And subsequently two final ciphertexts are generated by utilizing two-step phase-shifting interferometry. In the decryption process, a high-quality reconstructed image with relative error <TEX>$RE=7.6061{\times}10^{-31}$</TEX> can be achieved only when the summation of the two ciphertexts is Fresnel-transformed to the reconstructed plane. During the verification process, any silhouette information was invisible in the two reconstructed images from different single ciphertexts. Both of the two single REs are more than 0.6, which is better than in previous research. Moreover, this proposed scheme works well with gray images.