Phase-shifting Interferometer Research Articles

One-grating common-path phase-shifting interferometer for quantitative phase imaging

A one-grating common-path phase-shifting interferometer for quantitative phase imaging is proposed as an improvement over the original interferometer design. In the original version, the setup is long, involving a grating pair, and requires precise mechanical translation of the grating, which poses difficulties for practical applications. The proposed interferometer utilizes grating multiplexing to reduce the length of the optical setup and uses a phase-only spatial light modulator to implement pinhole filtering and phase shifting simultaneously without any moving parts, making it more conducive for the realization of a practical version.

Time-resolved measurement of acoustic density fluctuations using a phase-shifting Mach-Zehnder interferometer.

Phase-shifting interferometry is one of the optical measurement techniques that improves accuracy and resolution by incorporating a controlled phase shift into conventional optical interferometry. In this study, a four-step phase-shifting interferometer is developed to measure the spatiotemporal distribution of acoustic density oscillations of the gas next to a rigid plate. The experimental apparatus consists of a polarizing Mach-Zehnder interferometer with a polarization camera capable of capturing four polarization directions in one shot image and it is used to measure the magnitude and the phase of density fluctuations through a duct of rectangular cross section connected to a loudspeaker. The results are compared with the well-established thermoacoustic theory describing the thermal coupling between acoustic oscillations and rigid boundaries, and the results show a very good agreement for various ratios of the (frequency-dependent) thermal boundary layer thickness to the plate spacing. This measurement technique could be advantageously employed to analyze more complex heat transfer processes involving the coupling of acoustic oscillations with rigid boundaries.

Nanoscale surface metrology with a liquid crystal-based phase-shifting angular shearing interferometer.

In this Letter, a phase-shifting angular shearing interferometer has been proposed for the application in optical surface metrology (SM) by using a combination of a wedge-shaped liquid crystal (LC) cell and a polarization phase shifter. The demonstration of this angular shearing interferometer for step height measurement is accomplished with the help of a phase-shifting technique. Four phase-shifted interferograms produced by a geometrical phase shifter are subjected to a simplified Wiener deconvolution method, which resembles a simple analysis technique for shearing interferograms in comparison to alternative approaches. A simulation study has been conducted to validate the proposed technique. The experimental results show an accuracy of 5.56% for determining the step height, which also agrees with the results obtained by atomic force microscopy. Owing to the tunability of birefringence, the proposed LC-based angular shearing interferometry technique will be useful to control spatial resolution in optical metrology.

High-dynamical-range angular measurement based on Heterodyne Phase Shifting Interferometry

A novel Heterodyne Phase Shifting Interferometer (HPSI) with nano-radians resolution and tens of milli-radians measurement range is developed. Instead of using a PZT-driven mirror to generate phase shift, the interferometer uses the heterodyne scheme. Such that the non-linearity and hysteresis effects of PZT are overcome. The non-linearity within the full range is less than 0.0005. Applying an intensity-modulated beam to test the angular measurement performance, the angular measurement noise level is 1.5nrad/Hz1/2. Due to the large dynamical range and good linearity, this HPSI is a good choice for precision angular metrology.

Experimental evaluation of characteristics for high-speed moving source

In this study, we evaluate the characteristics of a moving sound source in frequency and amplitude by visualizing a sound field around it. This is because train noise propagates with the characteristics of a moving sound source. To evaluate the characteristics, the visualization measurement was conducted using an optical sound measurement system, parallel phase-shifting interferometer, and low-noise moving model test facility. In the frequency result, modulation was observed, and the frequencies agreed well with the theoretical values. To evaluate the amplitude of sound, we determined the correction coefficient by which the integral pressure fluctuations recorded with the parallel phase-shifting interferometer are converted to the ones in one cross-section. By applying the conversion coefficient to the measured sound field, the shadow zone generated by the boundary layer was observed.

High-precision phase-shift method for heavy-load reference mirror based on capacitance sensor monitoring

To address the problems associated with the high-precision phase shift of heavily loaded reference mirrors in large-aperture interferometry, this study proposes a high-precision phase-shifting method based on capacitance sensor monitoring. This method achieves spatial nanoscale phase shifting for a heavy-load reference mirror via spatial three-point synchronous driving of piezoelectric ceramics combined with flexible hinges. The highly sensitive spatial translation of a heavy-load reference mirror is achieved through the gravitational loading of air-floating support and the spatial translation motion modeling of the reference mirror. Nano precision phase-shifting feeds are achieved through spatial in-situ monitoring by combining capacitive sensors and piezoelectric ceramics with a proportion integration differentiation closed-loop drive. Finally, high-precision and high-stability mechanical phase shifting in large-aperture interferometry circumvent the principal shortcomings of existing large-aperture wavelength-tuned phase shifting. The experimental results reveal that the established heavy-load reference mirror mechanical phase shift system has a resolving power of 1.1 nm, frequency response of 120 Hz, and synchronization accuracy of 2° for three-phase shift quantities. This method provides a novel phase-shifting technique for constructing large-aperture phase-shifting interferometers.

Phase retrieval for single-frame interferogram with an irregular-shaped aperture based on deep learning.

This paper proposes a high-precision phase retrieval method based on deep learning to extract the Zernike coefficients from a single-frame interferogram with an irregular-shaped aperture. Once the Zernike coefficients are obtained, the phase distribution can be retrieved directly using the Zernike polynomials. For many apertures, the root mean square (RMS) of the residual wavefront between the true and estimated wavefronts reached the order of 10-3 λ. Simulations were conducted under different noise conditions, indicating that the proposed method has high measurement accuracy and robustness. Experiments demonstrated that the accuracy achieved by this method was comparable to that of commercial phase-shifting interferometers. We believe that this method is useful for measuring optical surfaces with irregular apertures.

Spatio-temporal measurement of density distribution in oscillatory boundary layer using phase-shifting interferometer

The objective of this study is to directly measure the acoustic density fluctuations of a gas confined between two parallel plates. For this purpose, we originally developed a phase-shifting interferometer, where a polarizing Mach–Zehnder interferometer was incorporated with a polarization camera. This enabled us to achieve a time resolution of 70 FPS in addition to the high-resolution measurement by the phase-shifting technique. The test beam passes through the two parallel plates of width 10 mm, where an acoustic field is generated using a loudspeaker that produces high acoustic pressures (up to 3000 Pa) at frequencies ranging from 0.5 to 5 Hz. In this work, we successfully measure the density fluctuations, which are otherwise difficult to obtain using conventional optical interferometers. Additionally, the experimental results are compared to the visco-thermal acoustic theory, where a good agreement was reached, affirming the effectiveness of this phase-shifting interferometer. Furthermore, using the proposed setup, we are able to measure and deduce the spatiotemporal variations of the pressure, density, and temperature without interfering with the acoustic field, which distinguishes it from measurement probes like thermocouples. This interferometer can be used to study more complex phenomena such as edge effects in thermoacoustic engines.

Correcting retrace and system imaging errors to achieve nanometer accuracy in full aperture, single-shot Fizeau interferometry.

X-ray mirrors with single-digit nanometer height errors are required to preserve the quality of ultra-intense photon beams produced at synchrotron or free electron laser sources. To fabricate suitable X-ray mirrors, accurate metrology data is needed for deterministic polishing machines. Fizeau phase-shifting interferometers are optimized to achieve accurate results under nulled conditions. However, for curved or aspheric mirrors, a limited choice of reference optic often necessitates measurement under non-nulled conditions, which can introduce retrace error. Using experimental measurements of a multi-tilted calibration mirror, we have developed an empirical model of Fizeau retrace error, based on Zernike polynomial fitting. We demonstrate that the model is in good agreement with measurements of ultra-high quality, weakly-curved X-ray mirrors with sags of only a few tens of microns. Removing the predicted retrace error improves the measurement accuracy for full aperture, single shot, Fizeau interferometry to < 2 nm RMS.

Visualization of dynamic phase mixing and equilibrium process in transcritical and supercritical conditions

The phase mixing and equilibrium process plays a crucial role in the energy cycle and fuel process of supercritical fluids. This study focuses on the dynamic phase mixing during equilibrium process. The transient high-precision measurement of the density field during the phase mixing and equilibrium process is realized by the phase-shifting interferometer based on the pixelated-array masked method. The slow density equilibrium between four typical phase transition routes is captured and quantified with the visualization images and local parameters. The interface spreading rate during the equilibrium between liquid and supercritical is at a rather high level as 0.82 mm/s on average. This phenomenon is controlled by mixing and diffusion, which are faster process than evaporation. Nevertheless, equilibrium can be retarded even in a supercritical environment during the phase mixing between liquid-like and gas-like crossing the pseudo-critical line (average value of interface spreading rate as 0.45 mm/s), where there is a significant density gradient that generates a “solid wall” (buffer layer) effect.

Optical visualization of heat transfer in supercritical carbon dioxide under near-critical, liquid-like, and gas-like conditions

Heat transfer in supercritical carbon dioxide (sCO2) is experimentally visualized, and a measurement method is proposed for evaluating the transport phenomena in near-critical, liquid-like, and gas-like conditions. There are various uses for sCO2 in engineering applications, such as abstraction, material processing, and soil remediation. However, the heat and mass transfer under supercritical conditions have not been fully revealed, and innovative measurement techniques with higher spatial and temporal resolutions are required. This study focuses on the evaluation of heat transfer in sCO2 using a high-speed phase-shifting interferometer. The density distribution of sCO2 under different temperature and pressure conditions is successfully visualized with the proposed interferometer. Characteristics of the density field patterns are observed near the critical point and in liquid-like and gas-like conditions. It is demonstrated that the sensitivity of the density (i.e., refractive index) to temperature changes is different for each condition. The transient heat transfer under gas-like condition is evaluated by the interferometer, and numerical simulations with 3D model are performed to evaluate the experimental results. Finally, the interference fringes pattern obtained by the interferometer is shown to be qualitatively in good agreement with the numerical temperature field change. Additionally, transient variations of optical path length difference obtained in the experiment, which means apparent temperature distribution, were compared with numerical simulations. Experimental results are quantitatively in good agreement with the numerical results under a thermal diffusivity of order 10−8 m2/s, confirming the feasibility of the proposed measurement technique for the transient heat transfer in sCO2.

Reference-frame-independent model of a collective-excitation atom interferometer

We theoretically analyze the operating principles of a proposed matter-wave Sagnac interferometer utilizing Bose-Einstein condensate (BEC) phonon modes as an interference medium. Previous work found that the orbital angular momentum phonon modes of a ring-trapped BEC are split in frequency by rotations, leading to a measurable rotation signal. We develop an alternate description in which an imbalance in the counter-propagating modes' amplitudes (populations) is induced by the rotation of the system during condensation. This description gives analytic forms for the interferometric phase shift in 1D and is readily generalized to include mean-field interactions. To validate our findings, we simulate a ring-trapped BEC Sagnac interferometer in one dimension and demonstrate that measurement of an unknown rotation rate can be performed using a modified analysis. Our simulation data show strong agreement with our analytic results, and we further employ simulations to explore and clarify the role of superfluidity in this matter-wave Sagnac interferometer.

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.

Separate the primary aberration for the deformation error of optical flat in 300-mm-aperture vertical Fizeau interferometer

As a key component in a phase-shifting interferometer, the optical flat is often affected by supporting methods and generates deformation. In this paper, we propose a method to separate the deformation error through rotation measurement and Zernike polynomial fitting. We studied the principles, simulations, and experiments of this method. First, a support model is established in Comsol, and the deformation of the optical flat is analyzed. Then, the numerical simulation of the separation method for the deformation is performed, and the corresponding experiment is carried out on a Φ300 mm vertical Fizeau interferometer to verify the correctness of the method. The results show that numerical calculation and experimental measurement results are in good agreement, the difference between the PV value and the RMS value are only 0.004λ and 0.002λ, respectively. Finally, the error term that affected the calculation accuracy was analyzed. This proves that the proposed method can effectively separate the deformation introduced by the support.

Imaging through scattering media using differential intensity transmission matrices with different Hadamard orderings.

A transmission matrix (TM) is a powerful tool for light focusing and imaging through scattering media. For measuring it, the normal way requires establishing a multiple-step phase-shifting interferometer, which makes the TM measurement not only complex and sensitive but also time-consuming. Imaging through scattering media using an intensity TM method can make the setup for TM measurement without the phase-shifting interferometer, thus it is much simple, more stable, and several times faster. Here, based upon a differential intensity TM method, we demonstrated it to do imaging through scattering media using different Hadamard orderings. To accelerate the TM measuring speed while degrading as little as possible of the imaging quality, a relatively reasonable strategy to plan Hadamard orderings for the TM measurement is designed since it can suggest us to preferentially measure the components in TM that are more important to the imaging quality. Thanks to the different Hadamard orderings, their influences on the imaging quality at different measuring ratios are investigated, thus an optimal measuring ordering for accelerating the TM measurement can be obtained, while only sacrificing as little as possible of the image fidelity. Simulations and experiments verify the effectiveness of the proposed method.

Preliminary measurements of transient boundary heat transfer process under supercritical pressures using pixelated phase-shifting interferometry

The boundary heat transfer dynamics of supercritical carbon dioxide (CO2) is vital to its application in solar heater, extraction, refrigeration, heat pumps, power generations, nuclear systems and many other occasions. In this study, an experiment system using pixelated-array masked phase-shifting interferometer has been established for the visualization of temperature of supercritical CO2. The laser diagnostic process has been operated with a carefully designed visualization chamber to give clear transient results of the density and temperature fields for the first time. Those field information has been used to calculate the heat transfer characteristic parameters of the supercritical CO2 fluid in different boundary conditions. The current quantitative visualization analysis by the utilization of pixelated methods is shown capable of giving transient analysis of detailed field evolutions. The results show that (1) small temperature differences are generated in supercritical boundary due to the typical large thermal capacity; (2) supercritical convection boundary is with much quicker thermal quenching than subcritical states, while the subcritical states boundary parameter show localized characteristics while supercritical cases show fluctuation phenomena; (3) the boundary fluctuations heat transfer field exist for supercritical states and the fluctuations affect the overall heat transfer behaviour and the field evolution in supercritical states.

Experimental characterization and analysis of supercritical jet dynamics by phase-shifting interferometer system

The dynamics of trans-/super-critical jet significantly influence system behaviors for applications in spray and combustion, power conversion and chemical reactions. In this study, a pixelated phase-shifting interferometer with high spatial and temporal resolution has been established to investigate the quantitative characteristics of the sub-/trans-/super-critical jets. Thermodynamic features of the jets have been revealed through the transient quantitative field data. Theoretical analysis through experimental results on density and its gradient data has been elaborated. Boundary fractal analysis and radial diffusion process are examined with parameters of mixing layer thickness and spreading rate. The radial distance of the maximum density gradient decreases with time, which indicates jet transition from liquid to supercritical state as approaching the dense core. The fractal dimension of the edge varies from 1.30 to 1.38 for liquid-into-supercritical jet, showing that the initial “liquid” dominated feature and gradually transform process to a supercritical state from the mixing layer.

Quantitative visualization of injection jet flow behaviors of transcritical and supercritical processes by pixelated phase-shifting interferometer

Precise measurement and visualization of trans-/supercritical jet processes are of great significance for the control of power engines, chemical reaction and industrial processes. The quantitative measurement of transcritical jet under the influence of high pressure effect and transient effect is the key issues in such analysis. In this study, an improved phase-shifting interferometer system with high temporal and spatial resolution (0.001 s, 3.45 μm) has been realized by pixelated-array masked method to investigate characteristics in trans/supercritical jet processes. The transient density field and boundary structure of the phase-transition interface during four jet processes under sub/trans/supercritical conditions were quantitatively measured, which is different from those under subcritical conditions. The results show that the characteristic of subcritical jet is phenomenon of fragmentation and atomization within the experimental cell due to the presence of interfacial tension and strong density pulsations. The atomization is suppressed by high pressure effect in supercritical jet. Instead, Instead, single-phase mixing occurs due to the absence of surface tension. The dense core is found gradually dissolved during diffusion process of the jet flow. The transition from liquid-like to gas-like status of fluid occurs when the injection fluid goes cross the pseudo-critical line, and the jet surface has a large density gradient (in the current test range, |∇ρ|max = 8 × 105 kg/m4) due to temperature rise and pseudo-boiling expansion.

High-precision fiber-optic temperature sensor based on Fabry-Perot interferometer with reflective thin-film multilayer structures

An embodiment of a fiber-optic temperature sensor based on a Fabry-Perot interferometer and a scheme for interrogating an experimental sample of the sensor are proposed. The proposed solution makes it possible not to use expensive spectral measuring devices (spectrum analyzer, interrogator). The region of free dispersion and the phase sensitivity of the developed Fabry-Perot interferometer were determined in the temperature range from 20 °C to 590 °C. The accuracy of measuring the ambient temperature is calculated. The long-term stability of the measuring setup at room temperature has been evaluated. The phase shift of the Fabry-Perot interferometer with temperature change was registered. The design of the Fabry-Perot interferometer is implemented using reflective thin-film multilayer structures obtained by stage-by-stage electron-beam deposition in vacuum on polished end cleavages of an optical fiber. The interferometer interrogation method is based on the use of a vertical-cavity surface-emitting laser (VCSEL) operating in a pulsed mode. The principle of registering the phase shift of the interferometer with a change in temperature is based on the use of auxiliary modulation of laser radiation along the wavelength due to modulation (periodic change) of the duration of optical pulses. Auxiliary modulation makes it possible to obtain additional harmonic components in the interferometer signal, which are further used in homodyne demodulation to restore the interferometer phase shift signal proportional to the change in the optical path difference between the interferometer mirrors. The design of the high-temperature sensor is based on a Fabry-Perot interferometer the reflecting mirrors of which are five alternating layers of thin films of TiO2 and Al2O3. Based on the results of the temperature experiment, it was concluded that an increase in the ambient temperature leads to a decrease in the free dispersion region of the Fabry-Perot interferometer. The conclusion made is consistent with the theoretical data. According to the results of the experiment, it is shown that the phase sensitivity of the interferometer to temperature changes is 0.94 rad/K. The accuracy of temperature measurements at the 3σ level was 0.017 K. The results of the study may be of great importance in creating systems for monitoring temperatures above 300 °C. The use of such an interferometer makes it possible to carry out high-precision relative temperature measurements.

Co-phase state detection for segmented mirrors by dual-wavelength optical vortex phase-shifting interferometry.

The application of large-aperture telescopes requires the support of co-phase measurement techniques for segmented mirrors. This paper proposes a novel method to detect the co-phase state of segmented mirrors by applying a dual-wavelength phase-shifting interferometer based on optical vortex. Theory and experiments indicate that the wrapped phase map edges obtained by phase-shifting interference of the vortex beam are distributed in the form of a Fermat spiral. The piston error of the segmented mirrors corresponds to the rotation of the standard Fermat spiral center. In contrast, the tip/tilt error corresponds to the alteration of the center position of the deformed Fermat spiral. The rotation angle and the center position of the spiral are obtained by curve fitting, and the co-phase errors can be inversely solved. The experiments achieved an accuracy of approximately 4.04 nm in the piston and 0.16″ in the tip/tilt. The method avoids using complex lens arrays and devices, has an extended measurement range, high accuracy, and allows the co-phase errors between all sub-mirrors to be obtained in real-time. This study provides a novel and general method for detecting co-phase errors in a segmented primary mirror.