Phase-stepping Interferometry Research Articles

Robust phase-stepping interferometry for traceable length measurements in a Hilger–Watts gauge-block interferometer

We demonstrate a simple adaptation of the venerable Hilger–Watts gauge-block interferometer capable of performing automated, traceable calibrations of gauge block length by phase-stepping interferometry (PSI). We discuss the selection of robust PSI filtering algorithms to counteract in software the flaws of the hardware implementation, including markedly non-sinusoidal fringes and inhomogeneous phase step size. We find that the length measurement errors attributable to the phase measurement can be reduced to 0.6 nm, even with a simple retrofit to an instrument not originally designed for PSI.

Thermodiffusion in Ternary Mixtures of Water/Ethanol/Triethylene Glycol: First Report on the DCMIX3-Experiments Performed on the International Space Station

We report on thermodiffusion experiments conducted on the International Space Station ISS during fall 2016. These experiments are part of the DCMIX (Diffusion and thermodiffusion Coefficients Measurements in ternary Mixtures) project, which aims at establishing a reliable data base of non-isothermal transport coefficients for selected ternary liquid mixtures. The third campaign, DCMIX3, focuses on aqueous systems with water/ethanol/triethylene glycol as an example, where sign changes of the Soret coefficient have already been reported for certain binary subsystems. Investigations have been carried out with the SODI (Selectable Optical Diagnostics Instrument) instrument, a Mach-Zehnder interferometer set up inside the Microgravity Science Glovebox in the Destiny Module of the ISS. Concentration changes within the liquids have been monitored in response to an external temperature gradient using phase-stepping interferometry. The complete data set has been made available in spring 2017. Due to additionally available measurement time, it was possible to collect a complete data set at 30∘C and an almost complete data set at 25∘C, which significantly exceeds the originally envisaged measurements at a single temperature only. All samples could be measured successfully. The SODI instrument and the DCMIX experiments have proven reliable and robust, allowing to extract meaningful data even in case of unforeseen laser instabilities. First assessments of the data quality have revealed six out of 31 runs with some problems in image contrast and/or phase step stability that will require more sophisticated algorithms. This publication documents all relevant parameters of the conducted experiments and also events that might have an influence on the final results. The compiled information is intended to serve as a starting point for all following data evaluations.

Determination of any pure spatial qudits from a minimum number of measurements by phase-stepping interferometry

We present a proof-of-principle demonstration of a method to characterize \textit{any} pure spatial qudit of arbitrary dimension $d$, which is based on the classic phase shift interferometry technique. In the proposed scheme a total of only $4 d$ measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum state tomography which require of the order of $d^2$. By using this technique, we have experimentally reconstructed a large number of states ranging from $d=2$ up to $14$ with mean fidelity values higher than $0.97$. For that purpose the qudits were codified in the discretized transverse momentum-position of single photons, once they are sent through an aperture with $d$ slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows to reduce the number of measurement settings to 4 since the $d$ slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d-3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing to implement the reconstruction method in a general experiment.

Subtractive two-frame three-beam phase-stepping interferometry for testing surface shape of quasi-parallel plates.

We present an effective method of testing the surface shape of quasi-parallel plates which requires only two phase-shifted three-beam interferograms. We derive a general formula for difference of two three-beam interferograms as a function of the phase shift value. The phase shift does not have to be precisely determined and uniform in the image domain. We show and compare results of extracting the fringe set and corresponding phase distribution related to the plate front surface shape using the two dimensional continuous wavelet transform, Hilbert-Huang transform and Fourier transform methods. Simulated and experimental data is used to verify the algorithm performance and robustness.

Vibratory strain field measurement by transverse digital holography.

A method is presented for measuring vibratory strain fields using phase-stepped, image-plane digital holography. An object surface is observed along its normal vector while illuminated at equal and opposite angles by two mutually coherent laser beams. One beam is phase stepped by quarter-wavelength increments between TV frames, and the resulting images are processed to yield holographic images. Object vibrations result in zero-order Bessel function fringes in the display. The second beam is modulated at the same frequency of the object vibration and is used to shift the fringes in a manner analogous to phase step interferometry. The resulting images are processed to yield a wrapped phase map, which is unwrapped and corrected for the error associated with using zero-order Bessel functions in place of cosine functions. The unwrapped images are processed to obtain the average slopes for image segments, and these slopes are multiplied by a scale factor to convert them to strain. The analysis program used here divides the field of view into five horizontal by four vertical segments, which provide a map of the vibratory strain field.

Humidity coefficient correction in the calculation equations of air refractive index by He-Ne laser based on phase step interferometry.

The refractive index of air (RIA) is an important parameter in precision measurement. The revisions to Edlen's equations by Boensch and Potulski [Metrologia 35, 133 (1998)] are mostly used to calculate the RIA at present. Since the humidity correction coefficients in the formulas were performed with four wavelengths of a Cd(114) lamp (644.0, 508.7, 480.1, and 467.9 nm) and at the temperature range of 19.6°C-20.1°C, the application is restricted when an He-Ne laser is used as the light source, which is mostly applied in optical precision measurement, and the environmental temperature is far away from 20°C as well. To solve this problem, a measurement system based on phase step interferometry for measuring the effect of the humidity to the RIA is presented, and a corresponding humidity correction equation is derived. The analysis and comparison results show that the uncertainty of the presented equation is better than that of Boensch and Potulski's. It is more suitable in present precision measurements by He-Ne laser, and the application temperature range extends to 14.6°C-24.0°C as well.

Measuring nanoparticle size using phase-stepping interferometry: quantifying measurement sensitivity to surface roughness.

A method for sizing nanoparticles using phase-stepping interferometry has been developed recently by Little et al. [Appl. Phys. Lett. 103, 161107 (2013)]. We present an analytical procedure to quantify how sensitive measurement precision is to surface roughness. This procedure computes the standard deviation in the measured phase as a function of the surface roughness power spectrum. It is applied to nanospheres and nanowires on a flat plane and also a flat plane in isolation. Calculated sensitivity levels demonstrate that surface roughness is unlikely to be the limiting factor on measurement precision when measuring nanoparticle size using this phase-shifting-interferometry-based technique. The need to use an underlying surface that is very smooth when measuring nanoparticles is highlighted by the analysis.

STUDY OF THE MAJOR CONDITIONS INFLUENCING STEP HEIGHT MEASUREMENT USING INTERFERENCE MICROSCOPE

Step height measurements in micro and nano scale are essential to the micro-technology. The interferometric measurement is carried out using two main techniques: white light scanning interferometry (WLI) and phase stepping interferometry (PSI). A step height standard sample of nominal step height value h of 90.1 nm has been studied using the PSI mode with an optical filter light of λ = 633.4 nm against three parameters: vertical distance (z) between objective and sample, intensity of the incident light and the inclination angles of the sample to the incident light. The effect of numerical aperture of objectives has been referenced to some literature. The total budget of studied parameters for the sample 90.1 nm leads to expanded uncertainty of 0.46 nm.

IMPROVING THE ACCURACY OF INTERFERENCE MICROSCOPE MEASUREMENTS BY APPLYING GENETIC ALGORITHM

The interference microscope measurements with white light scanning interferometry (WLSI) and phase stepping interferometry (PSI) have been improved by optimizing the experimental conditions such as vertical distance between the objective and the surface under test (z), the tilt angles of the incident light beam with the surface and incident light intensity (I). Finally an accurate surface parameters measurement is obtainable. A genetic algorithm code has been developed to determine the best values of experimental conditions instead of manual adjustment by the operator. The combined uncertainty has been successfully reduced from 3.01 nm to 0.84 nm for the WLSI roughness measurement. While it has been reduced from 0.18 nm to 0.08 nm for the PSI step height measurement.

Phase-stepping interferometry of GaAs nanowires: Determining nano-wire radius

Phase stepping interferometry is used to measure the size of near-cylindrical nanowires. Nanowires with nominal radii of 25 nm and 50 nm were used to test this by comparing specific measured optical phase profile values with theoretical values calculated using a wave-optic model of the Phase stepping interferometry (PSI) system. Agreement within 10% was found, which enabled nanowire radii to be predicted within 4% of the nominal value. This demonstration highlights the potential capability for phase stepping interferometry to characterize single nanoparticles of known geometry in the optical far-field.

Multispectral phase shifting interferometry algorithm

Standard phase shifting interferometry (PSI) algorithms are used to obtain phase map from fringe patterns. Their spectral properties can by analyzed using frequency transform function (FTF). In this paper new multispectral phase shifting interferometry algorithm (mPSI) has been proposed. It is combination of two simpler PSI algorithms and enables filtering out unwanted high frequency fringes from the image. It can be used to increase measurement range of standard interferometry by analyzing fringe pattern created by multiplication of two other fringe patterns recorded with the use two different wavelength. Numerical simulations of mPSI algorithm and results from real setup are shown in the paper. Full Text: PDF References J. Schmit, K. Creath,"Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry", Appl. Opt. 34 No. 19 3610 (1995) [CrossRef] J. Schmit, K. Creath,"Window function influence on phase error in phase-shifting algorithms", Appl. Opt. 35 No. 28 5642 (1996) [CrossRef] B. S. Lee,T. C. Strand,"Profilometry with a coherence scanning microscope", Appl. Opt. 29 No. 26 3784 (1990) [CrossRef] P. Hariharan, M. Roy,"White-light Phase-stepping Interferometry for Surface Profiling", J. Mod. Opt. 41 No 11 2197 (1994) [CrossRef] H. van Brug H, R. G., "On the effective wavelength in two-wavelength interferometry", Pure Applied. Opt. 7 1465 (1998) J. C. Wyant Optical Testing: Extended Range Two-Wavelength Interferometry (Optical Sciences Center University of Arizona) J. Schmit, P. Hariharan,"Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm", Opt. Eng. 45, 115602 (2006) [CrossRef] Y. Cheng, J. C. Wyant,"Two-wavelength phase shifting interferometry", Appl. Opt. 23, No 24 4539 (1984) [CrossRef] P. de Groot,"Measurement of Transparent Plates with Wavelength-Tuned Phase-Shifting Interferometry", Appl. Opt. 39, No 16 2658 (2000) [CrossRef] M. Servin, J. C. Estrada, J. A. Quiroga, "The general theory of phase shifting algorithms", Opt. Exp. 17 No 24, 21867 (2009) M. Wengierow, L. Sałbut, Z. Ramotowski,"Interferometric multiwavelength system for long gauge blocks measurements", Proc. SPIE 8082, 80822R (2011) [CrossRef]

Three-dimensional phase step profilometry with a multicore optical fiber

This paper demonstrates the feasibility of using phase stepping and a multicore optical fiber to calculate an object's depth profile. An interference pattern is projected by an optical fiber onto the object. The distorted interference pattern containing the object information is captured by a CCD camera and processed using a phase step interferometry method. The phase step method is less computationally intensive compared to two-dimensional Fourier transform profilometry and provides more accuracy when measuring objects of high frequency spatial variations.

Quantitative single-shot imaging of complex objects using phase retrieval with a designed periphery

Measuring transmission and optical thickness of an object with a single intensity recording is desired in many fields of imaging research. One possibility to achieve this is to employ phase retrieval algorithms. We propose a method to significantly improve the performance of such algorithms in optical imaging. The method relies on introducing a specially designed phase object into the specimen plane during the image recording, which serves as a constraint in the subsequent phase retrieval algorithm. This leads to faster algorithm convergence and improved final accuracy. Quantitative imaging can be performed by a single recording of the resulting diffraction pattern in the camera plane, without using lenses or other optical elements. The method allows effective suppression of the "twin-image", an artefact that appears when holograms are read out. Results from numerical simulations and experiments confirm a high accuracy which can be comparable to that of phase-stepping interferometry.

Fast localized wavefront correction using area-mapped phase-shift interferometry

We propose an innovative method for localized wavefront correction based on area-mapped phase-shift (AMPS) interferometry. In this Letter, we present the theory and then experimentally compare it with a previously demonstrated method based on spot-optimized phase-stepping (SOPS) interferometry. We found that AMPS outperforms SOPS interferometry in terms of speed by threefold, although in noisy environments the improvements may be larger. AMPS yielded similar point-spread functions (PSF) as SOPS for moderate system-induced aberrations, but yielded a slightly less ideal PSF for larger aberrations. The method described in this Letter may prove crucial for applications where the phase-stepping solution does not have sufficient speed.

Internal profile measurement of GaAs microstructures based on near-infrared light phase-stepping interferometry

Internal profile nondestructive measurement of MEMS microstructures based on semiconductor materials is urgently demanded with the dramatic development of MEMS industry. A new attempt was made to extend the interferometric method and the thickness-profile measurement theory of transparent thin film layers from the visible light region (400–800 nm) to near-infrared light region (900–1700 nm). The measurement system based on Linnik interferometer using a near-infrared halogen light source with 1127 nm center wavelength was established for internal profile measurement. The measurement method for internal profile of GaAs microstructures was realized. From a comparison of reflection interferometry and transmission interferometry, it can be concluded that the latter measurement results were consistent with those of the former and internal profile was determined within an error range of 8%.

A self-tuning phase-shifting algorithm for interferometry

In Phase Stepping Interferometry (PSI) an interferogram sequence having a known, and constant phase shift between the interferograms is required. Here we take the case where this constant phase shift is unknown and the only assumption is that the interferograms do have a temporal carrier. To recover the modulating phase from the interferograms, we propose a self-tuning phase-shifting algorithm. Our algorithm estimates the temporal frequency first, and then this knowledge is used to estimate the interesting modulating phase. There are several well known iterative schemes published before, but our approach has the unique advantage of being very fast. Our new temporal carrier, and phase estimator is capable of obtaining a very good approximation of their temporal carrier in a single iteration. Numerical experiments are given to show the performance of this simple yet powerful self-tuning phase shifting algorithm.

Phase-shifts nπ/2 calibration method for phase-stepping interferometry

A novel phase-shifts n pi/2 calibration method for phase-stepping interferometry, in which the sum intensities and the squared sum of a series of phase-shifted interferograms are firstly calculated, and then by the minimization of the variances of these sums perform the n pi/2 phase-shifts calibration, is proposed in this paper. The proposed method can overcome effectively the effect of the variations of background and modulation intensities in interferograms with any phase structure, and it is also insensitive to the nonlinearity of phase shifter. Numerical simulation and experiments are implemented to verify the effectiveness of this method.

Merging Phase Shifting Interferometry with Confocal Chromatic Microscopy

The implementation of the basic physical principle of Chromatic Confocal Microscopy in the field of Phase stepping interferometry (PSI) opens new opportunities for the development of an innovative surface metrology method specially dedicated to 3D nanotopography with subnanometric z axis resolution altogether with a very large measuring range: typically up to one hundred micrometers. The basic property of optical sectioning inherent to (chromatic) Confocal imaging is particularly well adapted to Phase stepping Interferometry since it automatically solves the critical and time consuming problem of phase unwrapping computation. The axial chromatic extension of the chromatic confocal setup offers a very fast and easy way to determine the height of the different elementary surfaces forming the measured object. It is then easy to carry out, for each one of those elementary surfaces, a measurement in phase shifting interferometry, at the wavelength corresponding to the altitude indicated by the confocal chromatic, in order to reach subnanometric axial resolutions. The four phases needed for implementing the phase stepping interferometric measuring procedure can be successively realized by adequate spectral shifts instead of the classical axial displacements of the reference mirror which then stands in a fixed position. Consequently this chromatic confocal phase stepping interferometer (CCPSI) has definitely no moving part, the spectral shifts being done by electrooptical means. Typical applications are MEMS and microoptics surface topography and/or roughness metrology. For this purpose we designed a new system incorporating confocal chromatic imaging and phase stepping interferometry. As a direct consequence of the optical sectioning property, this system allows measuring through any type of optical window (for example a cover glass).

Phase-stepped gauge block interferometry using a frequency-tunable visible laser diode

Frequency changes induced by bias or temperature modulation of injection diode lasers can provide an economical and effective method of applying phase-stepping interferometry to optical metrology. However, the intrinsic frequency instability of these devices limits their use in gauge block interferometry where precise and repeatable phase steps must be maintained simultaneously on two discontinuous surfaces and over relatively long path lengths. We demonstrate a method using a visible injection diode laser, the frequency of which is locked by using a Fabry-Perot interferometer. Small changes to the length of the Fabry-Perot interferometer shift the frequency of the laser producing proportional and repeatable phase steps to the gauge block interferogram. This method has been successfully implemented with a Fizeau-type gauge block interferometer with a phase measurement resolution of 0.005 lambda. The phase data are then processed to map the surface form of gauge blocks up to 100 mm in length and to objectively assess surface shape parameters.

Three-dimensional profilometry by symmetrical fringes projection technique

A precise interferometric method for three-dimensional surface measurements in a wide dynamic range is presented. The method is based on symmetrical two-spacing projection phase-stepping interferometry. Two approaches are proposed—one with double symmetrical illumination and the other with double symmetrical observation—which reduce influence of shadowing at investigation of complex surfaces and allow absolute coordinates estimation. The theoretical background, experimental check as well as accuracy and sensitivity assessment of the method are discussed. The proposed technique is especially useful for remote, non-destructive in-situ measurements of real objects—for example historical objects and monuments.