Fizeau Interferometer Research Articles

Novel analysis of alignment error on spherical Fizeau interferometer and uncertainty evaluation of sphericity calibration system based on random ball test

The National Metrology Institute of Japan (NMIJ) has developed a high-precision sphericity calibration system utilizing a Fizeau interferometer and conducted a thorough evaluation of the system's measurement uncertainty. Generally, there are two principal sources of uncertainty in the measurement outcomes when utilizing a spherical Fizeau interferometer. First, the accuracy of the system is significantly influenced by the prior knowledge of the reference sphere surface's absolute profile. To address this, the study established a straightforward yet effective calibration system for the reference sphere surface using a random ball test method. Second, the system's precision is affected by misalignment aberration, which occurs when there is any lateral or longitudinal displacement of the test sphere from the confocal position relative to the reference sphere. This misalignment can introduce both high-order shape errors (misalignment aberrations) and low-order shape errors (alignment errors). Through analytical consideration of an observation coordinate system on the camera plane, this study delves into misalignment aberrations, suggesting that the impact of misalignment should be determined experimentally for each reference sphere unit due to potential imperfections that may be revealed by misalignment. Furthermore, the study proposes that uncertainties related to misalignment aberrations are theoretically confirmed to be smaller than previous studies by conducting an in-depth uncertainty analysis of the misalignment, with a focus on the observation coordinate system on the camera plane. Notably, the research demonstrates that a measurement uncertainty level of λ/100 is achievable, maintaining a broader tolerance for misalignment than previously reported studies. The uncertainty of calibration of the reference sphere unit's absolute profile was 4.2 nm and the uncertainty of sample measurement was determined to be 4.6 nm with misalignment tolerance of ±λ/10. This advancement marks a stride toward improving the accuracy and reliability of sphericity measurements, offering potential for widespread application in precision engineering and metrology.

An optimization of the Fourier self-calibration algorithm for improved noise suppression capability

Abstract Self-calibration emerges as a promising approach in ultra-precision field, overcoming the limitations imposed by instrument capabilities in conventional calibration. In the self-calibration research, the algorithm based on the Fourier transform firstly established a rigorous theoretical framework. However, despite advancements in alternative self-calibration methods that have improved noise suppression, the traditional Fourier algorithm still faces challenges, particularly in mitigating random measurement noise in the separation results. In this study, we introduce methods on enhancing the noise suppression capabilities of the Fourier self-calibration algorithm. We investigate the hypothesis in the traditional algorithm and find that the computation of weighted averages had minimal effects. Then, we present our method by augmenting measurement positions. Considering both the increase in the number of measurement positions and the enhancement of noise suppression capabilities, augmenting translation positions in the opposite direction is proved to be the most effective strategy. Our algorithm can effectively suppress measurement noise when one-sided point number of the artifact is below 21. While the additional data processing slightly increases runtime, it does not alter the algorithm’s computational complexity, yet it significantly improves noise suppression. The effectiveness of our method is validated through a Monte Carlo comparison of uncertainties with the original algorithm. Experiments on Fizeau interferometer further prove robustness and practical feasibility of our proposed algorithm, with the measurement repeatability being reduced by 26.34% and 25.15% in the errors separated from the stage and the artifact.

Investigation of error sources in absolute flatness measurement using Fizeau interferometer

Investigation of error sources in absolute flatness measurement using Fizeau interferometer

High precision wavefront correction method in interferometer testing.

A wavefront correction method is proposed for high-precision optic surfacing, addressing the discrepancy between wavefront and real surface errors in Fizeau interferometer testing. We believe this to be a proposed novel method that encompasses optical surface function parameters fitting, lateral distortion correction, misalignment error removal, and sag surface error calculation. The method's error has been thoroughly analyzed, including aspects of function parameters fitting, ray tracing, and interpolation. The effectiveness of the method was demonstrated by correcting the wavefront of an off-axis parabolic mirror in null testing configurations, significantly reducing artificially created annular errors and improving off-axis direction errors from 0.23λ to 0.05λ (λ=632.8 nm), with the PV of aspheric departures exceeding 8.5 mm.

Laboratory Demonstration of Grating Embedded Mirrors for Single-Shot Heliostat Optical Metrology

Abstract Concentrated solar power (CSP) plants need to monitor the surface slope error of thousands of heliostats with sub-milliradian accuracy. Large numbers of heliostats installed in harsh environments mean that measurement speed and durability are key design considerations for metrology systems. We present a compact, accurate, and high-speed heliostat slope error metrology system that is robust to the harsh environmental conditions at CSP plants. The system is composed of (1) grating embedded mirrors (GEMs), which have multiple different phase diffraction gratings written within a glass mirror substrate, and (2) a compact optical system, the diffractive auto-stigmatic Hartmann camera (DASHCam). Using focused ultrafast laser pulses, we write phase gratings within second-surface float glass mirror without damaging the reflective coating on the backside. The gratings direct a small fraction of incident light to non-specular directions, which the DASHCam senses from a virtual center of curvature to measure the mirror slopes at each grating patch. In a CSP plant, each heliostat mirror would be a GEM, and the DASHCam would rapidly measure slope and canting errors during heliostat manufacturing, installation, or operation. We fabricated 0.1 meter-diameter GEMs with 24 grating patches, built a prototype DASHCam system, and tested the system repeatability and accuracy by comparing against a Fizeau interferometer. Our results show, in a laboratory environment, 24μrad root mean square (RMS) measurement repeatability and 47μrad RMS accuracy, with single-shot image capture. The combination of GEMs and DASHCam is a promising metrology approach that could lead to improved optical accuracy of heliostats throughout their life cycle. This work serves as a proof of principle for this system.

Linear systems characterization of the topographical spatial resolution of optical instruments

Lateral resolving power is a key performance attribute of Fizeau interferometers, confocal microscopes, interference microscopes, and other instruments measuring surface form and texture. Within a well-defined scope of applicability, limited by surface slope, texture, and continuity, a linear response model provides a starting point for characterizing spatial resolution under ideal conditions. Presently, the instrument transfer function (ITF) is a standardized way to quantify linear response to surface height variations as a function of spatial frequency. In this paper, we build on the ITF idea and introduce terms, mathematical definitions, and appropriate physical units for applying a linear systems model to surface topography measurement. These new terms include topographical equivalents of the point-, line-, and edge-spread functions, as well as a complex-valued transfer function that extends the ITF concept to systems with spatial-frequency-dependent topography distortions. As an example, we consider the experimental determination of lateral resolving power of a coherence scanning interference microscope using a step-height surface feature to measure the ITF directly. The experiment illustrates the proposed mathematical definitions and provides a direct comparison to theoretical calculations performed using a scalar diffraction model.

Measuring the radius of curvature of spherical surfaces with actively tunable Fizeau and Twyman-Green interferometers

Accurate and repeatable measurement of the radius of curvature (RoC) of spherical sample surfaces is of great importance in optics. This importance lies in the ubiquitous use of spherical optical elements such as curved mirrors and lenses. Due to a high measurement sensitivity, interferometric techniques are often deployed for accurate characterization of the sample surface RoC. One method by which a typical commercial Fizeau or Twyman-Green (TG) interferometer measures surface RoC is via scanning between two principal retroreflective optical configurations—namely, the confocal and catseye configurations. Switching between these two configurations is typically achieved by moving an optical head along the axis of the propagating laser beam and the RoC is estimated by measuring the magnitude of mechanical motion to switch between the two principal configurations. In this paper, we propose a motion-free catseye/confocal-imaging-based sample RoC measurement system. The necessity of bulk motion to switch between the two configurations is circumvented via the use of an actively controlled varifocal lens. We demonstrate the usefulness of the proposed innovation in RoC measurements with either the TG or the Fizeau interferometer. Furthermore, we convert a commercial motion-based Zygo RoC measurement system into a motion-free one by introducing a tunable lens inside the apparatus and using it to accurately characterize the RoC of different test samples. We also compute the wavefront aberrations for all spherical sample surfaces from the recorded measurement data.

A dual-stage correction approach for high-precision phase-shifter in Fizeau interferometers

This paper proposes a dual-stage correction approach for a high-accuracy phase-shifter in Fizeau laser interferometers, effectively reducing the phase shift errors in PSI (Phase-Shifting interferometry) and improving the precision of surface profile measurements. The precision of PSI depends on the displacement accuracy of the reference mirror, and the function of the phase shifter is to drive the reference mirror to generate wavelength-level displacements with nanometer-level accuracy. The first-stage correction aims to reduce the nonlinearity error of the reference mirror by ULPS (Ultra-High Linearity Phase Shifter). The second-stage correction aims to reduce the average velocity error and plane velocity uniformity error of the reference mirror by AVIC (Auto Velocity Iterative Correction). The experimental results indicate that, after the first-stage correction, ULPS exhibits a 0.05 % nonlinearity, a 2.5 nm repeatability, and a 0.2 nm resolution under a load of 20 kg, and its nonlinearity surpassing all phase-shifters documented in the literature. After the second-stage correction, the average velocity error and plane velocity uniformity error of the reference mirror are reduced to 0.07 %.

Stitching interferometry for long X-ray mirrors with lateral multi-shift-based absolute calibration.

To eliminate the reference mirror (REF) error of the Fizeau interferometer for measuring X-ray mirrors, the reference calibration method of lateral multi-shift measurements at the hundreds-micrometer pixel level is presented. Because of the high aspect ratio of long X-ray mirrors, by shifting the surface under test (SUT) along the tangential direction with integer multiple pixels, we extend the calibration method by using the difference between multiple shifted measurements to build an augmented multi-matrix for extracting the two-dimensional (2D) absolute surface. The method can be applied to arbitrary measurement regions of the test optics, and the measurement for both the benchmark sub-aperture and calibration of the REF is accomplished in a single measuring process. Furthermore, by adjusting the shift to the millimeter scale, reference-subtracted sub-apertures can be stitched to obtain the absolute 2D map of the X-ray mirror. Experimental results show that all the 2D discrepancies reach sub-nanometer repeatability, and comparison results between the long trace profiler (LTP) and the proposed method have been performed. Therefore, the results demonstrated that the proposed method meets the requirements of X-ray mirror measurements.

Absolute tests of three flats for interferometer with 800 mm aperture.

In this study, absolute tests of three flats were performed by using an 800 mm Fizeau interferometer. The flats were installed by employing double-wedge rubber, and the deformation of the flats, primarily involving power and spherical aberrations, was analyzed by conducting finite element analysis. The measurement results revealed an astigmatism component that did not follow the rotation. The astigmatism was separated from the measurement result, and the remaining aberration was consistent with the simulation results, confirming that the astigmatism deformation did not originate from the rubber installation. Subsequently, an absolute test of an 800 mm flat was completed, and the results were compared with those of the traditional three-flat absolute test and Zygo interferometer. The directions of the vertical diameter lines were consistent. The peak-to-valley difference in the full-surface shape was less than 4 nm, and the root mean square was less than 1 nm. The surface errors of flats A and C were consistent with the result of replacing the reference flat, B, with the fourth flat, D, to perform an absolute test. The difference between flats A and C was similar to Zygo's results, thus eliminating the influence of the transmission flat. These results verified the accuracy of the results.

Simultaneous interferometric profilometry of surface and thickness of transparent plate using wavelength-modulation

Wavelength-modulation interferometers have been widely used to measure the geometric characteristics of transparent plates. However, several errors can be generated by multiple reflections of the laser beams and environmental uncertainties when the wavelength is modulated. In this research, two novel phase-calculation formulas with the same number of interference fringes (15) are proposed to measure the geometric characteristics of a transparent plate. First, a 15s-sample phase-calculation formula for measuring the surface shape of a transparent wedge plate was developed using a system of linear equations. Second, a 15t-sample phase-calculation formula for measuring the optical thickness irregularity of transparent parallel plate was developed by using the characteristic-polynomial theory. The two 15-sample formulas were visualized in the frequency domain and complex plane to confirm the properties of their error-compensation ability. Furthermore, the two formulas were compared with conventional phase-calculation formulas through the numerical analysis. Finally, the surface profile of the wedge plate and optical thickness irregularity of the parallel plate were measured via the two proposed formulas using wavelength-modulation Fizeau interferometer. The standard deviations of 15s-sample formula and the 15t-sample formula were confirmed as 4.3616 nm and 1.2875 nm, respectively. Using these formulas to measure the geometric properties of transparent plate on nanoscale can contribute to the production of high-performance semiconductor devices.

Thin diameter optical displacement sensor for grasping force sensing of surgical robot

AbstractThe purpose of this study is to realize a grasp force measurement of forceps for robotic surgery systems. Generally, surgical robot forceps is actuated by traction wire, and we have suggested a newly force sensing system by measuring the wire expansion utilizing micro displacement sensors. For measuring the elongation of the wire, the white‐light interference principle has been utilized on sensor system, and the sensor consists of a tip‐tilted optical fiber, a Fizeau interferometer and a sensor holder. In this report, displacement sensor has been successfully fabricated and mounted onto the wire to measure the elongation of the wire under tension and discussed the experimental results.

Assessment and verification of thickness homogeneity in mask blank by utilizing main harmonics of triple-surface interferometry

To assess the thickness homogeneity of a mask blank using wavelength-modulating interferometry, novel methods for determining and verifying the thickness homogeneity using triple-surface interferometry are proposed. First, based on the optical path difference in triple-surface interferometry, the phases of main harmonics were defined, and the determination method of the back thickness variation was derived. Next, a verification method for thickness assessment was derived using the theoretical formula of the triple-surface interferometry. And then, the thickness assessment of the mask blank was conducted by wavelength-modulating Fizeau interferometer. The thickness homogeneity of the mask blank was determined by the phases of main harmonics extracted using harmonic interferometric-iterative analysis. Finally, the applicability of the verification method was evaluated by comparing the theoretical errors and standard deviations of the experimental results.

Scanning interferometric phase-calculation formula for simultaneous topographic profiling of thickness and surface of optical flats

With the rapid development of industrial technologies, both the thickness irregularity and surface shape of transparent optical flats should be profiled precisely to maintain the performance of industrial equipment. When measuring the optical flat thickness using an interferometer, the profiling uncertainty can deteriorate owing to a constant phase error. In this study, a new 3N phase-extraction formula that can compensate for the Zernike piston error was developed for the simultaneous measurement of thickness and surface. Using Fourier representation, the characteristics of the 3N formula were shown in the frequency domain. Subsequently, the combined and Zernike piston error suppression ability of the 3N formula was verified by comparison with the other formulas. Finally, the thickness and surface of the optical flats were measured using the 3N formula and a wavelength-scanning Fizeau interferometer. The standard deviations of the thickness and surface profiling measurements were 4.435 nm and 9.654 nm, respectively, which were much smaller than those calculated using other phase-calculation formulas.

Modeling multiply scattered light in full-field optical coherence tomographic imaging of biological tissues

In this work, we have constructed a handheld full-field optical coherence tomography (FFOCT) system with high spatial resolution, and the Michelson interferometer (served as compensating optical path) and Fizeau interferometer (served as detection) were mounted in series in this system. The expressions of the multiple scattered light decomposed as the sum of incoherent and coherent contribution were derived based on the diffusion equation and mutual coherence function, and the variations of resolution with imaging depth were derived by using the model of multiple scattering. Experiments were conducted on samples composed of the USAF 1951 resolution target covered by the onion slice and human finger-skin slice, respectively, to validate and evaluate the accuracy of our model. A method of image restoration is then presented in which the relation between the resolution and measuring depth and the simulated PSF in the absence of multiple scattering are employed. With the help of this method the lateral resolution of 2.2 μm and 3.9 μm were separately achieved in the onion slice (measuring depth of 64 μm) and human finger-skin slice (measuring depth of 60 μm), and more tiny structures in the en face image of human finger-skin and moth wings were visible. In addition, a novel method for estimating the values of mean free path and anisotropy of scattering medium was proposed.

Solutions Mixing Visualization in Continuous-Flow Microreactors via Interferometric Technique

The paper demonstrates the potential of interferometry as a visualization technique for studying the mixing processes of liquid reagents in continuous-flow microreactors. We visualized two types of instabilities: double-diffusion convection and Marangoni soluto-capillary convection. We employed two optical schemes, depending on how large values of refractive index inhomogeneities resulting from instabilities needed to be visualized, including the shear interferometer and the Fizeau interferometer. The interferometry enabled a qualitative investigation of the structure and dynamics of the generated convection. In addition, by implementing the phase shift method by the IntelliWave program, we quantified the efficiency of the mixing process between pumped liquids. These quantitative findings complemented our qualitative visualization, providing further evidence of the effectiveness of interferometry in studying mixing processes. Our results confirm that the interferometry technique is an effective tool for the visualization and analysis of convective flows in continuous-flow microreactors. Moreover, the insights gained from this research contribute to the broader understanding and optimization of mixing processes in microreactor systems.

Measuring the indication error of the Fizeau interferometer and evaluating its uncertainty

The indication error, as a characteristic of the interferometer, can be obtained through verification and calibration, and the uncertainty during measurement represents the reliability of the indication error. This paper employed the modified absolute measurement method to gain the indication error. First, the improved absolute algorithm was derived. Then, the absolute measurements were conducted with a 150-mm-aperture Fizeau interferometer. The measured indication error conforms to the maximum allowable error of the interferometer. Finally, the measurement uncertainty evaluation was carried out based on the method to characterize the indication error’s dependability, which was much less than one-third of the maximum allowable error.

Non-iterative phase tilt interferometry for the tilt shift calculation and phase extraction.

This study proposed a non-iterative phase tilt interferometry (NIPTI) for the tilt shift calculation and phase extraction, which can solve the random tilt-shift problem caused by external vibration. The method approximates the higher order terms of the phase to make it suitable for linear fitting. Then according to an estimated tilt, the accurate tilt shift can be obtained by least squares method without iteration, and the phase distribution can be calculated by it. The simulation results indicated that the root mean square error of the phase calculated by the NIPTI can reach up to 0.0002λ. The experimental results showed that the calculated phase exhibited no significant ripple when using the NIPTI for cavity measurements in a time-domain phase shift Fizeau interferometer. Moreover, the root mean square repeatability of the calculated phase reached up to 0.0006λ. It indicates that the NIPTI provides an efficient and high-precision solution for random tilt-shift interferometry under vibration.

X-ray mirrors with sub-nanometre figure errors obtained by differential deposition of thin WSi2 films.

Differential deposition by DC magnetron sputtering was applied to correct for figure errors of X-ray mirrors to be deployed on low-emittance synchrotron beamlines. During the deposition process, the mirrors were moved in front of a beam-defining aperture and the required velocity profile was calculated using a deconvolution algorithm. The surface figure was characterized using conventional off-line visible-light metrology instrumentation (long trace profiler and Fizeau interferometer) before and after the deposition. WSi2 was revealed to be a promising candidate material since it conserves the initial substrate surface roughness and limits the film stress to acceptable levels. On a 300 mm-long flat Si mirror the average height errors were reduced by a factor of 20 down to 0.2 nm root mean square. This result shows the suitability of WSi2 for differential deposition. Potential promising applications include the upgrade of affordable, average-quality substrates to the standards of modern synchrotron beamlines.

Compact lensless Fizeau holographic interferometry for imaging domain patterns in ferroelectric single crystals.

Domain patterns in ferroelectric single crystals are physical systems that are fascinating from a theoretical point of view and essential for many applications. A compact lensless method for imaging domain patterns in ferroelectric single crystals based on a digital holographic Fizeau interferometer has been developed. This approach provides a large field-of-view image while maintaining a high spatial resolution. Furthermore, the double-pass approach increases the sensitivity of the measurement. The performance of the lensless digital holographic Fizeau interferometer is demonstrated by imaging the domain pattern in periodically poled lithium niobate. To display the domain patterns in the crystal, we have used an electro-optic phenomenon, which, when an external uniform electric field is applied to the sample, produces a difference in refractive index values in domains with different polarization states of the crystal lattice. Finally, the constructed digital holographic Fizeau interferometer is used to measure the difference in the index of refraction in the antiparallel ferroelectric domains in the external electric field. The lateral resolution of the developed method for ferroelectric domain imaging is discussed.