Fizeau Interferometer Research Articles

Image recognition of interference fringes in polishing by convolutional neural network with data augmentation by deep convolutional generative adversarial network

With the growing need for high specification requirements for the latest manufacturing processes and optical designs of glass lenses, the technical requirements for lens polishing have increased. The manufacturing parameters must be adjusted in a timely manner to meet the required specifications. We use a Fizeau interferometer to classify and analyze interference fringes measured in the actual polishing process of glass lenses. Given the low incidence of interference fringes in practice, the training dataset contained a disproportionate ratio of data for each data type. To reduce the manufacturing cost and data collection time, this study focused on three common types of interference fringes in the manufacturing processes and integrated a deep convolutional generative adversarial network with convolutional neural networks (CNNs) for fringe type classification. The deep convolutional generative adversarial network was used to establish a data augmentation generator, and Jensen–Shannon divergence was employed to identify the epoch number that yielded distributions of interference fringe numbers closest to the real distributions; this approach could achieve the diversity of interference fringes in the generated images. Finally, the generated data were used to train the CNN models, and the accuracy of image recognition reached above 86%.

Spectral performance analysis of the Aeolus Fabry–Pérot and Fizeau interferometers during the first years of operation

Abstract. In August 2018, the European Space Agency (ESA) launched the first Doppler wind lidar into space, which has since then been providing continuous profiles of the horizontal line-of-sight wind component at a global scale. Aeolus data have been successfully assimilated into several numerical weather prediction (NWP) models and demonstrated a positive impact on the quality of the weather forecasts. To provide valuable input data for NWP models, a detailed characterization of the Aeolus instrumental performance as well as the realization and minimization of systematic error sources is crucial. In this paper, Aeolus interferometer spectral drifts and their potential as systematic error sources for the aerosol and wind products are investigated by means of instrument spectral registration (ISR) measurements that are performed on a weekly basis. During these measurements, the laser frequency is scanned over a range of 11 GHz in steps of 25 MHz and thus spectrally resolves the transmission curves of the Fizeau interferometer and the Fabry–Pérot interferometers (FPIs) used in Aeolus. Mathematical model functions are derived to analyze the measured transmission curves by means of non-linear fit procedures. The obtained fit parameters are used to draw conclusions about the Aeolus instrumental alignment and potentially ongoing drifts. The introduced instrumental functions and analysis tools may also be applied for upcoming missions using similar spectrometers as for instance EarthCARE (ESA), which is based on the Aeolus FPI design.

Dynamic Fizeau interferometer with assistant defocused source.

We report and demonstrate a dynamic Fizeau interferometer (DFI) capable of high-precision measurements under severe vibration. The method involves two interferometric passes: a null-interference pass (NP) and a circular carrier-frequency pass (CP). The NP adopts an on-axis Fizeau design, preserving the common path of the reference and test beams. The CP uses an assistant defocused source to generate circular fringe patterns, assisting the NP in phase extraction and sign correction. The feasibility and performance of the proposed method are demonstrated experimentally. To the best of our knowledge, this is the first proposal of a general asynchronous phase-shifting DFI that can not only demodulate null-fringe patterns and correct phase-sign ambiguity but can also maintain the common-path advantages of the Fizeau interferometer.

Broadband Variable Transmission Sphere for Fizeau Interferometry

Transmission spheres used in interferometry are specified by f-number and source wavelength. In this paper, we explore a broadband variable transmission sphere (BVTS) system based on freeform Alvarez lenses that enables variable operation across a broad range of f-numbers and wavelengths. Potential applications and performance tradeoffs are discussed in comparison to conventional spherical transmission spheres. Simulation results are presented for f/15 to f/80 configurations from visible to long-wave infrared sources in a Fizeau interferometer. Simulation results highlight that spherical, coma, and astigmatism impose limits on surface measurement quality.

Parameter mismatch phase extraction method for spatial phase-shifting interferograms

Mismatch of interferogram parameters, such as the contrast and the fringe frequency inconsistency, can lead to phase error in simultaneous phase-shifting interferometry. In the dynamic Fizeau interferometer (DFI) based on the lateral displacements of light sources, an extended light source is used to suppress coherent noise. Aiming at the contrast mismatch generated by the extended light source and the inherent fringe frequency mismatch of the DFI, We propose a phase-extraction method to solve these problems. Based on this dynamic interferogram model, a linear relationship is constructed between the tilt angle and interferogram contrast, carrier frequency error, phase-shift error. Use the two steps of initial estimation and fine iterations to extract the phase. The RMS of the residuals was better than 1.5 × 10–5λ for the simulation and 0.004 λ for the experimental calculation, which demonstrates that the ripple error was suppressed during phase extraction.

Research of laser ultrasonic defect statistics recognition technology based on radial basis function neural network

In this study, we detected the depth of surface defect at a statistic on rate of 95.8% or more by using the Radial Basis Function Neural Network, a Q-switched and pulsed Nd:YAG laser and optical fiber Fizeau interferometer. In the experiment, Nd:YAG pulsed laser was used for generating ultrasound in a sample, while an optical fiber Fizeau interferometer was used to detect the ultrasound generated by a laser line source. Our results showed that this system has a high sensitivity as well as a high spatial resolution, what is more important is that this system is applicable to practice field.

Surface measurement of silicon wafer using harmonic phase-iterative analysis and wavelength-scanning Fizeau interferometer

Wavelength-scanning interferometry has been widely used for surface measurements of silicon wafers with a high reflection factor owing to its non-contact and high-resolution measurement. However, the measurement results are degraded by the second-order harmonics of the signal and nonlinearity of phase modulation in wavelength scanning. In this study, a harmonic phase-iterative technique using five-frame experimental interferograms was developed for the surface measurement of a silicon wafer, while simultaneously compensating for the second-order harmonics and nonlinear phase modulation. Considering the reflection factor of the silicon wafer and the harmonics of the signal, a harmonic phase-iterative technique was derived. Furthermore, by combining the proposed iterative technique with the harmonic convergence condition and the selected pixel technique, the harmonic phase-iterative technique was insensitive to divergence errors and considerably reduced the computational time of phase extraction. The surface of the silicon wafer was measured using the harmonic phase-iterative technique and five-frame interferograms obtained using a wavelength-scanning Fizeau interferometer. The repeatability error of the silicon wafer surface measurement was 2.617 nm. Finally, it was confirmed that the harmonic phase-iterative technique had superior phase-extraction ability in terms of residual ripples and repeatability errors compared to the conventional iterative technique and phase-modulation technique.

Simulation analysis of a lidar performance for atmospheric temperature profile measurements based on the Fizeau interferometer and multichannel photomultiplier tube

AbstractIn this study, a lidar system based on the Fizeau interferometer and photomultiplier tube array is designed to analyze the entire Rayleigh‐Brillouin spectrum aiming at obtaining the atmospheric temperature profiles. The laser wavelength emitted by the lidar is 355 nm with the pulse energy of 20 mJ, and the diameter of the receiving telescope is 0.6 m. We analyze the influence of the cumulative average of different pulse numbers on the accuracy of temperature inversion. The temperature inversion results in different height ranges are calculated and the results show that if the integration time is 100 s, the temperature deviation is less than 3 K in the altitude range below 24 km.

Simultaneous Thickness and Surface Profiling of Blank Mask and its Experimental Verification Using Robust Harmonic Fringe-Iterative Algorithm and Fizeau Interferometer

Simultaneous Thickness and Surface Profiling of Blank Mask and its Experimental Verification Using Robust Harmonic Fringe-Iterative Algorithm and Fizeau Interferometer

Simultaneous Thickness and Surface Profiling of Blank Mask Using Robust Harmonic Phase-Iterative Algorithm and Fizeau Interferometer

Simultaneous Thickness and Surface Profiling of Blank Mask Using Robust Harmonic Phase-Iterative Algorithm and Fizeau Interferometer

Tilted Wave Fizeau Interferometer for flexible and robust asphere and freeform testing

Tilted Wave Interferometry (TWI) is a measurement technique for fast and flexible interferometric testing of aspheres and freeform surfaces. The first version of the tilted wave principle was implemented in a Twyman-Green type setup with separate reference arm, which is intrinsically susceptible to environmentally induced phase disturbances. In this contribution we present the TWI in a new robust common-path (Fizeau) configuration. The implementation of the Tilted Wave Fizeau Interferometer requires a new approach in illumination, calibration and evaluation. Measurements of two aspheres and a freeform surface show the flexibility and also the increased stability in both phase raw data and surface measurements, which leads to a reduced repeatability up to a factor of three. The novel configuration significantly relaxes the tolerances of the imaging optics used in the interferometer. We demonstrate this using simulations on calibration measurements, where we see an improvement of one order of magnitude compared to the classical Twyman-Green TWI approach and the capability to compensate higher order error contributions on the used optics.

Separate the Primary Aberration for the Deformation Error of Optical Flat in 300-Mm-Aperture Vertical Fizeau Interferometer

Separate the Primary Aberration for the Deformation Error of Optical Flat in 300-Mm-Aperture Vertical Fizeau Interferometer

System error calibration for Φ300 mm vertical Fizeau interferometer based on liquid reference

System error calibration for Φ300 mm vertical Fizeau interferometer based on liquid reference

A Convenient Way to Determine the Optimum Angle of Incidence of Fizeau Interferometer

In a Fizeau interferometer, off-axis illumination will lead to fringe optimization. Primarily due to the unique structure of our interferometer, we first analyze the influence of the optical properties of the parallel plate as a part of the interferometer on the optimal incident angle. Generally, the incident angle determination is mainly based on the graphing method proposed by Langenbeck and the estimation formula proposed by Kajava. However, Langenbeck’s method is cumbersome, and the error of Kajava’s estimation formula is large. Based on the predecessors, this paper proposes a modified method of determining the optimal angle of incidence and further derives more accurate optimal angle expressions than Kajava’s. By simply substituting the wedge angle of the wedge cavity and the reflectivity of the cavity, the optimum incidence angle can be obtained immediately. Thus, it eliminates the tedious and complex process of finding the optimum incident angle by graphing method and makes the formula method the simplest method to find the optimum incident angle. Finally, the comparison of the interference intensity at the optimum incidence angle calculated by the improved method and normal incidence is given. It is found that the beam has a good suppression effect on the sub-peak when it is incident at the optimum incident angle calculated by the method in this paper.

A Self-Calibration Stitching Method for Pitch Deviation Evaluation of a Long-Range Linear Scale by Using a Fizeau Interferometer.

An interferometric self-calibration method for the evaluation of the pitch deviation of scale grating has been extended to evaluate the pitch deviation of the long-range type linear scale by utilizing the stitching interferometry technique. Following the previous work, in which the interferometric self-calibration method was proposed to assess the pitch deviation of the scale grating by combing the first-order diffracted beams from the grating, a stitching calibration method is proposed to enlarge the measurement range. Theoretical analysis is performed to realize the X-directional pitch deviation calibration of the long-range linear scale while reducing the second-order accumulation effect by canceling the influence of the reference flat error in the sub-apertures’ measurements. In this paper, the stitching interferometry theory is briefly reviewed, and theoretical equations of the X-directional pitch deviation stitching are derived for evaluation of the pitch deviation of the long-range linear scale. Followed by the simulation verification, some experiments with a linear scale of 105 mm length from a commercial interferential scanning-type optical encoder are conducted to verify the feasibility of the self-calibration stitching method for the calibration of the X-directional pitch deviation of the linear scale over its whole area.

Wavelength-tuning interferometry with suppression of dispersive error

In measurements of the variations in the thickness and surface profile of optical flats using wavelength-tuning fringe analysis, an interference signal can be detected based on the modulation frequency of the signal. However, the modulation frequency can deviate from the predetermined value through the refractive-index dispersion of the optical flat. In this study, two types of noble 25-frame phase-detection algorithms that can compensate for the effect of refractive-index dispersion were developed using Fourier analysis. The behaviors of the 25-frame algorithms were visualized on both the frequency domain and complex plane. The dispersion-resistance ability and signal-to-noise ration of the 25-frame algorithm were estimated by calculating the PV errors and using the frequency-transfer-function paradigm, respectively. Finally, the variations in the thickness and surface profile of the optical flat were measured using a wavelength-tuning Fizeau interferometer and the proposed 25-frame algorithms. The standard deviations of the thickness and surface measurements were 1.654 nm and 8.368 nm, respectively, which were much smaller than those of the conventional phase-detection algorithms.

General iterative algorithm for phase-extraction from fringe patterns with random phase-shifts, intensity harmonics and non-uniform phase-shift distribution.

Advanced iterative algorithm (AIA) is a flexible and effective phase-shifting algorithm (PSA) which can extract phase from fringe patterns with random unknown phase-shifts, making it attractive in the scenarios where phase-shifts are unknown or not accurate. However, accuracy of AIA degrades when intensity harmonics and/or phase-shift non-uniformity are presented. To solve this problem, multiple PSAs have been proposed, but they restrict their fringe model in one way or another, and thus sacrifice the immunity to certain error source(s). In this paper, a general iterative algorithm (GIA) which adopts a most general fringe model is proposed. In GIA, the many unknowns in the fringe pattern model are divided into three groups including: (i) the fringe amplitudes, (ii) the phase and (iii) the phase-shifts related parameters, and alternatively optimized through univariate search technique group by group to improve accuracy and convergence. The Levenberg-Marquart method is used for the optimization of each group of unknowns due to its excellent accuracy and robustness. GIA is shown to have better accuracies than all of its relevant competitors through both a large number of simulations as well as real experiments with a Fizeau interferometer.

Wavelength-tuning multiple-surface interferometric analysis with compression of Zernike piston phase error

In this study, we propose a 6N – 3 algorithm for suppressing the Zernike piston phase error by expanding 6N – 5 algorithm using algorithmic polynomial theory. We demonstrate that the proposed algorithm can be used for simultaneous measurement of the surface profile and thickness homogeneity by rotating the characteristic diagram on the complex plane. We applied Fourier analysis to confirm the properties of 6N – 3 algorithm in the frequency domain. We then estimate the error-compensating capability of 6N – 3 algorithm, after which we compare this capability to that of classical algorithms using numerical error analysis. To verify the effectiveness, we conducted an experiment using 6N – 3 algorithm and Fizeau interferometer to measure the thickness homogeneity and surface profile of a glass plate. The standard deviation of the measured thickness homogeneity was 1.965 nm, which was significantly smaller than that of the other classical algorithms.

Effect of retrace error on stitching coherent scanning interferometry measurements of freeform optics.

We report on the effect of retrace error during measurement of freeform optics using a commercial coherence scanning interferometer (CSI), and its in-built stitching capabilities. It is shown that measuring segments of freeform optics under non-null conditions, results in artifacts on the measured zone, similar to the Seidel aberrations. An experimental approach is used to quantify the induced aberrations based on the local slopes of the surface. Simulation of surfaces containing different order aberrations is shown to have a significant effect on the measurement data. A correction method is proposed that uses experimental measurements to determine the required correction based on local slope and position in the aperture. These corrections reduce the measurement difference from a comparison measurement using a Fizeau interferometer.

NOA61-Polymer Fiber Fizeau Interferometer with a Flexible NOA65-Polymer Taper for Simultaneous Measurement of Tilt Angle and Temperature.

This study proposes and experimentally demonstrates a NOA61-polymer fiber Fizeau interferometer (PFFI) connected to a flexible NOA65-polymer taper (PT) for simultaneous measurement of tilt angle and temperature (T). The PT/PFFI fiber sensor consists of a taper-shaped flexible NOA65 polymer and single-mode fiber with an endface that is attached to a NOA61-polymer. The NOA61-polymer of PFFI is highly sensitive to variations of T with high repeatability and enables the simultaneous measurement of tilt angle by connecting with the highly flexible NOA65-PT. the interference fringe visibility of optical spectra in the PFFI can be highly controlled by the tilt angle of the PT and is thus capable of measuring tilt angles with high sensitivity. On the other hand, wavelength shifts of the spectra in the PFFI only occur when T varies. The proposed PT/PFFI can simultaneously detect the tilt state and the variation of surrounding T by measuring the optical spectral responses and eliminating cross sensitivity. Experimental results demonstrate the PT/PFFI can simultaneously measure tilt angles and T with good sensitivities and obtain averages of 0.4 dB/° and 0.17 nm/°C, respectively.