Fringe Interpolation Research Articles

Robust wavenumber and dispersion calibration for Fourier-domain optical coherence tomography.

Many Fourier-domain optical coherence tomography (FD-OCT) systems sample the interference fringes with a non-uniform wavenumber (k) interval, introducing a chirp to the signal that depends on the path length difference underlying each fringe. A dispersion imbalance between sample and reference arms also generates a chirp in the fringe signal which, in contrast, is independent of depth. Fringe interpolation to obtain a signal linear in k and compensate dispersion imbalance is critical to achieving bandwidth-limited axial resolution. In this work, we propose an optimization-based algorithm to perform robust and automated calibration of FD-OCT systems, recovering both the interpolation function and the dispersion imbalance. Our technique relies on the fact that the unique function that correctly linearizes the fringe data in k space produces a depth-independent chirp. The calibration procedure requires experimental data corresponding to a single reflector at various depth locations, which can easily be obtained by acquiring data while moving a sample mirror in depth. We have tested both spectral domain OCT and swept source OCT systems with various nonlinearities. Results indicate that the proposed calibration method has excellent performance on a wide range of data sets and enables nearly constant resolution at all imaging depths. An implementation of the algorithm is available online.

Un télémètre transportable de résolution submicrométrique

This paper describes a transportable system based on synthetic wavelength interferometry for absolute distance measurement. The synthetic wavelength is generated by means of two frequency-doubled Nd:YAG lasers. A superheterodyne technique has been implemented to detect the synthetic phase, enabling a fringe interpolation of ∼2π/5 600. A ∼5 μm accuracy (1σ) is demonstrated over 25 m, based on an indoor comparison with a classical fringe counting interferometer. An outdoor comparison has been carried out on a 864 m-long standard baseline located in Finland (Nummela), yielding an uncertainty of 700 μm (1σ). This work was carried out within the framework of the “Long distance” EMRP project that ended on June 2011.

Spindle Error Motion Measurement Using Concentric Circle Grating and Phase Modulation Interferometers

In the conventional methods of measuring radial, axial, and angular motions of a spindle concurrently, complicated reference artifacts with relative large volume, e.g., two balls linked to a cylinder, are required. A simple and small artifact is favorable from the viewpoint of accurate measurement. This paper describes a concurrent measurement of radial, axial, and angular spindle motions using concentric circle grating and phase modulation interferometers. In the measurement, concentric circle grating with fine pitch is installed on top of the spindle of interest. The grating is a reference artifact in the method. Three optical sensors are fixed over the concentric circle grating, to observe its proper positions. The optical sensor consists of a frequency modulated laser diode as a light source as well as two interferometers. One interferometer observes an interference fringe between reflected light from a fixedmirror and 0-th order diffraction light from the grating to measure the axial motion. Another interferometer observes an interference fringe between ± 2nd order diffraction lights from the grating tomeasure the radialmotion. Using three optical sensors, three axial displacements, and three radial displacements of the proper observed position of the grating can be measured. From these measured displacements, radial, axial, and angular motions of the spindle can be calculated concurrently. In this paper, a measurement instrument, a novel fringe interpolation technique using sinusoidal phase modulation, and experimental results are discussed.

Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers

We describe a transportable distance measurement system based on synthetic wavelength interferometry. Two frequency-doubled Nd:yttrium aluminum garnet lasers at 532 nm are used to generate a synthetic wavelength of approximately 2.5 cm. A nonpolarizing interferometric system has been set up to eliminate polarization cross-talk issue. A superheterodyne detection was performed to measure the synthetic phase and to determine absolute distances. The capability to achieve fringe interpolation of 2pi/5600 has been demonstrated and an agreement in distance measurement at the 4 microm level has been achieved, compared to an optical interferometric 3 m long displacement bench.

Common path two-wavelength homodyne counting interferometer development

The linearity of displacement measurement is an important parameter in nanometrology. The development of the common path two-wavelength (633 nm and 532 nm) homodyne counting interferometer within the Nanotrace project is described. These two differential interferometers are also designed for the elimination of optical path drifts and fluctuations. Their short-term instability is projected to be in the order of tens of pm in the single wavelength path difference and below 10 pm in the difference of these two wavelength path differences. This solution enables better real-time validation of interferometer fringe interpolation. The whole system could be transportable and measure displacements up to 100 µm under ambient air conditions under cover. Refractive index compensation will also be included.

Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer

Currently, the standard length measurement method with nanometre resolution is laser interferometry. However, it is difficult to determine an arbitrary length with an accuracy of sub-nanometre or less order using interferometers because they have a nonlinearity problem in fringe interpolation. A phase modulation homodyne interferometer (PMHI) that can be used to determine the optical path difference of an integer multiple of the wavelength (n × λ) with picometre resolution was proposed by IMGC (former Italian Standard Institute). The lattice spacing of approximately 0.246 nm for graphite regular crystalline lattices is uniform and stable over a long range, when the crystals are stress free. These crystals can be used as a reference scale with sub-nanometre resolution. The scanning tunnelling microscope (STM) is emerging as a powerful tool in surface engineering, and enables us to image atoms on a crystalline surface. Therefore, such a crystalline surface can be used as ‘the crystalline scale’ using the STM.In this study, an instrument for calibrating optical encoders is developed by combining a graphite crystalline lattice as a fine scale and the optical fringe of the PMHI as a coarse scale. The instrument consists of a precise linear X-axis sample stage, on which the reference graphite crystal and the optical encoder scale are set, a head of the STM with a YZ tip scanner and a PMHI. The relative displacement of the X-axis sample stage between optical interference dark fringes (= null points) of the PMHI, which is λ/16 times the integer value in the design, can be measured with picometre resolution using the phase modulation technique. A lattice spacing of 0.246 nm on the graphite crystalline surface is derived as the fine scale from the STM image and the optical fringes of the PMHI. In the experiment, the periodical error of the optical encoder, whose minimum resolution is less than a nanometre, is measured using both the lattice spacing of graphite and the optical fringes of the PMHI. The results show that the proposed instrument has the feasibility to calibrate optical encoders with an uncertainty of 10 pm order.

Improving the accuracy of homodyne Michelson interferometers using polarisation state measurement techniques

We demonstrate a simple homodyne Michelson interferometer design capable of fringe interpolation accuracy of one part in 36,000 leading to a potential accuracy of 10 pm. This improvement becomes possible thanks to the significant advances in polarimetry that permit measurement of the ellipsometric parameters ψ and Δ with an accuracy of 0.01° with readily available commercial equipment. The proposed method has been set up and we make a direct comparison with a commercial heterodyne interferometer by measuring the displacement of the same target mirror. This latter has been displaced by steps of 20 nm using a high accuracy position control method.

Michelson interferometry with 10 pm accuracy

We demonstrate a new heterodyne Michelson interferometer design for displacement measurements capable of fringe interpolation accuracy of one part in 36 000. Key to this level of accuracy are the use of two acousto-optic modulators for heterodyne frequency generation and digital signal processing demodulation electronics. We make a direct comparison of our interferometer to a commercial interferometer based on a Zeeman-stabilized laser, and show that the residual periodic errors in ours are two orders of magnitude lower than those in the commercial unit. We discuss electronically induced optical cross talk and optical feedback as sources of periodic error. Our new interferometer is simple, robust, and readily implemented.

OddRydberg spectrum of 40Ar(I): high-resolution laser spectroscopy andMQDT analyses of the nd, J = 4 levels and the ng´, J = 4resonances

Using transverse resonant two-photon, single-mode laser excitation of metastable 40Ar(4s 3P2) atoms in a collimated beam in combination with a calibrated travelling Michelson interferometer involving digital fringe interpolation, we have measured the energies for the Ar(4p,J = 3 nd,J = 4) transitions with principal quantum numbers n = 12-100 with a relative uncertainty of 8 × 10-8. The Rydberg atoms are formed in a region of low electric and magnetic fields and detected by electron transfer to SF6 molecules in a skimmed supersonic beam. A single-channel quantum defect analysis of the experimental data is performed, revealing weak perturbations of the n = 12 and 20 levels by interaction with the Ar(2P1/2 7g´ J = 4) and Ar(2P1/2 8g´J = 4) levels, respectively. The value for the ionization energy of the 40Ar+(2P3/2) limit has been determined as 21 647.076(2) cm-1 relative to the 40Ar(4p,J = 3) level and as 127 109.836(3)(±0.05) cm-1 relative to the ground state of 40Ar (see also note added in proof). Low-lying autoionizing ng´,J = 4 resonances (n = 9-13) have been investigated and their quantum defects and reduced width determined for the first time. A multichannel quantum defect theory analysis yields an estimate (0.76 cm-1) for the energy separation between the perturbed 20d level and the perturbing 8g´ level, predicts the transition intensity to the 8g´ level to be much weaker than that to the 20d level and shows that the g´-g coupling in the autoionization region is much stronger than the g´-d coupling determined from the 8g´-induced perturbation in the discrete nd, J = 4 spectrum.

High precision two-dimensional spatial fringe analysis method

Fringe analysis is used in engineering fields as a precise measurement technology. In this paper, a method for improving the accuracy of spatial fringe analysis is discussed. A new method, based on fringe scanning and interpolation techniques, is put forward. Two-dimensional fringe analysis based on this new method is also proposed. Measured results obtained using the new two-dimensional method and those obtained using the FFT method are compared. Results showed that while the accuracy of the two methods is equivalent, the calculation time for the new method is five times as speedy as that for the FFT method.

Isochromatic fringe sharpening and interpolation along an isoclinic countour, with application to fracture mechanics

A method is presented for photoelastic analyses whereby fractional fringe orders are determined with enhanced accuracy. It utilizes data from the Tardy compensation method and combines the data by a robust algorithm to produce highly sharpened isochromatic fringe contours in the immediate vicinity of an isoclinic line. Their sharpness provides accurate assessment of fractional fringe orders at a point, relative to that achieved with the broad fringes of the standard Tardy method. Errors introduced by small isoclinic deviations are quantified and found to be inconsequential. The method is well suited to the application of linear elastic fracture mechanics in fracture studies of threedimensional photoelasticity models of complex structures. It is demonstrated for the determination of stress intensity factor at a crack in the solid fuel element of a rocket motor, when the solid fuel is subjected to internal pressure. Other applications are practical, too.

A passive homodyne fibre vibrometer using a three-phase detector array

A non-contact, optical fibre-based vibrometer is described with output in the form of a set of spatial fringes. The fringe system is sampled at three different phases using a non-multiplexed linear detector array. No source frequency modulation is necessary, allowing convenient use of a wide variety of lasers, and surface motion is measured free from directional ambiguity. Fringe interpolation and multiple-fringe unwrapping are effected using a digital fringe processor. With a received power of 4 nW, a minimum noise-equivalent phase sensitivity of 150 mu Rad Hz-1/2 is demonstrated.

Strain analysis by mismatch moire method and grid method using Fourier transform

We have formerly presented a new method of the moire analysis of strain using the Fourier transform. It uses the phase information of the moire fringe brightness. By shifting the Fourier spectrum of the image of deformed grating lines, we obtain the “complex moire pattern”. Strain distribution is given as the derivatives of the phases of the complex moire fringes. The analysis is completely automated by digital image processing. All of the laborious and subjective procedures required in the conventional analysis such as fringe sign determination, fringe ordering and fringe interpolation are thus eliminated, and objective, fast and accurate analysis can be made.

Two-dimensional moiré method and grid method using Fourier transform

Formerly, the authors presented one-dimensional strain analysis by a moire method using a Fourier transform. In the present work, the method is extended to two-dimensional strain analysis. The analysis is completely automated by introducing digital image processing. All of the laborious and subjective procedures required in the classical and conventional moire method, such as separation of the two gratings, fringe-sign determination, fringe ordering and fringe interpolation, are completely eliminated; and objective, fast and accurate analysis can be made.

Application Of Moire Analysis Of Strain Using Fourier Transform

By shifting the discrete Fourier spectrum of the image of a deformed grating, we obtain the moire pattern, from which strain distribution is given as the derivatives of the phases of the complex moire fringes. The analysis is completely automated by digital image processing. All of the laborious and subjective procedures required in the conventional analysis such as fringe sign determination, fringe ordering, and fringe interpolation are thus eliminated, permitting objective, fast, and accurate analysis. Some applications for rubber plates are shown.

Moire analysis of strain by Fourier transform.

By shifting discrete Fourier spectrum of image of a deformed grating, we can obtain the complex moire Strain distribution is given as derivative of argument of complex moire pattern. The analysis is completely automatized by introducing digital image processing. All of laborious and subjective procedures reguired in conventional analysis such as fringe sign determination, fringe ordering, and fringe interpolation are completely eliminated and objective, fast and accurate analysis can be made.

Strain measurement by heterodyne holographic interferometry

Heterodyne holographic interferometry provides automated interference phase measurement with high resolution and allows the quantitative evaluation of 3-D displacement and strain on solid objects. The fundamentals of heterodyning in double-exposure holography are reviewed and its possibilities and limitations are discussed in detail. Experimental results of strain measurement on a curved object surface are reported. Three double-exposure hologram recordings with different illuminations are used to get the vector displacement. Based on locally calculated derivatives of the displacement field, a sensitivity for the strain components of 1 microm/m with a spatial resolution of a few millimeters has been achieved. The features of heterodyne holographic interferometry are compared with those of quasiheterodyne (phase stepping) fringe interpolation.

Phase modulation moire topography

Moire topography is a method of tridimensional measurement. It is easy to use but it is often not accurate enough. To resolve this problem, a method of tridimensional measurement by moire phase modulation has been suggested. In this particular method, the lines projected on the object are modulated interference fringes, and the observation grating corresponds to a set of curves; it must be calculated and fabricated. A study about the interpretation of the moire fringes produced through this grating is developed. The measurement of the phase of the modulation moire in different points allows fringe interpolation and therefore improves accuracy. An experiment has been carried out in order to test the method: the profile of a plate has been established with 20 mu m accuracy. This method can be applied to tridimensional measurement of diffusing objects.

Holographic Contouring Using Electronic Phase Measurement

The application of heterodyne and quasi-heterodyne (stepwise phase shifting) fringe interpolation techniques to contouring for 3-D object shape measurement is discussed. A setup with two spherical illumination sources for contour fringe generation is described. The contour fringe pattern can be observed in real time or from a holographic reconstruction. The use of a microcomputer for automated data acquisi-tion and processing allows one to determine the object shape for a large number of points and for arbitrarily shaped contour fringe surfaces. Experimental results are reported for both heterodyne and quasi-heterodyne phase measurement, offering a typical accuracy for object depth measurement of 0.2% of the lateral extension of the object.

HETERODYNE HOLOGRAPHY FOR VISUALIZATION OF SURFACE ACOUSTIC WAVES

Abstract A technique for recording and visualizing acoustic wavefronts over a finite field of view and with a resolution on the order of Angstroms has been developed. Multiple exposure holograms of an aluminum test block were used to record the surface displacements resulting from the detonation of a small explosive charge placed on the object surface. The primary wavefronts resulting from the explosive charge were sufficiently large as to be visible upon reconstruction of the holographic interferograms. In addition, details of the surface wavefronts which could not be deduced using conventional holographic interferometry were measured and displayed by modifying the pulsed recording geometry and subsequently applying heterodyne interferometric analysis to the holograms to interpolate between the observed interferometric fringes. Although theory predicts that still greater resolution is possible, fringe interpolation to 1/ 900 of a fringe ( about 3A displacement) was demonstrated.