Interference fringes are registered by detectors. All detectors absorb energy from a single or multiple superposed fields through the process of "square modulus" of the sum of the complex amplitudes. The detected energy becomes proportional to the total relative phase difference for all the superposed fields. The process creates ambiguity in discerning the effects due to frequency and phase modulations. We underscore that fringe detection being a physical interaction process between superposed fields and detecting molecules (including beam splitter boundary), the dipolar properties of atoms and molecules should be used to help us discern the effects due to frequency and phase modulations. We traditionally accept that orthogonally polarized light beams do not "interfere". Or, light beams of different frequencies are "incoherent" to each other; but we have highly developed heterodyne interferometry for which the wave fronts of the superposed beams must be matched. Yet, we do not explicitly recognize the roles of the molecules of detectors and beam splitters that really carry out the real functions. Besides, understanding the various processes behind their dipolar response can help us innovate more precision interferometric techniques. As for examples: (i) How precisely the polarization should be parallel to produce perfect visibility fringes? (ii) How precisely equal the optical frequencies of superposed beams should be to create perfectly steady-state energy redirection by a beam splitter in an interferometer with collimated and collinear beams. (iii) How small the wave front mis-match can be tolerated to produce perfect heterodyne fringes while superposing beams of different frequencies?

NASA's Space Interferometer Mission (SIM) PlanetQuest requires, among other things, very precise retroreflectors. The CSIRO Australian Centre for Precision Optics (ACPO) has developed Double Corner Cubes (DCCs) to meet the requirements. The DCC consists of an assembly of three 30^o wedged prisms optically contacted to a 132 mm diameter flat base plate. The material for all components was Zerodur. The specifications for the DCC were extremely challenging and posed considerable difficulties in the fabrication, coating, assembly, alignment and metrology. Some of the key specifications included: flatness of all reflecting surfaces to be ~ 10 nm peak to valley (P-V); dihedral angle errors < 0.5 arc seconds; collocation of the vertices of the two corner cubes within a circle of 5 um radius; all reflecting surfaces to be gold coated for a final microroughness < 0.5 nm rms; the clear or working aperture extended to within 0.2 mm of all physical edges; and the assembly had to withstand large vibrational forces. CSIRO delivered to JPL a DCC that was used as the primary unit in the so-called Kite testbed to satisfactorily meet the demonstration requirements of the SIM Milestone 8. This paper will discuss some of the procedures used to realize the DCCs and will show examples of results achieved.

The Space Interferometry Mission (SIM) requires the control of the optical path of each interferometer with picometer accuracy. Laser metrology gauges are used to measure the path lengths to the fiducial corner cubes at the siderostats. Due to the geometry of SIM a single corner cube does not have sufficient acceptance angle to work with all the gauges. Therefore SIM employs a double corner cube. Current fabrication methods are in fact not capable of producing such a double corner cube with vertices having sufficient commonality. The plan for SIM is to measure the non-commonalty of the vertices and correct for the error in orbit. SIM requires that the non-common vertex error (NCVE) of the double corner cube to be less than 6 μm. The required accuracy for the knowledge of the NCVE is less than 1 μm. This paper explains a method of measuring non-common vertices of a brassboard double corner cube with sub-micron accuracy. The results of such a measurement will be presented.

Interferometer wavefront bias is a complicated function of the optics imperfections in the instrument. An interferometric measurement of the figure error on a micro-refractive lens requires careful calibration to separate instrument bias from errors on the part. A self-calibration method such as the random ball test is effective in accomplishing this task without the need for a high-quality calibration artifact. The test, an averaging technique applied to a series of sphere surface patches, allows for calibration of the interferometric wavefront bias. Recent studies of the random ball test have shown that the calibration is affected by ray-trace errors that depend on the curvature of the ball used for the test and becomes significant in the micro-optic range (radius less than 1 millimeter). A comprehensive ray-trace simulation of the random ball test with modifiable variables was created using MATLAB^® and ZEMAX^® to allow for further investigation into the relationship between test lens misalignment, curvature, numerical aperture, ball figure error, and interferometer bias. The basis for the model hinges on defining a sphere in terms of a set of spherical harmonic functions, and varying the amplitudes and the number of functions to adjust the figure error on the sphere. The flexible simulation can be fine-tuned and used to model a variety of interferometers with different specifications. Our ultimate goal is to confirm the validity of the RBT, determine an efficient method of implementation, and understand the aspects impacting calibration uncertainty.

Self-calibration techniques in interferometry imply the absolute determination of the interferometer's systematic error without the need for a high-quality, precisely-known calibration standard. By taking separate measurements at different positions and under different orientations, the interferometer's error can be determined in an absolute manner even with non-perfect optical elements. This paper discusses the versatile possibilities of using diffractive optical elements as calibration tools in wavefront testing interferometry.

A new traceable method has been developed and investigated to experimentally determine the total amount of measuring deviations arising through the capture and demodulation of plane-mirror interferometer signals. The basic principle for such an analysis is the precise specification of length variations. However, either a measuring system of excellent accuracy or accurately defined movements within a stable platform are required. A common measuring motion can be achieved through the displacement of a reflecting wedge plate, which creates a constant step-down. The interferometer to be analyzed is used to determine the change in the wedge plate's thickness, which is caused by lateral movements controlled by another interferometer. The wedge's sampled surfaces demand high planarity as the change of thickness acts as the material measure. These conditions can be achieved by using the Nanopositioning and Nanomeasuring Machine in conjunction with a 0.5-degree tilted mirror placed on it. The interferometer to be analyzed is aligned with this mirror. To provide the highest possible linearity for lateral motion, the only measuring points are in nearly error-free lambda/2 steps of the interferometer. The NPM machine's already small deviations in positioning will only affect the evaluation of measuring errors of the reduced interferometer by a factor of about one-hundredth. This is one of the main advantages of the method. The interferometer to be analyzed - like the entire measuring setup - features a compact assembly and high mechanical and thermal stability. The measured deviations in linearity provide excellent verification of the prospected error influences.

The paper deals with the calibration error of unequal phase changes across the interferogram in phase shifting interferometry, i.e., tilt-shift error. For its detection the lattice-site representation of phase shift angles is used. The error can be readily discerned using (N+1) algorithms. Four and five frame algorithms are considered. The influence of experimental parameters on the error detection sensitivity is discussed. Numerical studies are complemented by experimental results.

The temporal phase shifting method (TPS) is the most powerful technique in phase-shifting shearography. Numerous different phase-shifting algorithms have been developed to determine phase distribution for different applications. Most of them are applied with known phase shifts. In these cases, the accuracy of quantitative measurement using phase-shifting technique depends strongly on the accuracy of the phase shifter e.g. a piezoelectric transducer (PZT) which is responsible for generating the known phase shifts. Thus, phase shifter has to be calibrated. A new calibration method based on the so-called advanced iterative algorithm which provides stable convergence and accurate phase distribution with minimum three randomly shifted interferograms will be proposed in this paper. A fully automatic calibration procedure for calibrating of desired phase shifts and for evaluating the characteristic curve of phase shift over voltage change will also be presented.

The National Institute of Standards and Technology as the U.S. National Metrology Institute has the fundamental responsibility to continuously push the limits of measurement science (metrology) to promote U.S. innovation and industrial competitiveness. In 2004, NIST finished construction of a $235 million, 49 843 m² Advanced Measurement Laboratory to enhance its measurement capabilities in response to the fast-growing metrology needs of the scientific and industrial community.

This work is devoted to two-beam interferogram intensity modulation decoding using spatial carrier phase shifting interferometry. Single frame recording, simplicity of experimental equipment and uncomplicated data processing are the main advantages of the method. A comprehensive analysis of the influence of systematic errors (spatial carrier miscalibration, non-uniform average intensity profile, nonlinear recording) on the modulation distribution determination using automatic fringe pattern analysis techniques is presented. The results of searching for optimum calculation algorithm are described. Extensive numerical simulations are compared with laboratory findings obtained when testing vibrating silicon microelements under various experimental conditions.

Sequential methods like the regularized phase tracker (RPT) are commonly used for fringe pattern demodulation with closed fringes. The only drawback of the RPT method is the necessity to implement a two-dimensional (2D) fringe following scanning in order to obtain the expected modulated phase. In this article we present a new method to demodulate single fringe patterns with closed fringes which use a simple 2D row by row scanning strategy. This is an important contribution because the 2D row by row scanning is extremely fast and easy to implement unlike the fringe following scanning. We have called this method the phase curvature tracker (PCT) because it uses the frequency curvature as regularizer to obtain the expected phase as a C² function with continuous curvature.

In optical measurement techniques, such as holographic interferometry, shearographic interferometry, and 3D-computer vision, phase is used as a carrier for different physical properties. For example, phase could stand for the information of shape, deformation, or strain. The useful information is coded and wrapped into -π to +π range in phase map. Phase unwrapping is the basic technique to resolve 2π ambiguities and recover physical properties. In this paper, we describe a novel method for unwrapping phase in a noise-disturbed phase map, which is characterized with broken fringes. In this method, phase map is well analyzed before phase unwrapping. Broken fringes are detected and fixed. Phase map is segmented into different regions based on phase quality. Those regions with good phase quality will be unwrapped first and errors introduced by noise will be minimized. Several applications will be presented to testify our new method.

An algorithm for phase determination from a single interferogram with closed fringes using Fourier transforms is proposed. Traditionally, the Fourier transform method has been used for interferograms containing only open fringes. This is done performing a pass-band filtering of one half of the frequency spectrum and taking the arctangent function of the real and imaginary parts of the filtered spectrum. The proposed method performs multiple pass-band filtering resulting in several wrapped phases with wrong sign changes corresponding to the orientation of the filters. Among the different wrapped phases, regions are expected to have the same sign enabling the correct reconstruction of the phase. The corrections of the sing changes are made by comparing the derivatives of the wrapped phases by means of a modified process of phase unwrapping taking into account small areas of the interferogram field. This is done under the assumption of that adjacent areas won't have abrupt changes in their phase derivatives as corresponding to smooth continuous surfaces. The procedure is robust against noise since low-pass filtering can be easily performed in the calculation of the wrapped maps and during the modified process of phase unwrapping.

Based on multi-lateral shearing interferometry, a powerful technique, called the Quadri-Wave Lateral Shearing Interferometer (QWLSI) is used to evaluate the wavefront in an accurate and precise way. Our device can be used for the characterization of complex and very aberrant optical devices, the control of optical components and also for laser beam evaluation. This communication will detail the response of the QWLSI and its metrological performances, such as its high resolution, its adjustable sensitivity and dynamic. It will then be focused on two innovative applications of the QWLSI. The first application concerns the evaluation of infrared lenses dedicated to high-performance cameras. We will present experimental results recently completed by our prototype dedicated to the LWIR domain (λ=8-14μm). In a second part, we will study the possibility to analyze wave-fronts with discontinuities. Such wave-fronts can be produced by segmented mirrors, diffractive components or also bundle of single-mode fibers. We will finally present simulation results for this latter application.

An on-axis, vibration insensitive, polarization Fizeau interferometer is realized through the use of a novel pixelated mask spatial carrier phase shifting technique in conjunction with a low coherence source and a polarization delay-line. In this arrangement, coherence is used to effectively separate out the orthogonally polarized test and reference beam components for interference. With both the test and the reference beams on-axis, the common path cancellation advantages of the Fizeau interferometer are maintained. The interferometer has the unique ability to isolate and measure any surface that is substantially normal to the optical axis of the cavity. Additionally, stray light interference is substantially reduced due to the source's short coherence. An expression for the fringe visibility on-axis is derived and compared with that of a standard Fizeau. Using a 15 mW source, the maximum camera shutter speed, used when measuring a 4% reflector, was 150 usec, resulting in very robust vibration insensitivity. We experimentally demonstrate the measurement of both sides of a thin glass plate without the need to modify the plate between measurements. Experimental results show the performance of this new interferometer to be within the specifications of commercial phase shifting interferometers.

It is well known that coherent illumination causes dust diffraction patterns which limit the ultimate accuracy of phase measurements. By the use of an extended monochromatic light source the phase can considerably be smoothened allowing a better accuracy for phase measurements. Before the invention of the laser interferometry had to use extended monochromatic sources like spectrum lamps. For plane mirror interferometers illuminated with an extended monochromatic incoherent source the localization plane and the contrast distribution in space have been described in extenso in the literature (see e.g. G. Schulz). In the 40-th of last century G. Hansen derived the condition for high fringe contrast in the Twyman-Green interferometer by rectifying the Twyman-Green geometry of sphericity tests to a plane mirror interferometer. Experiments will be presented using an extended source in form of a laser spot on a rotating scatterer demonstrating the theoretical background. These experiments demonstrate the necessity to image the surface to be tested as well as the reference surface sharply onto the detector plane in order to maintain high contrast. Due to the extended source the fringes are smooth and represent a cos-type intensity distribution.

We present a method for a cascading null test using twin computer-generated holograms to calibrate errors in null correctors. This will allow us to test large aspheres an order of magnitude better than current limits. We discuss various sources of CGH errors and how to calibrate them. We also mention some ways to measure and calibrate the errors in the test optics.

We describe a thorough processing procedure of experimental data from annular subaperture measurents, in which the crucial reconstruction algorithm is based on Zernike annular polynomials and matrix method. The result of simulation is satisfactory, and the proposed method is easier to implement than that of those overlapping annular subaperture interferometry. In addition, experimental results of testing parabolic mirror with about 700-mm diameter and 0.228 obscuration are also provided, and some limitations and considerations are discussed.

The main part of the Fluid Science Laboratory (FSL) that is under development for the Columbus Orbital Facility is the Optical Diagnosis Module (ODM). It provides quantitative measurements of fluid science experiments with the help of different interferometric techniques. The assembly of these methods is based on a Mach-Zehnder-Interferometer. Because of the employment on the FSL the design of this interferometer has several requirements regarding volume, mass, modularity, operational needs and especially an environment with thermal and mechanical distortions of the interferometer. At BIAS an interferometer was designed that detects the misalignment due to external distortions and compensates it. The design is based on a simplified Hartmann-Sensor and detects wavefront tilt and curvature errors. The active compensation of these errors is realized by piezoelectric driven optical components and operates in realtime but cannot compensate the curvature errors completely. In this paper we describe the construction of a modified setup that compensates wavefront tilt and curvature errors in a temperature range from 15°C to 35°C sufficiently for the planned applications. The new design is substantial simpler and hence the passive stability is better as well. The results of thermal tests with the new setup are demonstrated and show the enhanced stability of the new actively stabilized interferometer.

We present the application of wavefront sensing to 3-dimensional particle metrology for measuring the 3-component velocity vector field in a fluid flow across a volume. The technique is based upon measuring the wavefront scattered by a tracer particle from which the 3-dimensional tracer location can be calculated. Using a temporally resolved sequence of 3-dimensional particle locations the velocity vector field is obtained. In this paper we focus on an anamophic technique to capture the data required to measure the wavefront. Data is presented from a reconstruction of the phase of the wavefront as well as from a more pragmatic approach that examines only the defocus of that wavefront. The methods are optically efficient and robust and can be applied to both coherent and incoherent light in contrast to classical interferometric methods. A focus of this paper has been the filtering techniques in order to reliably extract the particle images from the overall image field. The resolution and repeatability of the depth (or range) measurements have been quantified experimentally using a single mode fiber source representing a tracer particle. A first proof of principle experiment using this technique for 3-dimensional PIV on a sparsely seeded gas phase flow is also presented.

Using our original method of interferometric data processing we have studied a radial density distribution in the flow separation range of a sphere flown in air at Mach number 2. This range contains a shock wave, a contact surface and a part of adjacent wake. The method is based on measurement of interferometric fringes and definition of light vector refraction component distributions normal to the symmetry axis. It is assumed that the interferometric picture has vertical fringes outside of the bow shock wave. The radial distributions are calculated according to the schlieren data processing. They were used for density field simulation and interferometric pictures visualization by the digital interferometry, some aspects of which were developed by authors early. The comparison of experimental interferometric pictures with the calculated pictures shows a possibility to use the considered method for study of the flow separation range. Results of the comparison are illustrated by examples of such interferometric pictures.

High-resolution, real-time 3-D absolute coordinate measurement is highly important in many fields. This paper presents such a system that measures 3-D absolute geometric shapes and positions at 30 frames per second (fps), with an image resolution of 532 × 500. The system is based on a digital fringe projection and fast three-step phase-shifting method. It utilizes a digital-light-processing (DLP) projector to project color encoded computer generated phase-shifted fringe patterns in grayscale, a high-speed CCD camera synchronized with the projector to capture fringe images at 90 fps. Based on the three-step phase-shifting algorithm, any successive three fringe images can be used to reconstruct one 3-D shape. Therefore, the 3-D shape measurement speed is 30 fps. To obtain absolute coordinates, absolute phase is required. In this research, a tiny marker is encoded in the projected fringe pattern, and detected by software from captured fringe images accurately. Absolute coordinates can be obtained when the marker position is located and the measurement system is calibrated. To demonstrate the performance of the system, we measured human faces. The details of facial geometries and dynamic changes of facial expressions are measured clearly. We also measured a hand moving over a depth distance of approximately 700mm, again both hand shapes and positions changes are clearly captured. By using the fast three-step phase-shifting algorithm and a parallel processing technique, we successfully developed a system that acquires, reconstructs, and displays 3-D shape in real-time. Applications of such a system include manufacturing, inspection, reverse engineering, computer vision and computer graphics.

We present a new scheme of dispersive interferometry utilizing a femtosecond pulse laser for the dispersion-insensitive measurement of the refractive index of an optical material. Not only the group refractive index but also the variation of the phase refractive index with wavelength is determined without prior knowledge. Experiment results obtained from specimens of BK7 and UV silica are discussed.

We report an exploitation of the optical comb of a femtosecond pulse laser as the wavelength ruler for the task of absolute length calibration of gauge blocks. To that end, the optical comb was stabilized to an Rb atomic clock and an optical frequency synthesizer was constructed by tuning an external single-frequency laser to the optical comb. The absolute height of gauge blocks was measured by means of multi-wavelength interferometry using multiple beams of different wavelengths consecutively provided by the optical frequency synthesizer. The wavelength uncertainty was measured 1.9 × 10^-10 that leads to an overall calibration uncertainty of 17 nm (k=1) in determining the absolute length of gauge blocks of 25 mm nominal length.

Scanning White Light Interferometry (SWLI) is a tool for measuring discontinuous surface profile. A short coherence length white light source is used in the interferometer so that the fringes are localized in the vicinity of zero Optical Path Difference (OPD). The object surface is scanned along the height axis to get the height variation over the object field. Another technique using white light is Spectrally Resolved White Light Interferometry (SRWLI) in which the white light interferogram is spectrally decomposed by a spectrometer. The interferogram displayed at the exit plane of the spectrometer has a continuous variation of wavelength along the chromaticity axis. This interferogram encodes the phase as a function of wave number. For a given OPD, the phase is different for different spectral component of the source. The OPD can be determined as the slope of the phase versus wave number linear fit. To determine the phase of the different spectral components, temporal phase shifting technique which typically uses five frames has been proposed. Since the OPD is related to the height of the test object at a point, a line profile of the object can be determined. In this paper we discuss the Hilbert Transform method for determination of phase in SRWLI. This procedure requires only one spectrally resolved white light interferogram.

Profiling of optical surfaces with discontinuous steps by monochromatic interferometry has the ambiguity of multiples of a quarter wavelength. Wavelength-tuning interferometry can measure these surfaces with a unit of synthetic wavelength that is usually much larger than that of the original source. In order to solve this problem, the fractional phases of the interferograms before and after wavelength tuning should be carefully estimated. Phase-shifting interferometry with a mechanical phase shift by a PZT transducer determines the fractional phases of the interferograms with a resolution of better than one part in 250 of the wavelength. After subtracting the mechanical drift of the test surface during wavelength tuning, the absolute distance between the test surface and the reference surface is measured with an uncertainty better than a quarter wavelength. An optical flat with two gauge blocks 1 mm in height contacting the surface is measured by a Fizeau interferometer. Experimental results demonstrate that the surface profile can finally be measured with an accuracy of 20 nm.

Structured light system using a digital video projector is increasingly used for a 3-D shape measurement because of its digital nature. However, the nonlinear gamma of the projector causes the projected fringe patterns to be non-sinusoidal, which results in phase error therefore shape measurement error. Previous work showed that, by using a small look-up-table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. In this research, we prove that this type of phase error compensation method is not limited to a three-step phase-shifting algorithm. It is generic for any phase-shifting algorithm. The phase error compensation algorithm is able to theoretically eliminate the phase error caused by the gamma of the projector completely. It is based on our finding that in phase domain, the phase error due to the projector's gamma is preserved for arbitrary object's surface reflectivity under arbitrary ambient light condition. The phase error can be pre-calibrated and stored into a LUT for phase error reduction. In this research, we captured a set of fringe images of a uniform flat surface board for such a calibration. The phase error of the flat board is analyzed and stored in a LUT for phase error compensation. Our experimental results show that by using this method, the measurement error can be reduced by at least 13 times for any phase-shifting algorithm.

In this paper, we present a new color fringe projection system based on optimum frequency selection and discuss its implementation. Recent results in optimum frequency selection in temporal phase unwrapping have shown that the resolution limit of commercial data projectors is reached using 3 different sets of projected fringes with different pitch. Therefore, there is a synergy between optimum fringe projection and commercial color (RGB) projectors and cameras offering the potential for parallel data acquisition and simultaneous measurement of surface color parameters. The hardware of the system is comprised of a Digital Light Processing (DLP) video projector, a color 3-chip CCD camera and a personal computer supporting two monitors. The software is developed in Microsoft Visual C++ and OpenGL. The phase calculation algorithm is based on the optimum three-frequency selection, so it has a maximum reliability to determine the fringe order and can obtain 3-D shape of an object with large slope changes or discontinuities on the surface. Since each RGB color channel carries a fringe pattern of a certain pitch, any coupling between the color channels from the projector to the CCD camera affects the phase measurements obtained. Commercially available systems inherently contain crosstalk and we compare some methods to decrease the coupling effect. As absolute fringe order is determined by heterodyning between fringes with different pitch that are imaged in separate color channels, chromatic aberration can cause incorrect calculation of fringe order. We have investigated the most sensitive heterodyne process and the color channels with minimum chromatic aberration to mitigate these effects. We also give a novel software method to further compensate for chromatic aberration. Results show that our system has the advantages of fast acquisition, large dynamic range, robustness in discontinuities, and potential color extraction.

In industry, there are needs to accurately measure the 3-D profile of edges parts in order to evaluate edge condition. Optical methods are increasingly used for this purpose due to its advantages such as being non-contact, fast, accurate, and easy to integrate with software for data acquisition and feature analysis. We utilized structured line projection technology to measure edge's profile. In this method a structured light distribution, created by the transmission of a sinusoidal grating, was projected onto the inspected parts at a certain incidence angle. The projected light lines were deformed due to the depth change on the edge surface. A CCD camera sitting at a different angle was used to record the deformed fringes. From the deformed fringes the 3D surface profile was extracted based on triangulation principal. Because the projected grating pattern can be interpreted as an interference pattern, we used the spatial carrier phase shifting and phase unwrapping method as in classic interferometry to extract the phase information from the intensity distribution of fringes. Due to the limited depth-of-range of the fringe image and depth-of-focus of the imaging lens on the CCD camera, the observed deformed fringes have different widths and frequencies at different depth. According to the definition of depth-of-focus, the fringe width out of focus can be on the order of the square root of 2 wider than that which is in focus. The width change can also be due to a tilt across the object. This width change affects the accuracy of the spatial carrier phase shifting and subsequently the accuracy of extracted profile. In this paper, we proposed to monitor the change of the fringe width with imaging depth. According to the width of the fringes, we defined a parameter called compressing rate, to use in computing the edge profile. For different edge types, the compressing rate was optimized in order to get the profile that can match the results from traditional the methods. By using this method, the system repeatability can be improved significantly.

For the first time, transmission digital holography microscopy is applied to observe coal palynofacies, which are organic fossil microcomponents contained in the coal grains. The recorded holograms were produced by using microscope lenses with 20x and 40x of lateral magnification respectively, and He-Ne laser of wavelength 594.5 nm. The results show that reflection digital holography microscopy is required for observing relative opaque particles, because the phase recovery is strong diminished by light transmission in those cases. On the other hand, the phase distribution is related to the relief of the particles and the variations of their refraction index. Therefore, a priori information should be necessary to properly relate the phase information to physical features of the particles. Numerical unwrapping procedures are also crucial. Procedures with special requirements can be needed for analysing fast varying phase distributions. However, digital holography microscopy becomes a high performance tool for 3D modelling of fossil particles if the above requirements are enough fulfilled.

A simple method for surface contouring by phase-shifting digital holography is reported. Diffusely reflected light from a coherently illuminated object is recorded by a CCD with coherent in-line superposition of a reference beam subject to phase-shifting. From three phase-shifted in-line holograms the complex amplitude of the object wave at the CCD plane is derived to reconstruct phase distributions before and after wavelength shift that is provided by a mode-hopping induced by a change of injection current of a laser diode. The difference of the reconstructed phases corresponding to each of the wavelengths is proportional to surface height from the reference plane that is normal to the incident beam. This setup permitting normal incidence introduces neither the carrier component corresponding to oblique reference plane nor the shadowing effect that arise in the fringe projection and the dual incident angle methods requiring oblique illumination. In experiments plane surfaces tilted by various angles are first measured to evaluate accuracy. Then a spherical surface and a miniature valve are employed. The results are compared with one-dimensional simulations using random number model of surface roughness to exhibit good agreement.

Complex paraxial optical systems, consisting of multiple lenses and sections of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Apertures are necessary in speckle based metrology systems to control the size of the speckle. We examine their effect on LCT systems and show using the "generalized Yamaguchi correlation factor" that a useful interpretation of the system's behavior, using the LCT, may still be obtained. Furthermore, we experimentally demonstrate our ability to determine simultaneous tilt and translation motion by capturing two, sequential, mixed doma in images with a single camera. We also show how localized deformations in an object may be measured using this system.

Contrary to the common belief of optical metrology, where the phase singularities are considered as obstacles in phase unwrapping, we propose a new technique that makes use of phase singularities in the complex signal representation of a speckle pattern as indicators of local speckle displacements. The complex signal representation is generated by Laguerre-Gauss filtering the dynamic speckle patterns. Experimental results are presented that demonstrate the validity and the performance of the proposed optical vortex metrology for lateral speckle displacement measurement with a large dynamic range.

This paper presents the concept and the experimental setup of an interferometric system that is designed to work without vibration isolation and uses the random mechanical vibrations as phase shifter. An additional high temporal resolution detector system consisting of three photodiodes is used for the determination of the phase shifts that occur at the recording of the interference images. Two orthogonally polarized laser beams of different wavelengths, a continuous He-Ne laser and a pulsed laser diode, are coupled and expanded to the test and reference plates. To resolve the ambiguity problem of the profile orientation, additional information is obtained from the signals of two adjacent optical fibers placed in the He-Ne interference field. The laser beams are also split and expanded to a plate with tilted surfaces. The fringes that occur at the reflection on the surfaces are used to analyze the wavelength stability of the laser diode, taking the He-Ne wavelength as reference. An adequate PSI algorithm for random phase shifts is proposed.

A heterodyne interferometer has been built in order to characterize vibrations on Micro-Electro-Mechanical Systems (MEMS). The interferometer offers the possibility of both phase and high resolution absolute amplitude vibrational measurements, which is of great importance. A frequency shift is achieved by introducing acoustooptic (AO) modulation in one of the interferometer arms. By using a lock-in amplifier a narrow bandwidth detection regime is achieved. This factor improves the amplitude resolution. By using two AO-modulators and varying the frequency inputs of both, the setup is designed to measure vibrations in the entire frequency range 0 - 1.2GHz. The absolute amplitude is obtained by performing two measurements at each sample point. The first step is to measure the first harmonic of the object vibration. The second step is to measure the frequency components of the light reflected from the test device corresponding to the frequency without object modulation. This is obtained by mixing the detector signal with an external signal generator, and adjusting the frequency of the latter. By combining these two measurements we are able to determine the absolute amplitude of the vibration. The interferometric setup can be used to characterize various kinds of micro- and nanostructures. The system is here demonstrated on a Surface Acoustic Wave (SAW) device and on Capacitor Micromachined Ultrasonic Transducers (CMUTs). We have measured absolute amplitudes with picometer resolution.

We report the use of the photo-emf effect in BTO photorefractive crystals to measure sub-micrometer-order amplitude transverse vibrations. The method is based on the illumination of the surface under analysis by a direct laser beam of wavelength λ = 532 nm and the collection of the back-scattered speckled patterns of light onto the photoconductor. A pattern of space-charge electric field is built-up in the photoconductive material volume in a time-scale corresponding to the response time of the material that is essentially controlled by charge-transport phenomena. A pattern of free electrons in the conduction band is simultaneously built-up with a much faster time-scale that depends on the excitation of electrons from photoactive centers inside the material band gap into the conduction band. If the illuminated target surface is static, the pattern of space-charge field and free electrons are in mutual equilibrium and no electric signal is detected. However, if the target is laterally vibrating, the speckle pattern of light is simultaneously moving and the fast pattern of free electrons follows. The pattern of space-charge field instead is comparatively much slower and is not able to follow it. In this way the free charge distribution and the pattern of electric field are mutually displaced proportionally to the amplitude of the target vibration and are not any more in equilibrium. An alternating current is therefore produced that can be detected to find out the size of the target vibration amplitude. We report experiments carried out with Bi₁₂TiO₂₀ crystals.

We present design of novel tool for end point detection of wafer thickness, and wafer topography employing low coherence fiber optic interferometer, which optical length of the reference arm of the interferometer is monitored by secondary long coherence length interferometer.

Digital stepping is desirable in optical metrology--operation is simple, absolute position is known, and random regions of interest can be skipped to, rapidly and accurately. However, in white-light interferometry, analog scanning has traditionally been employed because, in one operation, it achieves depth scanning of a sample and an electronically detectable optical carrier through a Doppler shift. This is not obligatory nor efficient in functional machine vision, especially if approximate preknowledge of the sample exists. Two methods, utilizing digital depth stepping and a superluminescent diode, are presented to decouple optical carrier generation from depth scanning in full-field white-light interferometry. One technique employs a complementary metal-oxide semiconductor camera and acousto-optic modulation to generate a frequency difference between two arms of a Mach--Zehnder interferometer. The other technique uses a Michelson interferometer with a piezoelectric transducer integrated to the digital stepper motor to facilitate 2λ analog scanning and an optical carrier of 4 periods, sampled with a standard charge-coupled device camera. In the former case, random depth access measurement of an engineering gauge block calibration sample is presented, while the latter demonstrates the application of the random depth access full-field white-light interferometry to a small punch test. A further benefit of these techniques is the possibility of interferometric phase retrieval on condition of path length matching; this is proven by the implementation of a heterodyne phase retrieval algorithm in the gauge block measurement. Both techniques represent an advance in optical metrology, offering an inexpensive and functional solution to machine vision and industrial measurement applications.

We propose a novel spectro-polarization modulator which generates radial liner polarized light sorted along wavelength concentrically. It consists of a polarizer, a retarder with high order retardation, and a quarter wave plate. If we set the radial polarizer after the spectro-polarization modulator, we can observe spectroscopic color concentrically. A displacement measurement method is proposed using chromatic aberration method.

Spectrally resolved white-light phase-shifting interferometry has been used for accurate measurements of the spectral phase of the wave reflected from a micromachined surface. The phase is linearly related to the wave number, and the slope of the graph of the phase vs. the wave number, for any point on the test surface, gives the absolute value of the optical path difference at this point. These values can be used to generate a line profile of the test surface. However, if the test surface is coated with a transparent thin film, multiple reflections affect the phase of the reflected wave. The values obtained for the phase then depend on the thickness and the refractive index of the film and exhibit an additional nonlinear variation with the wave number, which can be modeled using thin-film theory. We show that this additional nonlinear phase can be measured directly using spectrally resolved white-light interferometry. The thickness profile of the film can then be obtained by a least-squares fit to the experimental phase data.

In this paper, we report on the recent development of a novel low coherence interferometry technique for the purpose of 3D-topography measurements. It combines the well established techniques of spectral-interferometry (SI) and chromatic-confocal microscopy (CCM). Measuring the optical interference in the spectral-domain allows for the detection of a reflecting or scattering object's depth position, without the necessity of a mechanical axial-scan. Focusing the white-light detection field with a microscope objective combined with a diffractive optical element leads to an expansion of the axial-range of the sensor beyond the limited depth-of-focus, imposed by the numerical aperture (NA) of the focusing objective. Focusing with a high NA objective and confocally filtering the detection light field causes the reduction of the lateral dimension of the area sampled upon the object. By this, the lateral resolution of the sensor is enhanced and due to the high NA, a high light collection-efficiency is achieved as well. The attained interferometric signals consist of high-contrast wavelets, measured in the optical-frequency domain. The depth position of an investigated point of the object is given by the modulation-period of the wavelets. Therefore, unlike in CCM, positionwavelength referencing is not necessary.

The idea which initiated this work is to adapt the principle of phase coding to a Confocal Chromatic type sensor in order to enhance the dynamic range and the axial resolution of height measuring systems. When using a classical Confocal chromatic type measurement, we obtain a "coarse" value of the distance. Then by the means of a spectrally adjustable light source and a chromatic objective, we choose the wave length in order to focus in the plane of the object. Next, by measuring the phase coding, we can precisely define the profile of the object. So, in the end, we obtain the subnanometric preciseness of the phase coding while clearing up the ambiguity that is inherent to interferometric methods thanks to the confocal chromatic measurement. The initial idea is to use a Linnik interferometer, with a dedicated chromatic objective in the object arm, and an achromatic objective in the reference arm. A change of the wave length leads to a variation of the focus plane in relation to the object. This equivalence between a variation in wave length and a variation in distance, caused by the axial chromatism, leads us to imagine a system of phase coding in which the usual movement of the reference mirror is replaced by a variation of the wave length. We then obtain an interferometer without any moving part, which produces the extensive depth of field that is peculiar to chromatic systems.

Digital speckle pattern interferometry (DSPI) and digital shearography (DS) are full field, non-contact optical methods for deformation or its derivative measurement. These techniques are also well suited for the measurement of vibrations. For the present work we use a dual-function DSPI system for the visualization of resonance frequencies and their mode shapes. The system is convenient and also efficient to switch over from an out-of-plane sensitive configuration to shearography. A time-average technique using a refreshing reference frame used for better visualization of the modes. The technique suppresses ambient disturbances such as thermal noise and low freq. noise, etc so that measurement can be performed under harsh environmental condition. Vibration mode shapes are investigated for objects vibrating sinusoidally at different resonant frequencies, for both out-of-plane and shearography configurations.

A new design of an incremental optical encoder with a sub-micrometric resolution, composed of a divergent beam coming from an uncollimated laser diode, a sinusoidal phase diffraction grating and a detector consisting of a mask and a photodiode array is presented. It is modeled by the scalar theory of diffraction in the Fresnel approximation, obtaining the expression for the field at the detector plane. The tolerance of the system to rotations or displacements of its elements is analyzed. The output of the photodiode array is connected to an electronic interpolation circuit in order to verify the viability of the design.

We present a method of free-form surface profile measurement using white-light scanning interferometry. This method is based on the principle of curvature sensor which measures the local curvature under test along a line. The profile is then reconstructed from the curvature data on the each point. Unlike subaperture-stiching method and slope detection method curvature sensing have strong points from a geometric point of view in measuring the free-form surface profile. Curvature is related to second derivative terms of surface profile and an intrinsic property of the test piece, which is independent of its position and tip-tilt motion. The curvature is measured at every local area with high accuracy and high lateral resolution by using White-light scanning interferometry.

A high-resolution new fringe analysis method based on Hilbert transformation is proposed by using the features of speckle interferometry. Hilbert transformation that is used widely in communication systems as the filter technology is the complex signal processing technology. And, the transformation can produce the analytic signal that has phase difference; π/2 rad. This transformation is also an effective method in fringe scanning method. In the proposed fringe scanning method for speckle interferometry in this paper, the transformation is not applied to the speckle grams, but to directly speckle patterns. The simulation based on the principle of proposed fringe analysis is performed by using the simple intensity distribution model of the speckle patterns. The validity of the principle of the method and the usefulness of the method are shown in the simulation. The experiments show that the difference between the results by the new and the ordinary methods is about 0.54 rad (1/12wave) as standard deviation.

In this work, a study of a new type of optical fiber interferometer with three-beam system has been carried out. Some analyses of the performances of the interferometer have been considered and reported upon. A theoretical model of the optical fiber interferometer and demonstration that such system could be constructed by optical fiber structures are presented. To achieve the goal which ensures higher performance and stability of the interferometer the simulations for intensity noises, phase noise in the interference path and phase shift noise of the certain three-beam optical fiber interferometer have been carried out. The results indicate that the 3×3 coupler is key factor. The optical fiber interferometer could be used to constitute an optical fiber sensor such as optical fiber hydrophones array. Other applications of optical fiber sensor with the three-beam interferometer also have been discussed.

Two-dimensional phase unwrapping (PhU) is one of the most important processing steps in optical phase-shifting interferometry. It aims to reconstruct the continuous phase field, but in fact, it becomes very difficult to perform the PhU due to the existence of noise, low modulation, under sampling, discontinuities or other defects. For solving the problem, we present a new PhU method which is based on the local parameter-guided fitting plane. It relies on the basic plane-approximated assumption for phase value of local pixels and is guided by our proposed parameter map- Modulation-Gradient and Pseudo-Correlation (MGPC) parameter map. This new map is the combination of fringe modulation and wrapped phase, thus it's reliable in estimating the quality or goodness of phase data. Meanwhile, the adoption of fitting plane for the 3×3 window of pixels makes the process of PhU very fast. In applications, we offer the simulated and the experimental data to compare our method with the two previously reported path-following unwrapping algorithms. The unwrapped results show our proposed PhU method not only is efficient, but also has higher noise robustness in recovering the true phase field.

With digital holographic interferometry, we have investigated the phase differences of a plane wave transmitting through a partly-poled RuO₂:LiNbO₃ crystal sample. The holograms recorded by a CCD array are reconstructed by both the Fresnel transform method and the angular spectrum backward propagation algorithm. Comparison of the reconstructed results shows a higher resolution can be achieved by the angular spectrum backward propagation algorithm.

The development of the optic interferometric sensors is partly restricted with the demodulation technique. The Phase Generated Carrier (PGC) modulation scheme is a useful demodulation method for interferometric sensors, for it is simple and accurate. At first, the PGC scheme is demodulated by analog demodulation techniques. Its capability is limited by the electronic devices, and not fit to the large-scale sensors application. In the past few years, a number of digital demodulation processing approaches have been investigated. Another arctangent approach based on digital demodulator exhibits more advantages than the traditional Differentiate and Cross Multiply (DCM) approach, which has a small measurement range and rather complex circuit. In this paper, the arctangent approach on the PGC configuration used in the array of optical fiber interferometer are discussed and emulated. The measurement range, operation complexity, quality of operation, noise performance, and applicability are compared among the variant arctangent approaches and the DCM approach. The variant arctangent approaches and the DCM approach will be influenced by differently factors, such as, the intensity of the light source, the shift of visibility of the interference fringes due to the change of the state of polarization in fiber, the phase delay. The dependence on these influences of these approaches is analyzed in detail. Their effects and removal methods are validated through emulation.

The results of combining the wrapped phase with the fringe order of this phase to increase the precision of white-light interferometry at high scanning speed are presented. Monochromatic phase data are calculated using the Fourier method and the fringe order is determined using a general coherence peak sensing method. A wide scanning interval of 5λ/8 and a narrow-band color filter with a bandwidth of 70 nm are adopted to acquire interferograms. Experiments with an rms repeatability of step height measurement of below 1 nm and a scanning speed of 40 μm/s are performed.

In this work we present a sequential technique for temporal fringe pattern demodulation without carrier. The technique presented here, uses a temporal frequency estimator to obtain the temporal phase from a interferogram sequence. The restriction used to estimate the frequency is based on second order potentials in order to obtain the temporal phase as a phase function in an space C². The importance of this technique to demodulate temporal fringe patterns without carrier is mainly the simplification of experimental optical arrays in laboratory, where the experimental nature make it difficult to introduce a carrier frequency. This work present an on development technique for temporal demodulation, however, its projection on future work give us the possibility to obtain a robust demodulation method for temporal interferograms. To demonstrate the performance of this temporal fringe pattern demodulation technique, we are going to show a simulated interferogram sequence to demodulate it with this technique.

We present a theoretic approach to the characterization of low-power bright picosecond optical pulses with an internal frequency modulation simultaneously in both time and frequency domains. This approach exploits the joint Wigner time-frequency distribution, which can be determined and developed for these bright optical pulses by using a novel interferometric technique under our proposal. Either power or spectral densities inherent in such pulses can be obtained through integrating the Wigner distribution with respect to the corresponding conjugate variable. At first, the analysis and computer simulations are applied to studying the capability of Wigner distribution to characterize solitary pulses or train-average values for pulse string in practically much used case of the Gaussian shape, when the Wigner distribution is positive. In the context of this analysis, the relation between the spectrum density and the auto-correlation function is followed. Then, the simplest two-beam scanning Michelson interferometer is selected for shaping the field-strength auto-correlation function of low-power picosecond pulse trains. We are proposing and considering in principle the key features of a new interferometric experimental technique for accurate and reliable measurements of the train-average width as well as the value and sign of the frequency chirp of pulses in high-repetition-rate trains. This technique is founded on an ingenious algorithm for the advanced metrology, assumes using a specially designed supplementary semiconductor cell, and suggests carrying out a pair of additional measures with exploiting this semiconductor cell. Such a procedure makes it possible to construct the Wigner distribution and to describe the above-listed time-frequency parameters of low-power bright picosecond optical pulses. In the appendix, we follow the avenue of deriving the joint Wigner time-frequency distribution via choosing the Weil's correspondence between classical functions and operators.

We present a non-contact system for obtaining three-dimensional objects topography. The described system combines the fringe projection technique and the Talbot effect which is knowed like Talbot interferometry. In fringe projection technique, the digitalization is realized when black and white lines are projected over the object and this image is captured by the CCD. In Talbot interferometry, the object is collocated on one of the grating auto-image planes. The deformed grating image is captured by the CCD and superposed with other reference one that can be physical or computer generated (virtual/synthetic) for obtaining a moire pattern which gives information about the object topography. The topography of a coin and phalangeal articulation are obtained by using of this technique. The Spatial Synchronous Detection and Fourier Method were incorporate to retrieve the phase.

Effects of spatial coherence of optical field with wide frequency and angular spectra in the Michelson interferometer are studied. Manly effects of longitudinal spatial coherence are considered. It is shown that in classical configuration of the Michelson interferometer the effects of not temporal, but spatial coherence are taken place. Experimental data of observation of spatial and temporal coherence effects in image domain behind a lens at the output of the Michelson interferometer are shown.

There are many optical techniques to evaluate and calculate the shape of any concave surface, including lateral shear or phase lock interferometry tests. In this work we employ a Twymann Green interferometer to obtain several interferograms of any mirror under test, and by using a computer program and a piezo electric device, we apply the phase shifting technique to obtain several interferograms and with them we obtain the phase by applying several phase unwrapping methods, also we realize an experimental comparison between the methods employed to unwrap the phase. Finally results concerned with the advantages or disadvantages of the studied methods are discussed.

Nonlinear interference of two successively generated coherent anti-Stokes Raman scattering (CARS) signals from thin glass slabs is demonstrated, in which a collinear phase matching geometry is tried. We used 76 MHz mode-locked Nd:YAG pulsed laser at 1064 nm and its frequency-doubled optical parametric down-converted signal at 817.2 nm as Stokes and pump beams, respectively. The pulse duration time is around 7 ps for both incident laser beams. The relative phase of two CARS signals is controlled by a phase shifting element made of dispersive glass material of which thickness can be varied. The clear interference fringes are observed as the thickness of the phase shifting element changes. The interference effect is utilized to achieve better CARS image quality. We first try imaging the polystyrene beads immersed in water to estimate how the nonlinear interference could improve the contrast of the beads image. Performing the raster scan of the laser beams on the sample, we can get the CARS image and investigate the image quality as a function of the relative phase and amplitude.