A novel Vision ray metrology technique is reported that estimates the geometric wavefront of a measurement sample using the sample-induced deflection in the vision rays. Vision ray techniques are known in the vision community to provide image formation models even when conventional camera calibration techniques fail. This work extends the use of vision rays to the area of optical metrology. In contrast to phase measuring deflectometry, this work relies on differential measurements, and hence, the absolute position and orientation between target and camera do not need to be known. This optical configuration significantly reduces the complexity of the reconstruction algorithms. The proposed vision ray metrology system does not require mathematical optimization algorithms for calibration and reconstruction – the vision rays are obtained using a simple 3D fitting of a line.
In this paper, we demonstrate the recording of shear interferograms using diffractive polarization gratings and reconstruct complex wavefields using a known optimization algorithm. We validate the reconstruction of a complex wavefield by recording the object wave at a defocused distance and numerically refocusing the reconstructed wavefield. Finally, we discuss the impact of the shear selection strategy and aperture selection using the information density function and demonstrate this by reconstructing complex wavefields using experimental data.
This work presents a stable noise-robust numerical integration technique derived from a gradient representation of the Q-Forbes polynomials for surfaces with axial symmetry. This modal-integration technique uses an orthogonalization process through the Householder reflections to obtain a numerically orthogonal set for the surface slopes that is used to reconstruct the surface shape. It is shown that for typical Deflectometry measurements, the resulting random component of the uncertainty after numerical integration has a root mean square error well below 1nm.
We present a multicolor fluorescence microscope system, under a selective plane illumination microscopy (SPIM) configuration, using three continuous wave-lasers and a single-channel-detection camera. The laser intensities are modulated with three time-delayed pulse trains that operate synchronously at one third of the camera frame rate, allowing a sequential excitation and an image acquisition of up to three different biomarkers. The feasibility of this imaging acquisition mode is demonstrated by acquiring single-plane multicolor images of living hyphae of Neurospora crassa. This allows visualizing simultaneously the localization and dynamics of different cellular components involved in apical growth in living hyphae. The configuration presented represents a noncommercial, cost-effective alternative microscopy system for the rapid and simultaneous acquisition of multifluorescent images and can be potentially useful for three-dimensional imaging of large biological samples.
Multi-wavelength phase unwrapping techniques have traditionally been used to unwrap the phase at the shortest measurement wavelength, where numerous techniques have been developed with distinct advantages for a given application. Nevertheless, multi-wavelength techniques are more than phase unwrapping approaches: super-sensitive multiwavelength interferometers have a lower uncertainty than conventional interferometers, multi-wavelength techniques can break the Nyquist limit and thereby relax the requirements on the measurement system, and multi-wavelength techniques have also unconventional applications as e.g. optical encryption. This work discusses different multi-wavelength techniques, derives new noise criteria with no approximations, and outlines important, but still little researched areas of multi-wavelength phase unwrapping techniques.
Fringe Projection Profilometry and Phase Measuring Deflectometry Systems are well-established technologies for noncontact 3D surface measurements. A common challenge in those systems is to obtain the absolute surface information using few measurement frames. In practice both techniques often generate a series of sinusoidal fringe patterns with different frequencies and use a camera vision system to capture the (by the object) deformed patterns. The images containing the deformed patterns are then processed to obtain the 3D surface information. Sinusoidal multi-frequency techniques have been for years and are an on-going area of research, where various algorithms have been developed to measure the unwrapped phase map (at the shortest fringe period). Commonly, temporal phase shifting techniques are used to extract the phase at different frequencies. However, obtaining the phase map for every fringe period requires multiple measurement frames. In this work, the advantages of efficient grey level (GL) coding techniques are presented, where a focus is given on the reduction of frames for noncontact 3D surface measurements. A further focus is given on additional features of GL coding as e.g. the error-detecting and the error-correcting properties of those codes that make GL coding an interesting candidate for harsher measurement environments.
Quantitative Phase Imaging based on the Transport of Intensity Equation (TIE) has shown to be a practical tool for retrieving the phase information of biological and technical samples. When recovering the phase information with the TIE, the maximum lateral resolution that can be obtained is limited by the numerical aperture (NA) of the optical system. In order to overcome this limitation, a system that combines structured illumination and TIE-like techniques have been proposed. These methodologies enlarge synthetically the NA of the optical system, and thus, the lateral resolution of the retrieved phase can be improved. However, the employment of structured illumination may bring error amplifications in the retrieved phase due to its sensitiveness to phase discontinuities and shot noise. In this work, we propose a new methodology that improves the lateral resolution of the retrieved phase beyond the diffraction limit avoiding the problems related with the structured illumination. The methodology presented here uses tilted illumination and a TIE solver. We show that when using this configuration, we can extend the set of recovered frequencies by adjusting the angle of the tilted wave-front. Further, our methodology is designed to extend the NA by employing less tilted angles than other similar techniques. Hence, the final retrieved phase will have an enhanced lateral resolution without amplifying the numerical errors and employing a few tilted angles. Moreover, we show that the algorithm presented here can be combined with other TIE algorithms that are used for suppressing the Low Frequency Artifacts (LFAs) usually present when using TIE based techniques.
Single Beam phase reconstruction techniques are well established approaches for wavefront reconstructions in lens-less optical configurations. These approaches use a series of intensities captured at various defocus distances and rely on successful algorithms that use these data for the wave-field reconstruction. The usefulness of these methods has been demonstrated in the x-ray regime and the optical domain, i.e. in the area of life-sciences and the micro-optical metrology. However, the recovery of volume speckle fields has been made with varying degree of success, because conventional methods with equidistant measurement planes recover either only slowly varying (in case of deterministic paraxial methods) or fast varying wave-fields (in case of iterative wave-propagation based methods). A further critical aspect is the choice of the measurement planes, because specific sets of spatial frequencies may not be visible at the captured defocus planes. Usually this ill-posed problem is combated using regularization techniques giving increased computational effort with limited accuracy. This work reports an optimal single beam wave-field reconstruction with application to speckle field recovery for a wide range of spatial frequencies including both lower and higher spatial frequencies. The principle of this technique is based on cascading iterative and deterministic phase retrieval techniques as well as employing a plane selection strategy that ensures the support for all spatial frequencies for the given set of measurement planes. Reconstructions with this hybrid approach are fast, accurate, and unlike conventional methods do not require the use of regularization techniques.
Deterministic Phase Retrieval techniques (DPRTs) employ a series of paraxial beam intensities in order to recover the phase of a complex field. These paraxial intensities are usually generated in systems that employ plane-wave illumination. This type of illumination allows a direct processing of the captured intensities with DPRTs for recovering the phase. Furthermore, it has been shown that intensities for DPRTs can be acquired from systems that use spherical illumination as well. However, this type of illumination presents a major setback for DPRTs: the captured intensities change their size for each position of the detector on the propagation axis. In order to apply the DPRTs, reescalation of the captured intensities has to be applied. This condition can increase the error sensitivity of the final phase result if it is not carried out properly. In this work, we introduce a novel system based on a Phase Light Modulator (PLM) for capturing the intensities when employing spherical illumination. The proposed optical system enables us to capture the diffraction pattern of under, in, and over-focus intensities. The employment of the PLM allows capturing the corresponding intensities without displacing the detector. Moreover, with the proposed optical system we can control accurately the magnification of the captured intensities. Thus, the stack of captured intensities can be used in DPRTs, overcoming the problems related with the resizing of the images. In order to prove our claims, the corresponding numerical experiments will be carried out. These simulations will show that the retrieved phases with spherical illumination are accurate and can be compared with those that employ plane wave illumination. We demonstrate that with the employment of the PLM, the proposed optical system has several advantages as: the optical system is compact, the beam size on the detector plane is controlled accurately, and the errors coming from mechanical motion can be suppressed easily.
Phase retrieval techniques based on the Transport of Intensity Equation (TIE) use a sequence of through-focus intensity
images in order to recover the phase information. Classically, the capturing of these images have been made using
equally spaced plane separations. Recently, it has been shown that the phase retrieval techniques based on TIE can be
carried out using unequally plane separations. In this work we compare quantitatively the phase reconstruction of various
TIE solvers using the equal and unequal plane separation strategy.
In this work the stagnation problem in iterative single beam multiple intensity reconstruction algorithms is reduced by
combining deterministic and iterative phase retrieval techniques in order to compensate for paraxial artifacts. This
combined technique has a better convergence, because it suppresses better the stagnation problem present in iterative
phase retrieval techniques. The reported hybrid deterministic-iterative phase retrieval techniques can be successfully
employed for cases where an iterative solver is trapped in a local minimum, and moreover, allows increasing the
convergence of iterative solvers.
KEYWORDS: Signal to noise ratio, Phase retrieval, Neodymium, Error analysis, Solids, Computer simulations, Ions, Linear filtering, Chemical elements, Numerical analysis
The Transport of Intensity Equation (TIE) relates linearly the phase of an object to the intensity distribution in the Fresnel region and can be used as a phase retrieval technique. The key element in a TIE based solver is the calculation of the axial intensity derivative. This parameter is calculated from a series of captured intensities but its accuracy is subject to several parameters, such as e.g. the separation between the measurement planes, the Signal to Noise Ratio (SNR) in the captured intensities, the actual object phase distribution. Despite the importance of the estimation of this parameter, there is no general discussion how to optimize the axial intensity derivative. In this work, we developed the mathematical framework in which the retrieved phase can be obtained. An optimal separation is derived, which minimizes the error in the calculation of the axial derivative. Besides this, we study using a numerical analysis how the accuracy of the axial derivative influence the accuracy of the retrieved phase. Hence, we present a numerical procedure based in the Root Square Mean Error, which is able to minimized the error in the retrieved phase. It is later shown that this analysis is significant more accurate than available methods proposed in the literature. It is further shown, that the plane separation that minimizes the error in the axial intensity derivative is different to the plane separation that minimizes the error in the retrieved phase.
The transport of intensity equation (TIE) describes the relation between the object phase and the intensity distribution in
the Fresnel region and can be used as a non-interferometric technique to estimate the phase distribution of an object. A
number of techniques have been developed to solve the TIE. In this work we focus on one popular class of Poisson
solvers that are based on Fourier and the Multigrid techniques. The aim of this paper is to present an analysis of these
types of TIE solvers taking into account the effect of the boundary condition, i.e. the Neumann Boundary Condition
(NBC), the Dirichlet Boundary Condition (DBC), and the Periodic Boundary Condition (PBC). This analysis, which
depends on the location of an object wave-front in the detector plane, aims to identify the advantages and disadvantage
of these kinds of solvers and to provide the rules for choice of the best fitted boundary condition.
In 1948, Gabor proposed a lens-less in-line holographic microscope where the recorded hologram is a result of
interference of both reference wave and the wave diffracted from the object. In this work, an analysis of the hologram
formation and sampling issues are presented using the local-spatial frequency distribution at the camera plane.
Techniques are presented that enable the reconstruction of the complex object wave at the camera plane from a series of
defocused images. It is suggested to use the band-limited angular spectrum method for the back-propagation of the field
components, so that an accurate and quantitative estimate of the object-wave at the object plane is obtained. The
reconstruction procedure does not rely on the paraxial approximation and is therefore applicable to the case of high NA
beams.
Multi-wavelength interferometry (MWI) has a long tradition in the field of optical metrology and is used as a
solution to a number of applications. MWI phase unwrapping procedures are usually based on beat
wavelength approaches, Chinese Remainder Theorem (CRT) techniques, or the method of Excess Fractions
(EF). Each of these unwrapping approaches have a distinct advantage for a given application: Beat
wavelength and CRT based approaches offer a direct calculation of integer fringe order, and EF offers many
alternative sets of wavelengths to achieve a large unambiguous measurement range (UMR) with high
reliability. Nevertheless, a drawback of Beat wavelength and CRT based approaches is that they have a
limited UMR due to the available measurement wavelengths, and the alternative approach the EF is often
impractical in practice, because the calculation of the integer fringe order involves a large number of
computational steps. Recently, we have reported a unified theory of beat wavelength, EF and CRT
approaches, which enables the derivation of phase unwrapping approaches with low computational effort,
which hitherto had only been possible for CRT and beat wavelength approaches, whilst offering flexibility in
choosing the measurement wavelengths for a given UMR, which had previously only been the case for EF. In
this work, we briefly summarize the previous developed framework that determines the UMR and
measurement reliability and derive optimization criteria that are based on parameters, which are dependent
on the choice of the measurement wavelengths. The developed optimum wavelength selection strategies
maximize the dynamic range of interferometer for a given value of phase noise the dynamic range of
interferometer.
Multi-wavelength interferometry (MWI) has a long tradition and provides a solution to a number of applications in the
field of optical metrology. In MWI phase unwrapping procedures are usually based on beat wavelength approaches,
Chinese Remainder Theorem (CRT) techniques, or the method of Excess Fractions (EF). Each of these unwrapping
approaches has distinct advantages making it suitable for a given application. Beat wavelength and CRT based
approaches offer a direct calculation of integer fringe order, however, the unambiguous measurement range (UMR) is
limited by the available measurement wavelengths. On the other hand, EF offers many alternative sets of wavelengths to
achieve a large UMR with high reliability; however, the calculation of the integer fringe order involves a large number of
computational steps. In this work, a unified theory of beat wavelength, EF and CRT approaches is reported. It is shown
that the calculation of the integer fringe order requires a low computational effort, which hitherto had only been possible
for CRT and beat wavelength approaches, whilst offering flexibility in choosing the measurement wavelengths for a
given UMR, which had only been the case for EF. As the model can be used in a predictive way to determine the UMR
and measurement reliability it is possible to define optimization criteria that are based on parameters which are
dependent on the choice of the measurement wavelengths.
Interferometric metrology is well established for both single point and full field measurements.
However, absolute techniques for long range measurements, spanning 100's to 10,000's of
fringe orders whilst maintaining sub-fringe resolution have been reported with widely varying
levels of performance. In this paper, techniques for long range multi-wavelength
interferometry are reviewed with respect to applications of classical interferometry and fringe
projection profilometry. Whilst hierarchical geometric series methods provide a potential
solution it is shown that significantly greater freedom in wavelength selection is obtained by
applying excess fraction principles and a new predictive model for this technique is discussed.
The method of excess fractions has a long history in metrology. More recently excess fractions has been exploited to
resolve the fringe order ambiguity in interferometric metrology with varying degrees of success. There are a variety of
reports detailing the performance of excess fractions, for example, using 4 wavelengths an unambiguous measurement
range of 2.4 mm was achieved with a phase noise of 1/900th of a fringe. In an independent report a 4 wavelength
interferometer gave an unambiguous measurement range of 17 mm with a phase noise of 1/200th of a fringe. It has
been found that the unambiguous range of an excess fractions multi-wavelength interferometer depends on the
wavelengths used within the system. A theoretical model is reported in this paper that can be used in a predictive way
to determine the unambiguous measurement range based on three wavelength dependent parameters. The excess
fractions model is consistent with beat wavelength techniques but offers many alternative sets of wavelengths to
achieve, for a given phase noise, a particular unambiguous measurement range with a given reliability.
In this paper, we explore the optimization and implementation of multi-wavelength interferometers such
that measurements beyond the largest beat wavelength can be achieved reliably. A hybrid beat wavelength
approach is presented that also exploits wavelength coincidence between two beat wavelengths in order to measure
unambiguously over an extended range. The performance of the approach has been explored both through
simulations and experimental validation has been obtained using a fiber interferometer with 4 measurement
wavelengths. The initial results have demonstrated 1/200th of a fringe phase resolution giving absolute metrology
over 18.16 mm, or a dynamic range of 1 part in 2.4×106.
In this paper, we present an understanding of the failure modes of excess fractions solutions to multi-wavelength
interferometry. From this basis, an approach to select optimum measurement wavelengths has been
introduced. A practical fiber optic sensor has been constructed for simultaneous detection of the intensity at four
measurement wavelengths. The system has been demonstrated using two wavelength selections that are very near
the optimal configuration and the data analyzed using an excess fractions solver. Initial results have shown a
measurement range of 17 mm with reliable and robust absolute metrology from a system with a phase noise of
1/200th of a fringe. This corresponds to an overall dynamic range of 1 part in 2×106.
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