Wave field sensing by means of computational shear interferometry

Effective selection of shears in variable lateral shearing holography

The efficiency of reconstruction of complex wavefields in digital holography through shear interferometry has a direct correlation with the shears selected for image acquisition. Although studies to investigate the effect of shears have shown correlations between the selected shear set and the spatial and frequency contents of the reconstructed complex wavefield, to our best knowledge, not much information is available to provide a guide on how to select these shears optimally and what factors to be considered during this selection procedure. In this paper, we study the effect of shear parameters on the phase error through a series of simulations using a synthetic object wavefield and provide a range of shear parameters for optimal reconstruction. Further, we correlated the data by comparing the results with corresponding frequency information density maps.

Addressing phase-curvature in Fourier ptychography

In Fourier ptychography, multiple low resolution images are captured and subsequently combined computationally into a high-resolution, large-field of view micrograph. A theoretical image-formation model based on the assumption of plane-wave illumination from various directions is commonly used, to stitch together the captured information into a high synthetic aperture. The underlying far-field (Fraunhofer) diffraction assumption connects the source, sample, and pupil planes by Fourier transforms. While computationally simple, this assumption neglects phase-curvature due to non-planar illumination from point sources as well as phase-curvature from finite-conjugate microscopes (e.g., using a single-lens for image-formation). We describe a simple, efficient, and accurate extension of Fourier ptychography by embedding the effect of phase-curvature into the underlying forward model. With the improved forward model proposed here, quantitative phase reconstruction is possible even for wide fields-of-views and without the need of image segmentation. Lastly, the proposed method is computationally efficient, requiring only two multiplications: prior and following the reconstruction.

Terahertz referenceless wavefront sensing by means of computational shear-interferometry

In this contribution, we demonstrate the first referenceless measurement of a THz wavefront by means of shear-interferometry. The technique makes use of a transmissive Ronchi phase grating to generate the shear. We fabricated the grating by mechanical machining of high-density polyethylene. At the camera plane, the +1 and -1 diffraction orders are coherently superimposed, generating an interferogram. We can adjust the shear by selecting the period of the grating and the focal length of the imaging system. We can also alter the direction of the shear by rotating the grating. A gradient-based iterative algorithm is used to reconstruct the wavefront from a set of shear interferograms. The results presented in this study demonstrate the first step towards wavefield sensing in the terahertz band without using a reference wave.

Deep learning in optical metrology: a review

With the advances in scientific foundations and technological implementations, optical metrology has become versatile problem-solving backbones in manufacturing, fundamental research, and engineering applications, such as quality control, nondestructive testing, experimental mechanics, and biomedicine. In recent years, deep learning, a subfield of machine learning, is emerging as a powerful tool to address problems by learning from data, largely driven by the availability of massive datasets, enhanced computational power, fast data storage, and novel training algorithms for the deep neural network. It is currently promoting increased interests and gaining extensive attention for its utilization in the field of optical metrology. Unlike the traditional "physics-based" approach, deep-learning-enabled optical metrology is a kind of "data-driven" approach, which has already provided numerous alternative solutions to many challenging problems in this field with better performances. In this review, we present an overview of the current status and the latest progress of deep-learning technologies in the field of optical metrology. We first briefly introduce both traditional image-processing algorithms in optical metrology and the basic concepts of deep learning, followed by a comprehensive review of its applications in various optical metrology tasks, such as fringe denoising, phase retrieval, phase unwrapping, subset correlation, and error compensation. The open challenges faced by the current deep-learning approach in optical metrology are then discussed. Finally, the directions for future research are outlined.

Optical In-Process Measurement: Concepts for Precise, Fast and Robust Optical Metrology for Complex Measurement Situations

Optical metrology is a key element for many areas of modern production. Preferably, measurements should take place within the production line (in-process) and keep pace with production speed, even if the parts have a complex geometry or are difficult to access. The challenge for modern optical in-process measurements is, therefore, how to simultaneously make optical metrology precise, fast, robust and capable of handling geometrical complexity. The potential of individual techniques to achieve these demands can be visualized by the tetrahedron of optical metrology. Depending on the application, techniques based on interferometry or geometrical optics may have to be preferred. The paper emphasizes complexity and robustness as prime areas of improvement. Concerning interferometric techniques, we report on fast acquisition as used in holography, tailoring of coherence properties and use of Multiple simultaneous Viewing direction holography (MultiView), self reference used in Computational Shear Interferometry (CoSI) and the simultaneous use of several light sources in Multiple Aperture Shear Interferometry (MArS) based on CoSI as these techniques have proven to be particularly effective. The use of advanced approaches based on CoSI requires a transition of the description of light from the use of the well-known wave field to the coherence function of light. Techniques based on geometric optics are generally comparatively robust against environmental disturbances, and Fringe Projection (FP) is shown to be especially useful in very demanding measurement conditions.

Phase retrieval using bidirectional interference

In this paper we describe a phase retrieval algorithm using constraints given by diffraction patterns and phase difference obtained from bidirectional interference. Wave propagation and linear phase ramps are used to connect the recordings. At least three patterns are recorded and processed (two diffraction patterns and one interference pattern). The quality of the results can be improved when recording and processing more patterns. The method works well with non-sparse samples and short (few millimeter) recording distances. Simulations, comparisons with other methods, and experimental validations are presented.

Fast 3D form measurement using a tunable lens profiler based on imaging with LED illumination

We present a fast shape measurement of micro-parts based on depth discrimination in imaging with LED illumination. It is based on a 4f-setup with an electrically adjusted tunable lens at the common Fourier plane. Using such a configuration, the opportunity to implement a fast depth scan by means of a tunable lens without the requirement of mechanically moving parts and depth discrimination using the limited spatial coherence of LED illumination is investigated. The technique allows the use of limited spatially partially coherent illumination which can be easily adapted to the test object by selecting the geometrical parameters of the system accordingly. Using this approach, we demonstrate the approach by measuring the 3D form of a tilted optically rough surface and a cold-formed micro-cup. The approach is robust, fast since required images are captured in less than a second, and eye-safe and offers an extended depth of focus in the range of few millimetres. Using a step height standard, we determine a height error of ±1.75 μm (1σ). This value may be further decreased by lowering the spatial coherence length of the illumination or by increasing the numerical aperture of the imaging system.

Phase retrieval algorithms for lensless imaging using diffractive shearing interferometry

Diffractive shearing interferometry (DSI) is a method that has recently been developed to perform lensless imaging using extreme ultraviolet radiation generated by high-harmonic generation. In this paper, we investigate the uniqueness of the DSI solution and the requirements for the support constraint size. We find that there can be multiple solutions to the DSI problem that consist of displaced copies of the actual object. These alternative solutions can be eliminated by enforcing a sufficiently tight support constraint, or by introducing additional synthetic constraints. We furthermore propose a new DSI algorithm inspired by the analogy with coherent diffractive imaging (CDI) algorithms: the original DSI algorithm is in a way analogous to the hybrid input-output algorithm as used in CDI, and we propose a new algorithm that is more analogous to the error reduction algorithm as used in CDI. We find that the newly proposed algorithm is suitable for final refinement of the reconstruction.

Extreme ultraviolet lensless imaging without object support through rotational diversity in diffractive shearing interferometry

We report on a method that allows microscopic image reconstruction from extreme-ultraviolet diffraction patterns without the need for object support constraints or other prior knowledge about the object structure. This is achieved by introducing additional diversity through rotation of an object in a rotationally asymmetric probe beam, produced by the spatial interference between two phase-coherent high-harmonic beams. With this rotational diffractive shearing interferometry method, we demonstrate robust image reconstruction of microscopic objects at wavelengths around 30 nm, using images recorded at only three to five different object rotations.

Interference probe ptychography for computational amplitude and phase microscopy

We have developed an approach to Fresnel domain ptychography in which the illumination consists of an interference pattern. This pattern is conveniently created by overlapping two coherent beams at an angle. Only the phase and orientation of the interferometric fringe pattern needs to be scanned to reconstruct a high-fidelity object image, which alleviates the requirements for accurate sample positioning and system stability. As such, the resulting imaging systems can be constructed in an extremely simple and robust way. Object images are reconstructed from recorded Fresnel diffraction data using a modified ptychographical iterative engine. We demonstrate the capabilities of this imaging system by recording images of various biological samples, demonstrating quantitative phase contrast as well as a spatial resolution better than 2.2 μm.

Soft tissue elastography via shearing interferometry

Early detection of cancer can significantly increase the survival chances of patients. Palpation is a traditional method in order to detect cancer; however, in minimally invasive surgery the surgeon is deprived of the sense of touch. We demonstrate how shearing elastography can recover elastic parameters and furthermore can be used to localize stiffness imhomogenities even if hidden underneath the surface. Furthermore, the influence of size and depth of the stiffness imhomogenities on the detection accuracy and localization is investigated.

Diffractive shear interferometry for extreme ultraviolet high-resolution lensless imaging

We demonstrate a novel imaging approach and associated reconstruction algorithm for far-field coherent diffractive imaging, based on the measurement of a pair of laterally sheared diffraction patterns. The differential phase profile retrieved from such a measurement leads to improved reconstruction accuracy, increased robustness against noise, and faster convergence compared to traditional coherent diffractive imaging methods. We measure laterally sheared diffraction patterns using Fourier-transform spectroscopy with two phase-locked pulse pairs from a high-harmonic source. Using this approach, we demonstrate spectrally resolved imaging at extreme ultraviolet wavelengths between 28 and 35 nm.

High spatial resolution zonal reconstruction with modified multishear method in frequency domain

An exact multishear zonal algorithm is proposed to reconstruct two-dimensional wavefronts in frequency domain. The algorithm maintains the advantage of fast Fourier transform and loosens the "natural extension" requirement that the shear amounts must be divisors of sampling points N; therefore, it can be rapidly executed for large data arrays. The effect of tilt errors in multishear interferometry is analyzed and compensated in our method. The presented algorithm is applicable for a general aperture shape by using an iterative method. Application of large shears is allowed, and high resolution of the reconstructed wavefront can be achieved. Results of numerical simulations demonstrate the capability of our method.

Sparse light fields in coherent optical metrology [Invited]

In this publication, we demonstrate that recording the mutual intensity, instead of a wavefront, enables interferometric measurements with multiple independent light sources at the same time. This scheme can, for example, be used to overcome the problem of a limited acceptance angle of imaging systems in interferometry. We further show that, for a finite number of light sources, measuring a subspace of the mutual intensity equals the recording of the corresponding light field, which is sparse in phase space. This recording modality offers more flexibility with respect to the trade-off between angular multiplexing and spatial resolution than the state of the art, because it is not restricted by the geometric properties of a microlens array, but rather allows arbitrary sampling of the light field.

Dynamic wave field synthesis: enabling the generation of field distributions with a large space-bandwidth product

We present a novel approach for the design and fabrication of multiplexed computer generated volume holograms (CGVH) which allow for a dynamic synthesis of arbitrary wave field distributions. To achieve this goal, we developed a hybrid system that consists of a CGVH as a static element and an electronically addressed spatial light modulator as the dynamic element. We thereby derived a new model for describing the scattering process within the inhomogeneous dielectric material of the hologram. This model is based on the linearization of the scattering process within the Rytov approximation and incorporates physical constraints that account for voxel based laser-lithography using micro-fabrication of the holograms in a nonlinear optical material. In this article we demonstrate that this system basically facilitates a high angular Bragg selectivity on the order of 1°. Additionally, it allows for a qualitatively low cross-talk dynamic synthesis of predefined wave fields with a much larger space-bandwidth product (SBWP ≥ 8.7 × 10(6)) as compared to the current state of the art in computer generated holography.

Phase unwrapping with a virtual Hartmann-Shack wavefront sensor

The use of a spatial light modulator for implementing a digital phase-shifting (PS) point diffraction interferometer (PDI) allows tunability in fringe spacing and in achieving PS without the need for mechanically moving parts. However, a small amount of detector or scatter noise could affect the accuracy of wavefront sensing. Here, a novel method of wavefront reconstruction incorporating a virtual Hartmann-Shack (HS) wavefront sensor is proposed that allows easy tuning of several wavefront sensor parameters. The proposed method was tested and compared with a Fourier unwrapping method implemented on a digital PS PDI. The rewrapping of the Fourier reconstructed wavefronts resulted in phase maps that matched well the original wrapped phase and the performance was found to be more stable and accurate than conventional methods. Through simulation studies, the superiority of the proposed virtual HS phase unwrapping method is shown in comparison with the Fourier unwrapping method in the presence of noise. Further, combining the two methods could improve accuracy when the signal-to-noise ratio is sufficiently high.

Self-calibrating common-path interferometry

A quantitative phase measuring technique is presented that estimates the object phase from a series of phase shifted interferograms that are obtained in a common-path configuration with unknown phase shifts. The derived random phase shifting algorithm for common-path interferometers is based on the Generalized Phase Contrast theory [pl. Opt.40(2), 268 (2001)10.1063/1.1404846], which accounts for the particular image formation and includes effects that are not present in two-beam interferometry. It is shown experimentally that this technique can be used within common-path configurations employing nonlinear liquid crystal materials as self-induced phase filters for quantitative phase imaging without the need of phase shift calibrations. The advantages of such liquid crystal elements compared to spatial light modulator based solutions are given by the cost-effectiveness, self-alignment, and the generation of diminutive dimensions of the phase filter size, giving unique performance advantages.

On the speckle number of interferometric velocity and distance measurements of moving rough surfaces

The minimum achievable systematic uncertainty of interferometric measurements is fundamentally limited due to speckle noise. Numerical and physical experiments, regarding the achievable measurement uncertainty of Mach-Zehnder based velocity and position sensors, are presented at the example of the laser Doppler distance sensor with phase evaluation. The results show that the measurement uncertainty depends on the number of speckles on the photo detectors. However, while the systematic uncertainty due to the speckle effect decreases, the random uncertainty due to noise from the photo detector increases with increasing speckle number. This results in a minimal total measurement uncertainty for an optimal speckle number on the photo detector, which is achieved by adjusting the aperture of the detection optics.

Recovery of wavefront from multi-shear interferograms with different tilts

An improved multi-shear algorithm is proposed to reconstruct a two-dimensional wavefront from multiple phase differences measured by lateral shearing interferograms with different tilts. The effects of the tilt errors in the wavefront are analyzed and a compensation method is developed. Unbiased estimators are added to Fourier coefficients of the phase differences to eliminate the tilt errors adaptively. The algorithm is immune to the tilt errors and the wavefront under test can be recovered exactly. Computer simulation and optical test demonstrated that the proposed algorithm has higher recovery accuracy than the existing multi-shear algorithms.

Phase retrieval with resolution enhancement by using structured illumination

In this Letter, we present referenceless phase retrieval methods with resolution enhancement. Structured illuminations with different orientations and phase shifts are generated by a spatial light modulator and are used to illuminate the specimen. The generated diffraction patterns are recorded by a CCD camera, and the phase of the wavefront is reconstructed from these patterns.