Longitudinal spatial coherence applied for surface profilometry

Deep learning based coherence holography reconstruction of 3D objects

We propose a reconstruction method for coherence holography using deep neural networks. cGAN and U-NET models were developed to reconstruct 3D complex objects from recorded interferograms. Our proposed methods, dubbed deep coherence holography (DCH), predict the non-diffracted fields or the sub-objects included in the 3D object from the captured interferograms, yielding better reconstructed objects than the traditional analytical imaging methods in terms of accuracy, resolution, and time. The DCH needs one image per sub-object as opposed to N images for the traditional sin-fit algorithm, and hence the total reconstruction time is reduced by N×. Furthermore, with noisy interferograms the DCH amplitude mean square reconstruction error (MSE) is 5×104× and 104× and phase MSE is 102× and 3×103× better than Fourier fringe and sin-fit algorithms, respectively. The amplitude peak signal to noise ratio (PSNR) is 3× and 2× and phase PSNR is 5× and 3× better than Fourier fringe and sin-fit algorithms, respectively. The reconstruction resolution is the same as sin-fit but 2× better than the Fourier fringe analysis technique.

On z-coherence of Schell-model sources carrying a prescribed astigmatic phase

A simple expression for the correlations of beams radiated by Schell-model sources carrying a prescribed astigmatic phase (cross phase) in 3D space is derived. The z-coherence of such sources upon free-space propagation is investigated in detail. It is demonstrated that the z-coherence does not decrease to zero with an increasing separation of two axial points. Our results show that the initial cross phase, coherence, and correlation state of such sources affect the distribution of the z-coherence. Furthermore, the cross phase plays a role in maintaining z-coherence, which will be useful in applications where high z-coherence is required.

Tailoring on-axis spectral density with circularly coherent light beams

The on-axis cross-spectral density (CSD) of a beam radiated by a stationary source with a circular coherence state and a Gaussian spectral density is obtained in the closed form. It is revealed that the on-axis CSD is expressed via the Laplace transform of the source's degree of coherence or the Hilbert transform of the corresponding pseudo-mode weighting function. Such relations enable efficient tailoring of the on-axis spectral density, as we show with a slew of numerical examples.

On z-coherence of beams radiated by Schell-model sources with Gaussian profile

The degree of coherence and the intensity distribution on the axis of the beam radiated by a planar partially coherent source of the Schell-model type are investigated. We present an expression for the on-axis cross-spectral density which is valid for a very general Schell-model source, with the only constraint that the intensity distribution across the source is Gaussian. Furthermore, we show that such an expression takes very simple analytical forms for several commonly used degrees of coherence of the source.

Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy

Surface reconstruction for micro-samples with large discontinuities using digital holography is a challenge. To overcome this problem, multi-incidence digital holographic profilometry (MIDHP) has been proposed. MIDHP relies on the numerical generation of the longitudinal scanning function (LSF) for reconstructing the topography of the sample with large depth and high axial resolution. Nevertheless, the method is unable to reconstruct surfaces with large gradients due to the need of: (i) high precision focusing that manual adjustment cannot fulfill and (ii) preserving the functionality of the LSF that requires capturing and processing many digital holograms. In this work, we propose a novel MIDHP method to solve these limitations. First, an autofocusing algorithm based on the comparison of shapes obtained by the LSF and the thin tilted element approximation is proposed. It is proven that this autofocusing algorithm is capable to deliver in-focus plane localization with submicron resolution. Second, we propose that wavefield summation for the generation of the LSF is carried out in Fourier space. It is shown that this scheme enables a significant reduction of arithmetic operations and can minimize the number of Fourier transforms needed. Hence, a fast generation of the LSF is possible without compromising its accuracy. The functionality of MIDHP for measuring surfaces with large gradients is supported by numerical and experimental results.

Γ-profilometry: a new paradigm for precise optical metrology

We show that the shape of a surface can be unambiguously determined from investigating the coherence function of a wave-field reflected by the surface and without the requirement of a reference wave. Spatio-temporal sampling facilitates the identification of temporal shifts of the coherence function that correspond to finite height differences of the surface. Evaluating these finite differences allows for the reconstruction of the surface using a numerical integration procedure. Spatial sampling of the coherence function is provided by a shear interferometer whereas temporal sampling is achieved by means of a Soleil-Babinet compensator. This low coherence profiling method allows to determine the shape of an object with sub-micrometer resolution and over a large unambiguity range, although it does not require any isolation against mechanical vibration. The approach therefore opens up a new avenue for precise, rugged optical metrology suitable for industrial in-line applications.

Single-shot digital multiplexed holography for the measurement of deep shapes

This work develops a single-shot holographic profilometer that enables shape characterization of discontinuous deep surfaces. This is achieved by combining hologram frequency multiplexing and an illumination technique of complex amplitude in multi-incidence angle profilometer. Object illumination is carried out from seven directions simultaneously, where the radial angular coordinates of illumination plane waves obey the geometric series. It is shown that: (i) the illumination pattern provides the required frequency separation of all object wavefronts in transverse frequency space, which is necessary for hologram demultiplexing, and (ii) numerical generation of longitudinal scanning function (LSF) is possible, which has large measurement range, high axial resolution, and small side lobes. Low side lobes of LSF and the developed multiplexed field dependent aberration compensation method are essential to minimize the negative influence of speckle noise of single-shot capture on the measurement result. The utility of the proposed method is demonstrated with experimental measurement of heights of two step-like objects.

Spatial axial shearing common-path interferometer for natural light

A spatial axial shearing interferometer is proposed to obtain a mutual coherence function representing longitudinal spatial coherence of natural light. The modulation of the quadratic phase distribution displayed on a spatial light modulator generates a spatial axial shear without a radial one. Because the optical path lengths along the optical axis on the two paths are identical, the spatial axial shear can be greater than the coherence length derived by temporal coherence. Experimental results are given to confirm that the mutual coherence function obtained by the proposed interferometer has spatial distribution expected by the relation between coherence and diffraction formula.

High space-bandwidth in quantitative phase imaging using partially spatially coherent digital holographic microscopy and a deep neural network

Quantitative phase microscopy (QPM) is a label-free technique that enables monitoring of morphological changes at the subcellular level. The performance of the QPM system in terms of spatial sensitivity and resolution depends on the coherence properties of the light source and the numerical aperture (NA) of objective lenses. Here, we propose high space-bandwidth quantitative phase imaging using partially spatially coherent digital holographic microscopy (PSC-DHM) assisted with a deep neural network. The PSC source synthesized to improve the spatial sensitivity of the reconstructed phase map from the interferometric images. Further, compatible generative adversarial network (GAN) is used and trained with paired low-resolution (LR) and high-resolution (HR) datasets acquired from the PSC-DHM system. The training of the network is performed on two different types of samples, i.e. mostly homogenous human red blood cells (RBC), and on highly heterogeneous macrophages. The performance is evaluated by predicting the HR images from the datasets captured with a low NA lens and compared with the actual HR phase images. An improvement of 9× in the space-bandwidth product is demonstrated for both RBC and macrophages datasets. We believe that the PSC-DHM + GAN approach would be applicable in single-shot label free tissue imaging, disease classification and other high-resolution tomography applications by utilizing the longitudinal spatial coherence properties of the light source.

Multi-incidence digital holographic profilometry with high axial resolution and enlarged measurement range

In this work, multi-incident digital holographic profilometry for microscale measurements is presented. This technique assembles the set of object fields from captured holograms for generation of the longitudinal scanning function (LSF). Numerical propagation is used for refocusing, and thus, the LSF can be determined at any given plane along the optical axis. The LSF takes maximum value for in focus object points, which are used to obtain full-field height distribution of the sample. This principle is the base of proposed measurement technique. Three capturing holograms strategies, which give control over the shape of the LSF, unambiguous measurement range, axial resolution, and noise immunity, are discussed. The conclusions of this work are supported by numerical and experimental results.

Relationship between the source size at the diffuser plane and the longitudinal spatial coherence function of the optical coherence microscopy system

Coherence properties of light sources are indispensable for optical coherence microscopy/tomography as they greatly influence the signal-to-noise ratio, axial resolution, and penetration depth of the system. In the present paper, we report the investigation of longitudinal spatial coherence properties of a pseudothermal light source (PTS) as a function of the laser spot size at the rotating diffuser plate. The laser spot size is varied by translating a microscope objective lens toward or away from the diffuser plate. The longitudinal spatial coherence length, which governs the axial resolution of the coherence microscope, is found to be minimum for the beam spot size of 3.5 mm at the diffuser plate. The axial resolution of the system is found to be equal to an $\sim{13}\,\,{\rm \unicode{x00B5}{\rm m}}$∼13µm at 3.5 mm beam spot size. The change in the axial resolution of the system is confirmed by performing the experiments on standard gauge blocks of a height difference of 15 µm by varying the spot size at the diffuser plate. Thus, by appropriately choosing the beam spot size at the diffuser plane, any monochromatic laser light source can be utilized to obtain high axial resolution irrespective of the source's temporal coherence length. It can provide speckle-free tomographic images of multilayered biological specimens with large penetration depth. In addition, a PTS avoids the use of any chromatic-aberration-corrected optics and dispersion-compensation mechanism unlike conventional setups.

Investigation of longitudinal spatial coherence for electromagnetic optical fields

For light fields, the coherence in longitudinal direction is governed by both the frequency spectra and angular spectra they possess. In this work, we develop and report a theoretical formulation to demonstrate the effect of the angular spectra of electromagnetic light fields in quantifying their longitudinal spatial coherence. The experimental results obtained by measuring the electromagnetic longitudinal spatial coherence and degree of cross-polarization of uniformly polarized light fields for different angular spectra validate the theoretical findings.

Effect on the longitudinal coherence properties of a pseudothermal light source as a function of source size and temporal coherence

In the present Letter, a synthesized pseudothermal light source having high temporal coherence (TC) and low spatial coherence (SC) properties is used. The longitudinal coherence (LC) properties of the spatially extended monochromatic light source are systematically studied. The pseudothermal light source is generated from two different monochromatic laser sources: He-Ne (at 632 nm) and DPSS (at 532 nm). It was found that the LC length of such a light source becomes independent of the parent laser's TC length for a large source size. For the chosen lasers, the LC length becomes constant to about 30 μm for a laser source size of ≥3.3  mm. Thus, by appropriately choosing the source size, any monochromatic laser light source depending on the biological window can be utilized to obtain high axial resolution in an optical coherence tomography (OCT) system irrespective of its TC length. The axial resolution of 650 nm was obtained using a 1.2 numerical aperture objective lens at a 632 nm wavelength. These findings pave the path for widespread penetration of pseudothermal light into existing OCT systems with enhanced performance. A pseudothermal light source with high TC and low SC properties could be an attractive alternative light source for achieving high axial resolution without needing dispersion compensation as compared to a broadband light source.

Analysis of red blood cell parameters by Talbot-projected fringes

Red blood cell (RBC) anomalies are significant symptoms for identification of health disorders and several blood diseases, which involve the modification of the parameters and biophysical characteristics of such cells. The aim of this study is to measure the three-dimensional phase information of healthy RBCs and their parameters, such as cell diameter, thickness, and hemoglobin (Hb) content, using Talbot-projected fringes. The Talbot image of linear grating is projected onto an RBC slide. The deformed grating lines due to the shape and refractive index of RBCs are recorded by a CCD camera through a 20× microscope objective. Hilbert transform is used to extract the phase image from the deformed projected grating lines. Experimentally calculated values of diameter (8.2  μm), thickness (2.7  μm), and Hb content (28.7  pg/cell) are well within the limits available in the literature. The proposed system is robust and user-friendly and performs the imaging of RBCs with high axial and lateral resolution (2.19  μm).

Generation of stochastic electromagnetic beams with complete controllable coherence

We generate a stochastic electromagnetic beam (SEB) with complete controllable coherence, that is, the coherence degree can be controlled independently along two mutually perpendicular directions. We control the coherence of the SEB by adjusting the phase modulation magnitude applied onto two crossed phase only spatial light modulators. We measure the beam's coherence properties using Young's interference experiment, as well as the beam propagation factor. It is shown that the experimental results are consistent with our theoretical predictions.

Quantitative phase imaging of biological cells using spatially low and temporally high coherent light source

In this Letter, we demonstrate quantitative phase imaging of biological samples, such as human red blood cells (RBCs) and onion cells using narrow temporal frequency and wide angular frequency spectrum light source. This type of light source was synthesized by the combined effect of spatial, angular, and temporal diversity of speckle reduction technique. The importance of using low spatial and high temporal coherence light source over the broad band and narrow band light source is that it does not require any dispersion compensation mechanism for biological samples. Further, it avoids the formation of speckle or spurious fringes which arises while using narrow band light source.

Measuring the modulus of the spatial coherence function using an error tolerant phase shifting algorithm and a continuous lateral shearing interferometer

The modulus of the degree of coherence can be derived from interference patterns either by using fringes and next neighbour operations or by using several interferograms produced through phase shifting. Here the latter approach will be followed by using a lateral shearing interferometer exploiting a diffractive grating wedge providing a linearly progressive shear. Phase shifting methods offer pixel-oriented evaluations but suffer from instabilities and drifts which is the reason for the derivation of an error immune algorithm. This algorithm will use five π/2-steps of the reference phase also for the calculation of the modulus of the coherence function.

Image contrast reduction mechanism in full-field optical coherence tomography

Correct interpretation of image contrast obtained with full-field optical coherence tomography (FFOCT) technique is required for accurate medical diagnosis applications. In this work, first, the characteristics of microscopic structures of tissue that generate the contrast in en-face tomographic image obtained with FFOCT are discussed. Then an overview is given of the parameters that affect image contrast. Finally, the contrast correction factor for correct image interpretation and the contrast limits to practical FFOCT systems are outlined.

Manipulating the spatial coherence of a laser source

An efficient method for controlling the spatial coherence has previously been demonstrated in a modified degenerate cavity laser. There, the degree of spatial coherence was controlled by changing the size of a circular aperture mask placed inside the cavity. In this paper, we extend the method and perform general manipulation of the spatial coherence properties of the laser, by resorting to more sophisticated intra-cavity masks. As predicted from the Van Cittert Zernike theorem, the spatial coherence is shown to depend on the geometry of the masks. This is demonstrated with different mask geometries: a variable slit which enables independent control of spatial coherence properties in one coordinate axis without affecting those in the other; a double aperture, an annular ring and a circular aperture array which generate spatial coherence functional forms of cosine, Bessel and comb, respectively.

Coherence holography by achromatic 3-D field correlation of generic thermal light with an imaging Sagnac shearing interferometer

We propose a new technique for achromatic 3-D field correlation that makes use of the characteristics of both axial and lateral magnifications of imaging through a common-path Sagnac shearing interferometer. With this technique, we experimentally demonstrate, for the first time to our knowledge, 3-D image reconstruction of coherence holography with generic thermal light. By virtue of the achromatic axial shearing implemented by the difference in axial magnifications in imaging, the technique enables coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light.

Interferometric spatial coherence tomography: focusing fringe contrast to planes of interest using a quasi-monochromatic structured light source

Interferograms of plane parallel optical flats are hard to interpret when acquired with coherent illumination because of the complex fringe pattern resulting from the superposition of three main contributions, namely from the reference surface and the front and back sample surfaces. We illuminate the sample by a field of high temporal and specially tailored partial spatial coherence. This limits the fringe contrast to sheets of adjustable position and thickness along the axis of the interferometer. We outline the technique and demonstrate its application together with phase shifting interferometry to extract the topography of front and back surfaces of transparent samples.

Bayesian estimation for optimized structured illumination microscopy

Structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution by reconstructing high-resolution (HR) images from low-resolution (LR) images acquired through modulation of the transfer function of the microscope. The classical implementation has a number of drawbacks, such as requiring a large number of images to be acquired and parameters to be manually set in an ad-hoc manner that have, until now, hampered its wide dissemination. Here, we present a new framework based on a Bayesian inverse problem formulation approach that enables the computation of one HR image from a reduced number of LR images and has no specific constraints on the modulation. Moreover, it permits to automatically estimate the optimal reconstruction hyperparameters and to compute an uncertainty bound on the estimated values. We demonstrate through numerical evaluations on simulated data and examples on real microscopy data that our approach represents a decisive advance for a wider use of HR microscopy through structured illumination.

Scanning monochromatic spatial low-coherence interferometer

A scanning spatial low-coherence interferometer (S-LCI), using an off-axis converging single wavelength laser beam as the probe, resembles a conventional or temporal low-coherence interferometer (T-LCI) in signal formation and data processing. However, the S-LCI is advantageous over a T-LCI with the combination of angle resolving and depth discrimination capabilities. The S-LCI is demonstrated by measuring the angle dependent phase shifts among the multiple reflections of a glass plate, with incident angles accurately scaled in the Fourier domain. The refractive index and geometric thickness of the glass plate are simultaneously produced in this one-step measurement.

Spatial coherence effect on layer thickness determination in narrowband full-field optical coherence tomography

Longitudinal spatial coherence (LSC) is determined by the spatial frequency content of an optical beam. The use of lenses with a high numerical aperture (NA) in full-field optical coherence tomography and a narrowband light source makes the LSC length much shorter than the temporal coherence length, hence suggesting that high-resolution 3D images of biological and multilayered samples can be obtained based on the low LSC. A simplified model is derived, supported by experimental results, which describes the expected interference output signal of multilayered samples when high-NA lenses are used together with a narrowband light source. An expression for the correction factor for the layer thickness determination is found valid for high-NA objectives. Additionally, the method was applied to a strongly scattering layer, demonstrating the potential of this method for high-resolution imaging of scattering media.

Vectorial coherence holography

Extension of coherence holography to vectorial regime is investigated. A technique for controlling and synthesizing optical fields with desired elements of coherence-polarization matrix is proposed and experimentally demonstrated. The technique uses two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields.

Real-time coherence holography

Coherence holography capable of real-time recording and reconstruction is proposed and experimentally demonstrated with a generic Leith-type coherence hologram. The coherence hologram is optically generated in real-time using a Mach-Zehnder interferometer and reconstructed using a Sagnac radial shearing interferometer. With this method one can create an optical field distribution with a desired spatial coherence function, and visualize the coherence function in real-time as the contrast and phase variations in an interference fringe pattern. The reconstructed image of the complex coherence function has been quantified with the Fourier transform method of fringe-pattern analysis.

Improved illumination system for spatial coherence control

An improved illumination system is proposed for creating a temporally coherent and spatially incoherent extended source to be used for spatial coherence control and reconstruction of a coherent hologram. Taking into account the fact that a rotating ground glass does not behave as an ideal Lambertian diffuser, the new illumination system tailors the directivity of the scattered lights to direct the lights efficiently into an interferometer so that a spatial coherence function can be better controlled and detected with higher fidelity. Experimental results are presented that demonstrate improved performance of the proposed system.

Phase-shift coherence holography

We propose and experimentally demonstrate a new reconstruction scheme for coherence holography using computer-generated phase-shift coherence holograms. A 3D object encoded into the spatial coherence function is reconstructed directly from a set of incoherently illuminated computer-generated holograms with numerically introduced phase shifts. Although a rotating ground glass is used to introduce spatially incoherent illumination, the phase-shifting portion of the system is simple and free from mechanically moving components.

Spatial coherence profilometry on tilted surfaces

The influence of tilted surfaces on the measurement of shape by spatial coherence profilometry is investigated. Based on theoretical analysis and experimental results, the systematic measurement error caused by surface tilt is determined. The systematic measurement error depends not only on the tilt angle but also on the parameters of the experimental setup. The theoretical analysis and the experiments show the similarities and differences between spatial coherence profilometry and white-light interferometry. We also suggest the conditions to obtain correct measurements by use of spatial coherence profilometry.

3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift

A new image reconstruction scheme for coherence holography using a modified Sagnac-type radial shearing interferometer with geometric phase shift is proposed, and the first experimental demonstration of generic Leith-type coherence holography, which reconstructs off-axis 3-D objects with depth information, is presented. The reconstructed image, represented by a coherence function, can be visualized with a controllable magnification, which opens up a new possibility for a coherence imaging microscope.

Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function

By controlling the irradiance of an extended quasimonochromatic, spatially incoherent source, an optical field is generated that exhibits spatial coherence with phase singularities, called coherence vortices. A simple optical geometry for direct visualization of coherence vortices is proposed, and the local properties and the spatial evolution of coherence vortex are experimentally investigated. To our knowledge, this is the first direct and quantitative experimental measurement of a generic coherence vortex.

Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media

The synthesis of a multiple-peak spatial degree of coherence is demonstrated. This degree of coherence enables us to scan different sample points on different altitudes simultaneously and thus decreases the acquisition time. The multipeak degree of coherence is also used for imaging through an absorbing layer with different thicknesses or different indices of refraction along the layer. All our experiments are performed with a quasi-monochromatic light source. Therefore problems of dispersion and inhomogeneous absorption are avoided. Our experimental results are presented.

Variable-coherence tomography for inverse scattering problems

We propose a technique for determining the pair-correlation function of a quasi-homogeneous medium. The method uses the variation of the spatial-coherence properties of the incident beam to generate two separate volumes of coherence where the field is correlated. Using this specially prepared beam, we reconstruct experimentally the correlation function of a scattering potential by recording the scattered intensity in only one direction.

Variable coherence tomography

By adjusting the spatial coherence of a quasi-monochromatic beam, one can control the distance between two volumes of coherence. We propose to use such a beam in a typical scattering experiment and present a method for determining the degree of spatial correlation of a quasi-homogeneous medium by recording the scattered intensity in only one direction.

Influence of longitudinal spatial coherence on the signal of a scanning interferometer

The physical conditions for observing the longitudinal spatial coherence of light of an extended thermal source in interference experiments are defined. For experimental verification of these conditions a Michelson interferometer with a longitudinal scanning mirror was used. The influence of the longitudinal spatial coherence of thermal light on a fringe envelope is demonstrated experimentally.

Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications

Giving a new physical interpretation to the principle of longitudinal coherence control, we propose an improved method for synthesizing a spatial coherence function along the longitudinal axis of light propagation. By controlling the irradiance of an extended quasi-monochromatic spatially incoherent source with a spatial light modulator, we generated a special optical field that exhibits high coherence selectively for a specific pair of points at specified locations along the axis of beam propagation. This function of longitudinal coherence control provides new possibilities for dispersion-free measurements in optical tomography and profilometry. A quantitative experimental proof of principle is presented.