Complex-valued Refractive Index Research Articles

Subgradient-projection-based stable phase-retrieval algorithm for X-ray ptychography.

X-ray ptychography is a lensless imaging technique that visualizes the nano-structure of a thick specimen which cannot be observed with an electron microscope. It reconstructs a complex-valued refractive index of the specimen from observed diffraction patterns. This reconstruction problem is called phase retrieval (PR). For further improvement in the imaging capability, including expansion of the depth of field, various PR algorithms have been proposed. Since a high-quality PR method is built upon a base PR algorithm such as ePIE, developing a well performing base PR algorithm is important. This paper proposes an improved iterative algorithm named CRISP. It exploits subgradient projection which allows adaptive step size and can be expected to avoid yielding a poor image. The proposed algorithm was compared with ePIE, which is a simple and fast-convergence algorithm, and its modified algorithm, rPIE. The experiments confirmed that the proposed method improved the reconstruction performance for both simulation and real data.

First X-ray spectral ptychography and resonant ptychographic computed tomography experiments at the SWING beamline from Synchrotron SOLEIL.

X-ray ptychography and ptychographic computed tomography have seen a rapid rise since the advent of fourth-generation synchrotrons with a high degree of coherent radiation. In addition to quantitative multiscale structural analysis, ptychography with spectral capabilities has been developed, allowing for spatial-localized multiscale structural and spectral information of samples. The SWING beamline of Synchrotron SOLEIL has recently developed a nanoprobe setup where the endstation's first spectral and resonant ptychographic measurements have been successfully conducted. A metallic nickel wire sample was measured using 2D spectral ptychography in XANES mode and resonant ptychographic tomography. From the 2D spectral ptychography measurements, the spectra of the components of the sample's complex-valued refractive index, δ and β, were extracted, integrated along the sample thickness. By performing resonance ptychographic tomography at two photon energies, 3D maps of the refractive index decrement, δ, were obtained at the Ni K-edge energy and another energy above the edge. These maps allowed the detection of impurities in the Ni wire. The significance of accounting for the atomic scattering factor is demonstrated in the calculation of electron density near a resonance through the use of the δvalues. These results indicate that at the SWING beamline it is possible to conduct state-of-the-art spectral and resonant ptychography experiments using the nanoprobe setup.

Recovery of continuous 3D refractive index maps from discrete intensity-only measurements using neural fields

Intensity diffraction tomography (IDT) refers to a class of optical microscopy techniques for imaging the three-dimensional refractive index (RI) distribution of a sample from a set of two-dimensional intensity-only measurements. The reconstruction of artefact-free RI maps is a fundamental challenge in IDT due to the loss of phase information and the missing-cone problem. Neural fields has recently emerged as a new deep learning approach for learning continuous representations of physical fields. The technique uses a coordinate-based neural network to represent the field by mapping the spatial coordinates to the corresponding physical quantities, in our case the complex-valued refractive index values. We present Deep Continuous Artefact-free RI Field (DeCAF) as a neural-fields-based IDT method that can learn a high-quality continuous representation of a RI volume from its intensity-only and limited-angle measurements. The representation in DeCAF is learned directly from the measurements of the test sample by using the IDT forward model without any ground-truth RI maps. We qualitatively and quantitatively evaluate DeCAF on the simulated and experimental biological samples. Our results show that DeCAF can generate high-contrast and artefact-free RI maps and lead to an up to 2.1-fold reduction in the mean squared error over existing methods. Producing high-quality 3D refractive index maps from 2D intensity-only measurements is a long-standing objective in computational microscopy, with many applications involving the visualization of cellular and subcellular structures. A new method can reconstruct high-contrast and artefact-free images by employing the neural fields technique, which can learn a continuous 3D representation using a neural network that maps spatial coordinates to the refractive index values.

Phase retrieval based on a total-variation-regularized Poisson model for X-ray ptychographic imaging of low-contrast objects

Hard X-ray ptychography has become an indispensable tool for observing the microscopic structure of a thick specimen. It measures diffraction patterns by scanning an X-ray beam and visualizes the complex-valued refractive index of the specimen by a computational reconstruction called phase retrieval. The quality of imaging is dependent on the used phase-retrieval algorithm, especially when the intensity of the diffraction patterns in the high-spatial-frequency range is low and/or when the spatial overlap of the illumination area is small. In this paper, a phase-retrieval algorithm, AMPAM, based on the Poisson model and total variation (TV) is proposed. It applies alternating minimization using primal-dual splitting and gradient-descent algorithms to compute the result without matrix inversion. The imaging capability of the proposed algorithm from low-dose and/or sparsely scanned data was investigated by numerical simulations. The proposed algorithm was compared with ADPr, which is the state-of-the-art algorithm based on the TV-regularized Poisson model. The results indicated that AMPAM can provide good-quality images with a computational cost 7–11 times less than ADPr. In addition, ink toner and macroporous silica particles were imaged at SPring-8 BL24XU to confirm the applicability of the algorithm to actual measurements.

Classes of Balanced Gain-and-Loss Waveguides as Non-Hermtian Potential Hierarchies

In this work, we report the construction of different classes of complex-valued refractive index landscapes, with real spectra, in the framework of the factorization method. The particular case of guiding hyperbolic-type profiles is considered in the PT- and non-PT-symmetric configurations. In both schemes, the imaginary part of the refractive index satisfies the zero-total-area condition indicating that the total transverse optical power is preserved, allowing stable propagating modes to be obtained. The spectra and the guided modal field amplitudes are obtained and their orthogonality relations are established.

Electromagnetic waves in graded-index planar waveguides

The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

A partial-dithering strategy for edge-illumination x-ray phase-contrast tomography enabled by a joint reconstruction method

Edge-illumination x-ray phase-contrast tomography (EIXPCT) is a promising imaging technology where partially opaque masks are utilized with laboratory-based x-ray sources to estimate the distribution of the complex-valued refractive index. EIXPCT resolution is mainly determined by the period of a sample mask, but can be significantly improved by a dithering technique. Here, dithering means that multiple images per tomographic view angle are acquired as the object is moved over sub-pixel distances. Drawbacks of dithering include increased data-acquisition times and radiation doses. Motivated by the flexibility in data-acquisition designs enabled by a recently developed joint reconstruction method, a novel partial-dithering strategy for EIXPCT data-acquisition is proposed. In this strategy, dithering is implemented at only a subset of the tomographic view angles. The strategy can result in spatial resolution comparable to that of the conventional full-dithering strategy, where dithering is performed at every view angle, but the acquisition time is substantially decreased. Here, the effect of dithering parameters on image resolution is explored.

On the Wave Speed of Thermal Radiation Inside and Near the Boundary of an Absorbing Material

Abstract Planck's law describes thermal radiation into vacuum from a black body in thermal equilibrium. This law can be easily adapted to describe radiation into a transparent medium with a constant refractive index, and it admits a less trivial extension to radiation into a transparent medium with a nonconstant refractive index. However, this law cannot be straightforwardly generalized to describe thermal radiation into absorbing media and, in particular, to describe thermally exited electromagnetic fields inside the radiating body itself. We first analyze Planck's law and show why it cannot be straightforwardly extended to radiation into an absorbing medium. The derivation of this law relies on the assumption that a radiated field admits decomposition into normal modes, which cannot exist in absorbing media that are characterized by a complex-valued refractive index n=n′+in″, whose imaginary part describes the rate of energy dissipation. Correspondingly, the speed of electromagnetic waves in absorbing media c=c0/n, where c0 is the speed of light in vacuum, is also complex-valued, which suggests that the conventional concept of a complex valued wave speed is not suitable for modeling thermal radiation. We demonstrate that complex-valued wave speeds adequately describe waves that carry signals, such as radio waves and laser beams. Such waves decay because they pass some of their energy to the medium. The energy absorbed by the medium is eventually reradiated, but in studies focused on the transmission of signals, the reradiated fields are ignored as noise. In order to study thermal radiation in an absorbing material, one must treat the material and the radiation together as a closed system. The energy in such a system is conserved, and its distribution between the material and radiation does not change in time. This radiation admits decomposition into normal modes, which makes it possible to extend Planck's law to radiation into absorbing materials. This paper proposes a model of thermal radiation in an absorbing medium as a closed, energy conserving system. The radiation field has normal modes that correspond to an effective speed of wave propagation. Assuming that an absorbing material and the radiation in it are in thermal equilibrium, we show that deep inside the material, the average speed of photons is given by a frequency and temperature-dependent expression c∗=c0/(1+e−ℏω/κT). While this result is independent of the material, we further show that close to the boundary of the medium, the speed of thermal radiation depends in a complex way on the refractive index and the extinction coefficient of the material, as well as the direction of propagation and the distance from the material's surface.

Analysis of Artifacts in SAR Imagery Due to Fluctuation in Refractive Index

In this paper, we analyze the artifacts in synthetic aperture radar imagery due to fluctuations in the refractive index of the atmosphere, specifically due to absorption and variations in speed of light. SAR image reconstruction algorithms ignore absorption and assume unit refractive index. We assume a complex-valued refractive index for the atmosphere and analyze the errors in reconstructed images due to absorption and fluctuations in the speed of electromagnetic (EM) waves. Absorption leads to amplitude errors, which can be viewed as a filtering error. This error can be characterized as systematic variations in the ground reflectivity function. Fluctuations in the speed of EM waves lead to phase errors, which manifest as positioning errors or geometric distortions in synthetic aperture radar (SAR) imagery. We perform numerical simulations to verify our analysis.

Light scattering from volcanic-sand particles in deposited and aerosol form

The light-scattering properties of volcanic sand collected in Iceland are studied here to characterize the sand particles and develop a reference for future remote-sensing observations. While such sand is common in Iceland, the smaller-size fraction can be readily transported by winds and found in the atmosphere at distant locations. The sand appears dark when deposited on a surface due to the high optical absorption of the material. Therefore, atmospheric regions containing such particles during a dust storm may absorb sunlight considerably, causing redistribution of solar energy. Here, we measure the angular scattered-light intensity and degree of linear polarization from the sand. This is done with two experimental apparatuses, the Cosmic Dust Laboratory (CoDuLab) at the Institute de Astrofísica de Andalucía (IAA) and the goniospectropolarimeter (FIGIFIGO) at the Finnish Geospatial Research Institute (FGI). Two scattering-scenarios of practical interest for remote-sensing applications are considered: (1) single sand-particles suspended in aerosol as an optically thin cloud, and (2) the same particles deposited on a substrate. We also model the measurements with the discrete dipole approximation to estimate the complex-valued refractive index m, where we find that m ≈ 1.6 + 0.01i at λ = 647 nm. Lastly, we present a comparative analysis of the polarimetric response of the sand particles with that reported in the literature for carbon-soot, another highly absorbing atmospheric contaminant.

Retrieval of the complex-valued refractive index of germanium near the M4,5 absorption edge

The complex-valued index of refraction of germanium in the extreme ultraviolet (XUV) is measured by multi-angle reflectance of synchrotron radiation. The resulting index of refraction is higher resolution than previously measured values. It reveals new structures attributed to transitions from the 3d-core orbitals to the Σ5,2c and the X5,2c conduction bands. Additionally, it is shown that the problem of total external reflection, which renders multi-angle reflectance measurements insensitive to the complex-valued refractive index at grazing incidence, can be overcome by employing measurements at angles of incidence away from the critical angle.

Properties of a Joint Reconstruction Method for Edge-Illumination X-Ray Phase-Contrast Tomography.

Edge illumination X-ray phase-contrast tomography (EIXPCT) is an emerging technique for estimating the complex-valued refractive index distribution of an imaged object. A conventional EIXPCT system requires measurement of multiple images at each tomographic view angle, leading to prolonged data acquisition times. Recently, a joint reconstruction (JR) method has been developed to enable single-shot EIXPCT imaging without restrictive assumptions related to the object, imaging geometry or hardware. In the JR method, estimates of the refractive index distribution are recast as the solution to a nonlinear optimization problem. While a preliminary study has demonstrated the potential usefulness of the JR method, its numerical properties remain largely unexplored. In this project, the convexity, cross-talk properties and noise properties of the JR method are investigated.

Topological optimality condition for the identification of the center of an inhomogeneity

The inverse scattering problem for inhomogeneous media is considered within the topology optimization framework. Varying the complex-valued refractive index we derive a zero-order necessary optimality condition in minimizing the L2 misfit cost functional of the far-field measurement. The topology asymptotic expansion of the optimality condition leads to an imaging operator, which is used to identify the center of the unknown inhomogeneity using few far-field measurements. Numerical tests show high precision and stability in the reconstruction using our optimality condition based imaging both in two and three dimensions.

Single-shot edge illumination x-ray phase-contrast tomography enabled by joint image reconstruction.

Edge illumination x-ray phase-contrast tomography (EIXPCT) is an emerging x-ray phase-contrast tomography technique for reconstructing the complex-valued x-ray refractive index distribution of an object. Conventional image reconstruction approaches for EIXPCT require multiple images to be acquired at each tomographic view angle. This contributes to prolonged data-acquisition times and elevated radiation doses, which can hinder in vivo applications. In this work, a new "single-shot" method is proposed for joint reconstruction (JR) of the real and imaginary-valued components of the refractive index distribution from a tomographic data set that contains only a single image acquired at each view angle. The proposed method is predicated on a nonlinear formulation of the inverse problem that is solved by using a gradient-based optimization method. The method is validated and investigated using computer-simulated and experimental EIXPCT data sets.

Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to $0$. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

Hard X-ray phase-contrast tomography of non-homogeneous specimens: grating interferometry versus propagation-based imaging.

X-ray phase-contrast imaging is an effective approach to drastically increase the contrast and sensitivity of microtomographic techniques. Numerous approaches to depict the real part of the complex-valued refractive index of a specimen are nowadays available. A comparative study using experimental data from grating-based interferometry and propagation-based phase contrast combined with single-distance phase retrieval applied to a non-homogeneous sample is presented (acquired at beamline ID19-ESRF). It is shown that grating-based interferometry can handle density gradients in a superior manner. The study underlines the complementarity of the two techniques for practical applications.

Elementary signals in ptychography.

Ptychographic imaging has gained popularity for its high resolving power and sensitivity as well as for its ability to map simultaneously the sample's complex-valued refractive index and the illumination. Yet, despite significant progress that allows for reliable practical implementation, some of the technique's fundamentals remain poorly understood, and oftentimes successful data acquisition is either overly conservative or relies more on experimenters experience than on rational data acquisition strategies. Here, we propose a theoretical framework of ptychography, which is based on Gabor's notion of decomposition into elementary signals and the concept of frames. We demonstrate how this framework can straightforwardly be used to derive sampling requirements or to provide arguments on how to optimize the ptychographic scan. More generally, our theoretical framework can serve as a bridge between the experimental technique and the rich and well established mathematical disciplines of wavelets decomposition and spectrogram analysis.

Image reconstruction exploiting object sparsity in boundary-enhanced X-ray phase-contrast tomography

Propagation-based X-ray phase-contrast tomography (PCT) seeks to reconstruct information regarding the complex-valued refractive index distribution of an object. In many applications, a boundary-enhanced image is sought that reveals the locations of discontinuities in the real-valued component of the refractive index distribution. We investigate two iterative algorithms for few-view image reconstruction in boundary-enhanced PCT that exploit the fact that a boundary-enhanced PCT image, or its gradient, is often sparse. In order to exploit object sparseness, the reconstruction algorithms seek to minimize the l(1)-norm or TV-norm of the image, subject to data consistency constraints. We demonstrate that the algorithms can reconstruct accurate boundary-enhanced images from highly incomplete few-view projection data.

Boundary-enhanced region-of-interest image reconstruction in propagation-based x-ray phase-contrast tomography.

Propagation-based phase-contrast tomography is an imaging method that can reconstruct the distribution of the three-dimensional complex-valued refractive index distribution of an object. In many applications, a boundary-enhanced image is sought that reveals the locations of boundaries in the refractive index distribution. In this letter, we demonstrate that within certain regions-of-interest, the three-dimensional Laplacian of the real-valued component of the refractive index distribution can be reconstructed from knowledge of truncated phase-contrast projection data. The proposed reconstruction algorithm is demonstrated by use of experimental measurement data.

Refractive index and wave vector in passive or active media

Materials that exhibit loss or gain have a complex-valued refractive index $n$. Nevertheless, when considering the propagation of optical pulses, using a complex $n$ is generally inconvenient---hence the standard choice of real-valued refractive index, i.e., ${n}_{s}=\text{Re}(\sqrt{{n}^{2}})$. However, an analysis of pulse propagation based on the second-order wave equation shows that use of ${n}_{s}$ results in a wave vector different to that actually exhibited by the propagating pulse. In contrast, an alternative definition ${n}_{c}=\sqrt{\text{Re}({n}^{2})}$, always correctly provides the wave vector of the pulse. Although for small loss the difference between the two is negligible, in other cases it is significant; it follows that phase and group velocities are also altered. This result has implications for the description of pulse propagation in near resonant situations, such as those typical of metamaterials with negative (or otherwise exotic) refractive indices.