Complex Refractive Index Distribution Research Articles

Single-shot intensity diffraction tomography via polarization-multiplexed LED illumination.

We present a single-shot intensity diffraction tomography method via polarization-multiplexed LED illumination. Three LED elements covered with 0°, 45°, and 135° linear polarizers, respectively, are lit up simultaneously to illuminate the sample with illumination angles matching the numerical aperture of the objective. The scattering field of the sample is recorded on a single intensity image with a polarization sensor, and three intensity images corresponding to the three LED elements are decoupled from the intensity image by using a pre-calibrated intensity transform matrix. After a slice-wise deconvolution procedure, the 3D complex refractive index distribution of the sample can be recovered. To demonstrate the performance of our method, we perform experiments on a USAF absorption resolution target, rat hippocampal cell lines, and spongy spicule. These imaging results show that our method can achieve 3D tomography for various biomedical samples with a near incoherent diffraction-limited lateral resolution of 690 nm and an axial resolution of 4.68 μm.

Continuous-wave terahertz in-line holographic diffraction tomography with the scattering fields reconstructed by a physics-enhanced deep neural network

Diffraction tomography is a promising, quantitative, and nondestructive three-dimensional (3D) imaging method that enables us to obtain the complex refractive index distribution of a sample. The acquisition of the scattered fields under the different illumination angles is a key issue, where the complex scattered fields need to be retrieved. Presently, in order to develop terahertz (THz) diffraction tomography, the advanced acquisition of the scattered fields is desired. In this paper, a THz in-line digital holographic diffraction tomography (THz-IDHDT) is proposed with an extremely compact optical configuration and implemented for the first time, to the best of our knowledge. A learning-based phase retrieval algorithm by combining the physical model and the convolution neural networks, named the physics-enhanced deep neural network (PhysenNet), is applied to reconstruct the THz in-line digital hologram, and obtain the complex amplitude distribution of the sample with high fidelity. The advantages of the PhysenNet are that there is no need for pretraining by using a large set of labeled data, and it can also work for thick samples. Experimentally with a continuous-wave THz laser, the PhysenNet is first demonstrated by using the thin samples and exhibits superiority in terms of imaging quality. More importantly, with regard to the thick samples, PhysenNet still works well, and can offer 2D complex scattered fields for diffraction tomography. Furthermore, the 3D refractive index maps of two types of foam sphere samples are successfully reconstructed by the proposed method. For a single foam sphere, the relative error of the average refractive index value is only 0.17%, compared to the commercial THz time-domain spectroscopy system. This demonstrates the feasibility and high accuracy of the THz-IDHDT, and the idea can be applied to other wavebands as well.

Mathematical model of radiation scattering on a quasi – periodic structure in an optical fiber

The paper considers a quasi – periodic microheterogeneity structure formed under the effect of optical fiber melting, which can be used as a sensitive element for an optical sensor based on a Fabry – Perot interferometer or an optical radiation scatterer with improved flexibility and strength parameters. Mathematical modeling of the process of transmission and reflection of radiation on the microcavities of the intrafiber structure will make it possible to predict the characteristics of the developed sensors. The simulation was carried out in the COMSOL Multyphysics simulation package. A section of a single – mode SMF-28e fiber with microinhomogeneities located in the core, the dimensions of which were measured in the course of an empirical study, is considered. To solve the problem, a system of equations was compiled that describes the propagation of a plane electromagnetic wave in matter. The results obtained showed that the calculated reflection spectrum qualitatively coincides with the measured value, but does not describe it with high accuracy. This may be due to the fact that the model does not take into account intermode interference in the fiber, and does not have an introduced melt zone around microcavities, since such a melt zone has a complex refractive index distribution and composition

Transforming Space with Non-Hermitian Dielectrics.

Coordinate transformations are a versatile tool to mold the flow of light, enabling a host of astonishing phenomena such as optical cloaking with metamaterials. Moving away from the usual restriction that links isotropic materials with conformal transformations, we show how nonconformal distortions of optical space are intimately connected to the complex refractive index distribution of an isotropic non-Hermitian medium. Remarkably, this insight can be used to circumvent the material requirement of working with refractive indices below unity, which limits the applications of transformation optics. We apply our approach to design a broadband unidirectional dielectric cloak, which relies on nonconformal coordinate transformations to tailor the non-Hermitian refractive index profile around a cloaked object. Our insights bridge the fields of two-dimensional transformation optics and non-Hermitian photonics.

Quantitative super-resolution solid immersion microscopy via refractive index profile reconstruction

Solid Immersion (SI) microscopy is a modern imaging modality that overcomes the Abbe diffraction limit and offers novel applications in various branches of visible, infrared, terahertz, and millimeter-wave optics. Despite the widespread use, SI microscopy usually results in qualitative imaging. Indeed, it presents only the raw distributions (in the image plane) of the backscattered field intensity, while unlocking the information about the physical properties of an imaged object, such as its complex refractive index (RI) distribution, requires resolving the inverse problem and remains a daunting task. In this paper, a method for resolving the SI microscopy inverse problem is developed, capable of reconstructing the RI distribution at the object imaging plane with subwavelength spatial resolution, while performing only intensity measurements. The sample RI is retrieved via minimization of the error function that characterizes discrepancy between the experimental data and the predictions of analytical model. This model incorporates all the key features of the electromagnetic-wave interaction with the SI lens and an imaged object, including contributions of the evanescent and ordinary-reflected waves, as well as effects of light polarization and wide beam aperture. The model is verified numerically, using the finite-element frequency-domain method, and experimentally, using the in-house reflection-mode continuous-wave terahertz SI microscope. Spatial distributions of the terahertz RIs of different low-absorbing optical materials and highly absorbing biological objects were studied and compared to a priori known data to demonstrate the potential of the novel SI microscopy modality. Given the linear nature of the Maxwell’s equations, the developed method can be applied for subwavelength-resolution SI microscopy at other spectral ranges.

Resolution-enhanced intensity diffraction tomography in high numerical aperture label-free microscopy

We propose label-free and motion-free resolution-enhanced intensity diffraction tomography (reIDT) recovering the 3D complex refractive index distribution of an object. By combining an annular illumination strategy with a high numerical aperture (NA) condenser, we achieve near-diffraction-limited lateral resolution of 346 nm and axial resolution of 1.2 μm over 130 μ m × 130 μ m × 8 μ m volume. Our annular pattern matches the system’s maximum NA to reduce the data requirement to 48 intensity frames. The reIDT system is directly built on a standard commercial microscope with a simple LED array source and condenser lens adds-on, and promises broad applications for natural biological imaging with minimal hardware modifications. To test the capabilities of our technique, we present the 3D complex refractive index reconstructions on an absorptive USAF resolution target and Henrietta Lacks (HeLa) and HT29 human cancer cells. Our work provides an important step in intensity-based diffraction tomography toward high-resolution imaging applications.

Capabilities and limitations of the single-particle extinction and scattering method for estimating the complex refractive index and size-distribution of spherical and non-spherical submicron particles

Nondestructive inline measurements of submicron-sized particles and gas bubbles are required in many fields of study, including environmental, industrial, and biomedical research. The single particle extinction and scattering (SPES) method is a promising technique for this purpose because of its sensitivity to both the complex refractive index and the size of individual particles. Herein an original modification of the SPES instrument that suppresses measurement bias and background noise is presented. In addition, relevant theory and a practical algorithm for data analysis are presented. The inverse scattering problem of estimating the physical characteristics of a particle from SPES data using Bayesian theory for model selection is formulated, and the performance of the algorithm using simulated data for spherical and non-spherical particles is evaluated. The results suggest that a cluster of SPES data points for an ensemble of particles contains sufficient information to retrieve the complex refractive index and volume equivalent size-distribution of the particles, regardless of particle shape. The Bayesian inversion method also allows incorporation of informative data from other analytical methods (e.g., electron microscope, upstream size-sorting device) through the model and parameter priors. As practical examples, the SPES method was used to constrain the complex refractive index and size distribution of water-insoluble aerosols contained in rainwater. The results suggest that the SPES method itself or its use in combination with other conventional analytical methods can quantify the complex refractive index and size distribution of almost any type of submicron particle (including gas bubbles) in environmental, industrial, and biomedical samples.

Phase-contrast tomography from a series of single-shot measurements using a deterministic phase-retrieval method in X-ray imaging

An X-ray tomographic imaging method is proposed that is capable of reconstructing the complex refractive index of an object from a data set that involves only a single-shot image obtained at each tomographic view angle. This method is based on a deterministic phase retrieval method from a single diffraction intensity by use of an aperture array, and so the real and imaginary components, which are independent of each other, of the complex refractive index distribution can be reconstructed without object’s assumptions about negligible absorption or phase-attenuation duality, which have been frequently required in the other single-shot phase-retrieval methods for tomography. The validity of the method is shown in computer-simulated examples of three-dimensional object reconstruction.

Simplified tomographic diffractive microscopy for axisymmetric samples

Tomographic diffractive microscopy exhibits intrinsic features making it a method of choice for 3D high-resolution label-free imaging. However, these results are achieved at the cost of a heavy data acquisition/reconstruction process. This drawback can be circumvented for certain classes of samples. For example, axisymmetric samples, like optical or textile fibers, present geometrical properties that can be advantageously used to speed-up the acquisition process. We propose to take benefit of these properties to allow for full reconstruction of axisymmetric samples’ complex refractive index distribution, using four approaches, adapted to 3D samples. We applied the proposed reconstruction scheme, based on a numerical rotation of data, to both simulated and experimental data sets.

Weighted reconstruction of three-dimensional refractive index in interferometric tomography.

Interferometric tomography can reconstruct 3D refractive-index distributions through phase-shift measurements for different beam angles. To reconstruct a complex refractive-index distribution, many projections along different directions are required. For the purpose of increasing the number of the projections, we earlier proposed a beam-angle-controllable interferometer with mechanical stages; however, the quality of reconstructed distribution by conventional algorithms was poor because the background fringes cannot be precisely controlled. To improve the quality, we propose a weighted reconstruction algorithm that can consider projection errors. We demonstrate the validity of the weighted reconstruction through simulations and a reconstruction from experimental data for three candle flames.

Multiband infrared inversion for low-concentration methane monitoring in a confined dust-polluted atmosphere

A multiband infrared diagnostic (MBID) method for methane emission monitoring in limited underground environments was presented considering the strong optical background of gas/solid attenuation. Based on spatial distribution of aerosols and complex refractive index of dust particles, forward calculations were carried out with/without methane to obtain the spectral transmittance through the participating atmosphere in a mine roadway. Considering the concurrent attenuation and absorption behavior of dust and gases, four infrared wavebands were selected to retrieve the methane concentration combined with a stochastic particle swarm optimization (SPSO) algorithm. Inversion results prove that the presented MBID method is robust and effective in identifying methane at concentrations of 0.1% or even lower with inversed relative error within 10%. Further analyses illustrate that the four selected wavebands are indispensable, and the MBID method is still valid with transmission signal disturbance in a conventional dust-polluted atmosphere under mechanized mining condition. However, the effective detection distance should be limited within 50 m to ensure inversed relative error less than 5% at 1% methane concentration.

3D differential phase contrast microscopy.

We demonstrate 3D phase and absorption recovery from partially coherent intensity images captured with a programmable LED array source. Images are captured through-focus with four different illumination patterns. Using first Born and weak object approximations (WOA), a linear 3D differential phase contrast (DPC) model is derived. The partially coherent transfer functions relate the sample's complex refractive index distribution to intensity measurements at varying defocus. Volumetric reconstruction is achieved by a global FFT-based method, without an intermediate 2D phase retrieval step. Because the illumination is spatially partially coherent, the transverse resolution of the reconstructed field achieves twice the NA of coherent systems and improved axial resolution.

Deterministic versus stochastic level-set regularization in nonlinear phase contrast tomography

A new nonlinear level-set regularization method to reconstruct the complex refractive index distribution with in-line phase contrast tomography measurements is presented under the assumption that the index is piecewise constant. The nonlinear iterative approach is based on the Fréchet derivative of the intensity recorded at a single propagation distance and for several projection angles. The algorithm is successfully applied to a multi-material object for several noise levels. Better reconstruction results are achieved with a stochastic perturbation of the level-set function. This evolution corresponds to a stochastic evolution of the shape of the reconstructed regions. The reconstruction errors can be further decreased with topological derivatives. The different algorithms are tested on various multi-material objects.

Monomorphous decomposition method and its application for phase retrieval and phase-contrast tomography

We show that an arbitrary spatial distribution of complex refractive index inside an object can be exactly represented as a sum of two "monomorphous" complex distributions, i.e. the distributions with the ratios of the real part to the imaginary part being constant throughout the object. A priori knowledge of constituent materials can be used to estimate the global lower and upper boundaries for this ratio. This approach can be viewed as an extension of the successful phase-retrieval method, based on the Transport of Intensity equation, that was previously developed for monomorphous (homogeneous) objects, such as e.g. objects consisting of a single material. We demonstrate that the monomorphous decomposition can lead to more stable methods for phase retrieval using the Transport of Intensity Equation. Such methods may find application in quantitative in-line phase-contrast imaging and phase-contrast tomography.

Simultaneous retrieval of the complex refractive index and particle size distribution

A secondary optimization technique is proposed that allows the complex refractive index and particle size distribution (PSD) to be retrieved simultaneously by using the diffuse transmittance (T), diffuse reflectance (R), and collimated transmittance (T(c)) of a 1-D spherical particle systems as measured values. In the proposed method, two 1-D experimental samples of different thicknesses were exposed to continuous wave lasers of two different wavelengths. First, T, R, and T(c) were calculated by solving the radiative transfer equation. Then, the complex refractive index and PSDs were retrieved simultaneously by applying the inversion technique, quantum particle swarm optimization. However, the estimated results of the PSDs proved to be inaccurate. Hence, a secondary optimization was performed to improve the accuracy of the PSDs on the basis of the first optimization process. The results showed that the proposed technique can estimate the complex refractive index and particle size distribution accurately.

High-resolution in vivo imaging of mouse brain through the intact skull

Multiphoton microscopy is the current method of choice for in vivo deep-tissue imaging. The long laser wavelength suffers less scattering, and the 3D-confined excitation permits the use of scattered signal light. However, the imaging depth is still limited because of the complex refractive index distribution of biological tissue, which scrambles the incident light and destroys the optical focus needed for high resolution imaging. Here, we demonstrate a wavefront-shaping scheme that allows clear imaging through extremely turbid biological tissue, such as the skull, over an extended corrected field of view (FOV). The complex wavefront correction is obtained and directly conjugated to the turbid layer in a noninvasive manner. Using this technique, we demonstrate in vivo submicron-resolution imaging of neural dendrites and microglia dynamics through the intact skulls of adult mice. This is the first observation, to our knowledge, of dynamic morphological changes of microglia through the intact skull, allowing truly noninvasive studies of microglial immune activities free from external perturbations.

Stability analysis of FD-BPM applied in high power semiconductor laser models

We perform a stability analysis of the Crank-Nicolson based finite difference beam propagation algorithm for a medium with a complex refractive index distribution. We use typical waveguide parameters that correspond to an InGaAs–AlGaAs epitaxy. The results show that selecting the fundamental mode propagation constant as the BPM reference value, even though it allows reducing the errors, makes the FD-BPM scheme unstable when attempting to retrace the propagation.

Reconstruction of the complex refractive index in nonlinear phase contrast tomography

A new nonlinear method to reconstruct the complex refractive index distribution with in-line phase tomography measurements is presented. The inverse problem is regularized with the Tikhonov smoothing norm of the index. The original nonlinear iterative approach is based on the Fréchet derivative of the intensity recorded at a single propagation distance and on a Simultaneous Algebraic Reconstruction technique. The reconstruction method requires no a priori knowledge about the materials. The algorithm is successfully applied to some simulated data with noise.

Aerosol seasonal variations over urban–industrial regions in Ukraine according to AERONET and POLDER measurements

Abstract. The paper presents an investigation of aerosol seasonal variations in several urban–industrial regions in Ukraine. Our analysis of seasonal variations of optical and physical aerosol parameters is based on the sun-photometer 2008–2013 data from two urban ground-based AERONET (AErosol RObotic NETwork) sites in Ukraine (Kyiv, Lugansk) as well as on satellite POLDER instrument data for urban–industrial areas in Ukraine. We also analyzed the data from one AERONET site in Belarus (Minsk) in order to compare with the Ukrainian sites. Aerosol amount and optical depth (AOD) values in the atmosphere columns over the large urbanized areas like Kyiv and Minsk have maximum values in the spring (April–May) and late summer (August), whereas minimum values are observed in late autumn. The results show that fine-mode particles are most frequently detected during the spring and late summer seasons. The analysis of the seasonal AOD variations over the urban–industrial areas in the eastern and central parts of Ukraine according to both ground-based and POLDER data exhibits the similar traits. The seasonal variation similarity in the regions denotes the resemblance in basic aerosol sources that are closely related to properties of aerosol particles. The behavior of basic aerosol parameters in the western part of Ukraine is different from eastern and central regions and shows an earlier appearance of the spring and summer AOD maxima. Spectral single-scattering albedo, complex refractive index and size distribution of aerosol particles in the atmosphere column over Kyiv have different behavior for warm (April–October) and cold seasons. The seasonal features of fine and coarse aerosol particle behavior over the Kyiv site were analyzed. A prevailing influence of the fine-mode particles on the optical properties of the aerosol layer over the region has been established. The back-trajectory and cluster analysis techniques were applied to study the seasonal back trajectories and prevailing directions of the arrived air mass for the Kyiv and Minsk sites.

Study on Aerosol Model and Sources at Zhoushan, China Using Sun-sky Photometer Observation

Aerosol models are widely used in satellite remote sensing to derived aerosol mode from aerosol optical and microphysical properties. One year of ground-based aerosol remote sensing observations were carried out using sun-sky radiometer measurements in Zhoushan (122.1897E, 29.9944N), Zhejiang Province, Eastern China. At the same time column Aerosol Optical Depth (AOD), Ångström exponent (AE), Single Scattering Albedo (SSA), asymmetry factor (g), complex refractive index and column aerosol volume spectral distribution were retrieved by mature code as well as some procedures, such as radiometer calibration, cloud screening and data selection strategies. Aerosol size parameters were separated as fine effective radius (rvf) and coarse effective radius (rvc) due to the column aerosol size distribution is generally bimodal lognormal distribution. The relationship between these parameters and effective radius was shown and analyzed. It is shown that aerosol in Zhoushan is urban-industrial type dominate, mixed with marine aerosol and mineral dust aerosol. As a result, this study showed a part of aerosol comes from mainland industrial areas by using the backward trajectory model.