Optical Goniometer Paired with Digital Monte Carlo Twin to Determine the Optical Properties of Turbid Media.

We present a goniometer designed for capturing spectral and angular-resolved data from scattering and absorbing media. The experimental apparatus is complemented by a comprehensive Monte Carlo simulation, meticulously replicating the radiative transport processes within the instrument's optical components and simulating scattering and absorption across arbitrary volumes. Consequently, we were able to construct a precise digital replica, or "twin", of the experimental setup. This digital counterpart enabled us to tackle the inverse problem of deducing optical parameters such as absorption and scattering coefficients, along with the scattering anisotropy factor from measurements. We achieved this by fitting Monte Carlo simulations to our goniometric measurements using a Levenberg-Marquardt algorithm. Validation of our approach was performed using polystyrene particles, characterized by Mie scattering, supplemented by a theoretical analysis of algorithmic convergence. Ultimately, we demonstrate strong agreement between optical parameters derived using our novel methodology and those obtained via established measurement protocols.

Single scattering models for radiative transfer of isotropic and cone-shaped light sources in fog.

The simulation of rare edge cases such as adverse weather conditions is the enabler for the deployment of the next generation of autonomous drones and vehicles into conditions where human operation is error-prone. Therefore, such settings must be simulated as accurately as possible and be computationally efficient, so to allow the training of deep learning algorithms for scene understanding, which require large-scale datasets disallowing extensive Monte Carlo simulations. One computationally-expensive step is the simulation of light sources in scattering media, which can be tackled by the radiative transfer equation and approximated by analytical solutions in the following. Traditionally, a single scattering event is assumed for fog rendering, since it is the dominant effect for relatively low scattering media. This assumption allows us to present an improved solution to calculate the so called air-light integral that can be evaluated fast and robustly for an isotropic point source in homogeneous media. Additionally, the solution is extended for a cone-shaped source and implemented in a computer vision rendering pipeline fulfilling computational restrictions for deep learning uses. All solutions can handle arbitrary azimuthally symmetric phase functions and were tested with the Henyey-Greenstein phase function and an advection fog phase function calculated from a particle distribution using Mie's theory. The used approximations are validated through extensive Monte Carlo simulations and the solutions are used to augment good weather images towards inclement conditions with focus on visible light sources, so to provide additional data in such hard-to-collect settings.

Improved topographic reconstruction of turbid media in the spatial frequency domain including the determination of the reduced scattering and absorption coefficients.

The separation of scattering and absorption is of great importance for studying the radiative transfer in turbid media. Obtaining the corresponding coefficients for non-flat objects is difficult and needs special consideration. Building on our previous work [J. Opt. Soc. Am. A39, 1823 (2022)JOAOD60740-323210.1364/JOSAA.464007], we present an approach that takes the changing incident and detection angles relative to the surface normal of curved surfaces into account to improve the determination of the reduced scattering and absorption coefficients with measurements in the spatial frequency domain (SFD). The optical coefficients are reconstructed using a pre-calculated lookup table generated with Monte Carlo simulations on graphical processing units. With the obtained values, the error in the captured surface geometry of the object, which is due to the volume scattering, is compensated and reduced by 1 order of magnitude for measurements in the SFD. Considering the approximate surface geometry, the absorption and reduced scattering are accurately resolved for moderate object curvatures, with very low dependence on the tilt angle. In contrast to models that only correct the amplitudes of the SFD signal, our approach, in addition to the optical properties, predicts the phase values correctly, which is the reason why it can be used to correct the surface geometry.

Radiative transfer equation-based color prediction and color adjustment strategies.

This paper discusses different strategies for color prediction and matching. Although many groups use the two-flux model (i.e., the Kubelka-Munk theory or its extensions), we introduce a solution of the P N approximation for the radiative transfer equation (RTE) with modified Mark boundaries for the prediction of the transmittance and reflectance of turbid slabs with or without a glass layer on top. To demonstrate the capabilities of our solution, we have presented a way to prepare samples with different scatterers and absorbers where we can control and predict the optical properties and discussed three color-matching strategies: the approximation of the scattering and absorption coefficient, the adjustment of the reflectance, and the direct matching of the color valueL ∗ a ∗ b ∗.

Continuous Sizing and Identification of Microplastics in Water.

The pollution of the environment with microplastics in general, and in particular, the contamination of our drinking water and other food items, has increasingly become the focus of public attention in recent years. In order to better understand the entry pathways into the human food chain and thus prevent them if possible, a precise characterization of the particles concerning their size and material is indispensable. Particularly small plastic particles pose a special challenge since their material can only be determined by means of large experimental effort. In this work, we present a proof of principle experiment that allows the precise determination of the plastic type and the particle size in a single step. The experiment combines elastic light scattering (Mie scattering) with inelastic light scattering (Raman scattering), the latter being used to determine the plastic type. We conducted Monte Carlo simluations for the elastically scattered light for different kinds of plastics in a microfluidic cuvette which we could reproduce in the experiment. We were able to measure the Raman signals for different microplastics in the same measurement as the elastically scattered light and thereby determine their material. This information was used to select the appropriate Monte Carlo simulation data and to assign the correct particle size to different materials with only one calibration measurement.

Confocal Laser Scanning Microscope Imaging of Custom-Made Multi-Cylinder Phantoms: Theory and Experiment.

In this work, the image formation in a confocal laser scanning microscope (CLSM) is investigated for custom-made multi-cylinder phantoms. The cylinder structures were fabricated using 3D direct laser writing and consist of parallel cylinders with radii of 5 and 10 µm for the respective multi-cylinder phantom, with overall dimensions of about 200×200×200 µm3. Measurements were performed for different refractive index differences and by varying other parameters of the measurement system, such as pinhole size or numerical aperture (NA). For theoretical comparison, the confocal setup was implemented in an in-house developed tetrahedron-based and GPU-accelerated Monte Carlo (MC) software. The simulation results for a cylindrical single scatterer were first compared with the analytical solution of Maxwell's equations in two dimensions for prior validation. Subsequently, the more complex multi-cylinder structures were simulated using the MC software and compared with the experimental results. For the largest refractive index difference, i.e., air as the surrounding medium, the simulated and measured data show a high degree of agreement, with all the key features of the CLSM image being reproduced by the simulation. Even with a significant reduction in the refractive index difference by the use of immersion oil to values as low as 0.005, a good agreement between simulation and measurement was observed, particularly with respect to the increase in penetration depth.

Focusing Coherent Light through Volume Scattering Phantoms via Wavefront Shaping.

Manipulating the wavefront of coherent light incident on scattering media to enhance the imaging depth, sensitivity, and resolution is a common technique in biomedical applications. Local phase variations cause changes in the interference and can be used to create a focus inside or behind a scattering medium. We use wavefront shaping (WFS) to force constructive interference at an arbitrary location. The amount of light transmitted into a given region strongly depends on the scattering and absorption characteristics. These are described by their respective coefficients µs and µa and the scattering phase function. Controlling the scattering and absorption coefficients, we study the behavior of wavefront shaping and the achievable intensity enhancement behind volume scattering media with well-defined optical properties. The phantoms designed in this publication are made of epoxy resin. Into these epoxy matrices, specific amounts of scattering and absorbing particles, such as titanium dioxide pigments and molecular dyes, are mixed. The mixture obtained is filled into 3D-printed frames of various thicknesses. After a precise fabrication procedure, an integrating sphere-based setup characterizes the phantoms experimentally. It detects the total hemispherical transmission and reflection. Further theoretical characterization is performed with a newly developed hybrid PN method. This method senses the flux of light into a particular angular range at the lower boundary of a slab. The calculations are performed without suffering from ringing and fulfill the exact boundary conditions there. A decoupled two-path detection system allows for fast optimization as well as sensitive detection. The measurements yield results that agree well with the theoretically expected behavior.

Generic and Model-Based Calibration Method for Spatial Frequency Domain Imaging with Parameterized Frequency and Intensity Correction.

Spatial frequency domain imaging (SFDI) is well established in biology and medicine for non-contact, wide-field imaging of optical properties and 3D topography. Especially for turbid media with displaced, tilted or irregularly shaped surfaces, the reliable quantitative measurement of diffuse reflectance requires efficient calibration and correction methods. In this work, we present the implementation of a generic and hardware independent calibration routine for SFDI setups based on the so-called pinhole camera model for both projection and detection. Using a two-step geometric and intensity calibration, we obtain an imaging model that efficiently and accurately determines 3D topography and diffuse reflectance for subsequently measured samples, taking into account their relative distance and orientation to the camera and projector, as well as the distortions of the optical system. Derived correction procedures for position- and orientation-dependent changes in spatial frequency and intensity allow the determination of the effective scattering coefficient µs' and the absorption coefficient µa when measuring a spherical optical phantom at three different measurement positions and at nine wavelengths with an average error of 5% and 12%, respectively. Model-based calibration allows the characterization of the imaging properties of the entire SFDI system without prior knowledge, enabling the future development of a digital twin for synthetic data generation or more robust evaluation methods.

Chlorophyll- and anthocyanin-rich cell organelles affect light scattering in apple skin.

Apple skin contains several groups of strongly absorbing cell organelles with pigments that change dynamically in type and concentration during fruit maturation. Chlorophylls and carotenoids, both primarily involved in photosynthesis, are found in the grana of chloroplasts, while anthocyanin vacuolar inclusions (AVIs) accumulate for light protection in red-skinned cultivars. A Mie model describing light scattering by absorbing spherical particles in a non-absorbing medium allowed to theoretically investigate the explicit influence of grana and AVIs on the effective scattering coefficient [Formula: see text] and the absorption coefficient [Formula: see text]. The reconstruction of the complex refractive indices of the organelles predicted anomalous dispersion, i.e., a local increase in the real part of the refractive index in the spectral regions with high chlorophyll and anthocyanin absorption, in agreement with the Kramers-Kronig relations. As a result, peaks in [Formula: see text] were predicted to be shifted to longer wavelengths compared to the corresponding [Formula: see text] bands. This selective scattering effect was confirmed experimentally with integrating sphere measurements for red- or green-skinned apple samples of the cultivars 'Elstar', 'Gala' or 'Jonagold'. Comparison between simulations and measurements indicated that the Soret bands of chlorophyll a and chlorophyll b are at 435 nm and 469 nm, respectively, and overlap with the absorption of carotenoids, whose red-most edge is at 488 nm. For anthocyanin absorption, a pronounced blue shift from 550 to 520 nm was observed, indicating structural or chemical changes of AVIs.

Analytical solution of the radiative transfer theory for the coherent backscattering from two-dimensional semi-infinite media.

An analytical solution for coherent backscattering (CBS) in two dimensions was derived by solving the radiative transfer equation. Particularly, the single scattered radiance from a semi-infinite medium containing perpendicularly illuminated cylinders was obtained. At the boundary, a refractive index mismatch was taken into account. Furthermore, the link between the radiance and the CBS was shown in the small angle approximation. An excellent agreement was found between Monte Carlo simulations and the analytical solution. Additionally, it was shown that the often applied solution in the spatial frequency domain for quantifying the CBS delivered significantly different results compared to the derived exact analytical solution.

Analytical solution of the vector radiative transfer equation for single scattered radiance.

In this paper, derivation of the analytical solution of the vector radiative transfer equation for the single scattered radiance of three-dimensional semi-infinite media with a refractive index mismatch at the boundary is presented. In particular, the solution is obtained in the spatial domain and spatial frequency domain. Besides the general derivation, determination of the amplitude scattering matrix, which is required for the analytical solution, is given in detail. Furthermore, the incorporation of Fresnel equations due to a refractive index mismatch at the boundary is presented. Finally, verification of the derived formulas is performed using a self-implemented electrical field Monte Carlo method based on Jones formalism. For this purpose, the solution based on Jones formalism is converted to Stokes-Mueller formalism. For the verification, spherical particles are assumed as scatterers, whereby arbitrary size distributions can be considered.

Numerical simulation of phase-optimized light beams in two-dimensional scattering media.

Manipulating the incident wavefront in biomedical applications to enhance the penetration depth and energy delivery in scattering media such as biological tissue has gained a lot of attention in recent years. However, focusing inside scattering media and examining the electromagnetic field inside the medium still is an elaborate task. This is where electromagnetic field simulations that model the wavefront shaping process can help us understand how the focal near field evolves at different depths. Here we use a two-step beam synthesis method to simulate the scattering of complex incident wavefronts by well-characterized media. The approach uses plane wave electromagnetic near-field solutions in combination with an angular spectrum approach to model different light beams. We apply this approach to various two-dimensional scattering media and investigate the focus intensity over depth while scanning with and without phase optimization. We find that the scanned non-optimized beams have two regions characterized by exponential decays. The absolute progression of the focus intensity over depth for phase-optimized beams using all channels can be described by solutions of the radiative transfer theory. Furthermore, the average enhancement factor over depth of the phase-optimized focus intensity compared to that without optimization is investigated for different numerical apertures and scattering media. Our results show that, albeit the incident beam is diffusively scattered, the theoretical enhancement for a large number of optimization channels cannot be reached due to correlations between the channels. An increase in focus depth and an increase in the numerical aperture reduces the difference between the expected theoretical and simulated enhancement factors.

Analytical solution of the vector radiative transfer equation for single scattered radiance: erratum.

We have discovered a small error in our recently published work [J. Opt. Soc. Am. A39, 2045 (2022)JOSAAH0030-394110.1364/JOSAA.467890], which we correct in this erratum. The error is located in Section 2.D and affects the solution for polydisperse distributions.

Improved topography reconstruction of volume scattering objects using structured light.

The use of structured light projection enables the reconstruction of three-dimensional topography of surface reflecting objects. However, if the investigated object exhibits volume scattering, the obtained topography is erroneously caused by light undergoing volume scattering inside the object. In this theoretical study, we investigate these errors using Monte Carlo simulations. Additionally, a method is proposed to correct the errors by quantifying the light propagation in the scattering object based on the radiative transfer equation. Reconstructed surfaces with a small spatial variation of topography can be quickly corrected using a local correction method that depends only on the directions of the incident and detected light relative to the surface. For surfaces that show a large spatial variation of the surface geometry, another approach is introduced by simulating the light propagation in the whole scanned three-dimensional object using graphics processing unit (GPU)-accelerated Monte Carlo simulations. A cylindrical object and an incisor tooth are, exemplarily, investigated. The results show a major improvement in the reconstructed topography due to the correction with the proposed methods.

Imaging of custom-made single scatterers with the confocal laser scanning microscope.

In this work, we investigate image formation in the confocal laser scanning microscope for different single scatterers, both theoretically and experimentally. For spherical scatterers, an effective and fast algorithm was implemented to calculate the confocal image for different diameters and wavelengths. Measurements on a polystyrene sphere (PS) with a diameter of 20 µm confirmed the expected effects, for example, the appearance of a central signal similar to the point spread function of the optical system. Custom single scatterers were produced using 3D-direct laser writing (DLW), including a sphere with dimensions comparable to the aforementioned PS sphere. Despite an inevitably lower surface quality and symmetry, only minor differences were observed in the confocal image of the 3D-DLW sphere compared to a near-perfect PS sphere. Having verified the experimental images of spheres with the computed theoretical data, confocal measurements of four platonic bodies produced by 3D-DLW were measured with the goal to contribute to the understanding of image formation involving more complex scattering geometries.

Influence of Lambertian surface scattering on the spatially resolved reflectance from turbid media: a computational study.

The determination of the optical properties in turbid media plays an essential role in medical diagnostics and process control. The method of spatially resolved reflectance measurements is a frequently used tool to evaluate the reduced scattering coefficient as well as the absorption coefficient. In most cases a smooth interface is assumed between the medium under investigation and the surrounding medium. However, in reality, a rough surface is present at the interface, which alters the light interaction with the surface and volume of the turbid medium. Hence, the idea behind this paper was to investigate the influence of rough surfaces on the spatially resolved reflectance and thus on the determination of the optical properties of turbid media. Particularly, the influence of a Lambertian scattering surface on the result of Monte Carlo simulations of a spatially resolved reflectance setup is shown. In addition, we distinguish between the different interaction modes of surface scattering on the spatially resolved reflectance. There is a strong influence of roughness when the light enters and leaves the turbid medium. Furthermore, the simulations show that, especially for small reduced scattering coefficients and absorption coefficients, large errors in the determination of the optical properties are obtained.

Spatially resolved reflectance from turbid media having a rough surface. Part II: experiments.

Spatially resolved reflectance measurements are a standard tool for determining the absorption and scattering properties of turbid media such as biological tissue. However, in literature, it was shown that these measurements are subject to errors when a possible rough surface between the turbid medium and the surrounding is not accounted for. We evaluated these errors by comparing the spatially resolved reflectance measured on rough epoxy-based samples with Monte Carlo simulations using Lambertian surface scattering, the Cook-Torrance model, and the generalized Harvey-Shack model as surface scattering models. To this aim, goniometric measurements on the epoxy-based samples were compared to the angularly resolved reflectance of the three surface models to estimate the corresponding model parameters. Finally, the optical properties of the phantoms were determined using a Monte Carlo model with a smooth surface.

Spatially resolved reflectance from turbid media having a rough surface. Part I: simulations.

Determining the optical properties of turbid media with spatially resolved reflectance measurements is a well-known method in optical metrology. Typically, the surfaces of the investigated materials are assumed to be perfectly smooth. In most realistic cases, though, the surface has a rough topography and scatters light. In this study, we investigated the influence of the Cook-Torrance surface scattering model and the generalized Harvey-Shack surface scattering model on the spatially resolved reflectance based on Monte Carlo simulations. Besides analyzing the spatially resolved reflectance signal, we focused on the influence of surface scattering on the determination of the reduced scattering coefficients and absorption coefficients of turbid media. Both models led to significant errors in the determination of optical properties when roughness was not accounted for.

Distance insensitive reflectance setup for the spectrally resolved determination of the optical properties of highly turbid media.

A measurement system for a distance insensitive acquisition of the reflectance from turbid media is presented. The geometric relationships of the detection unit are discussed theoretically and subsequently verified using Monte Carlo simulations. In addition, an experimental setup is presented to prove the theoretical considerations and simulations. The use of the presented measurement system allows measurements of the reflectance in a distance range of approximately 2.5cm with a deviation of less than ±0.5% for highly scattering media. This contrasts with the use of a fiber in a classical detection unit placed at a defined angle and position relative to the sample surface, which results in deviations of ±30% in the measured reflectance over the same distance range.

Solutions for single-scattered radiance in the semi-infinite medium based on radiative transport theory.

In this paper, some explicit analytical solutions for single-scattered radiance in a half-space medium under consideration of a reflecting boundary are derived. We consider both a unidirectional beam source as well as an isotropic point source. In addition to direct applications within optical tomography and computer graphics, the obtained solutions are also needed when solving the radiative transport equation after the separation of the unscattered and single-scattered contribution. Comparisons between the derived analytical solutions and the Monte Carlo method display excellent agreement.