A four-flux radiative transfer model is used to evaluate diffuse reflectance and transmittance spectra of plane-parallel particulate materials. External parameters, including scattering and absorption cross sections, forward scattering ratios, and average path length parameters, are calculated utilizing Lorenz-Mie theory for independent scattering and a transition matrix approach for near-field dependent scattering evaluations. The formalism is applied to sub-micron-sized spherical titanium dioxide pigments in an aqueous-polymer matrix to evaluate their reflectance and transmittance spectra in the range of 0.3 to 1.5 μm in wavelength. The particle concentration, the degree of lumpiness, the number of particles per cluster, and the cluster compaction are varied to quantify the effect of dependent scattering. What we believe to be a novel scheme is introduced to model the morphology of samples in terms of a volume fraction occupied by particles that do not form aggregates and another fraction corresponding to particles that are part of aggregates, as well as the lumpiness of the sample and compaction of the clusters. A more detailed analysis is carried out for specific wavelengths in the ultraviolet, visible, and near infrared to correlate the variation of reflectance and transmittance due to dependent scattering with the propagating fluxes.

Reflectance spectra of Chrysinacupreomarginata's elytra are measured for visible wavelengths. The spectra consist of both left- and right-handed circularly polarized light, which suggests the presence of two chitin-based helicoidal structures separated by a unidirectional layer, as reported for the Chrysina resplendens scarabs. This fact has been corroborated by the analysis of scanning electron microscopy imaging. The structural and effective pitches of each helicoid have been obtained from these images. The height and width of the reflectance bands are linked with the presence of uric acid crystallites through the arrangement of oriented chitin fibrils embedded in the proteinaceous matrix. Calculated left- and right-handed circularly polarized reflectance spectra are evaluated from a radiative transfer matrix formalism. The photonic characterization of each helicoid is carried out in terms of the variation of the optical gap with depth through the twisted arrangements. A novel, to our knowledge, approach is developed to obtain the depth-dependence of both the uric acid volume fractions through each helical structure and the spectral positions and widths of the photonic band gaps. When modeling reflectance spectra to resemble measured ones, it is necessary to significantly increase the volume fraction of uric acid in the unidirectional layer for it to function as a half-wave plate.

Multilayered metamaterials with switchable optical properties are desirable for application purposes such as sensors and imaging systems. Some rare-earth metals (RE) could be used for this purpose as they undergo a metal-dielectric phase transition upon hydrogen absorption. Alternating layers of palladium and REs exposed to a hydrogen atmosphere are shown to exhibit a switchable and reversible reflectance band. The reflectance band is tunable across the visible spectrum by changing the thickness of the RE layers, while increasing the number of layers enhances the reflectance band amplitude. The result is a switchable multilayer metamaterial whose optical properties are tunable by changing the surrounding hydrogen pressure.

Two simplified versions of the four-flux radiative transfer model (the standard one and a novel extended form) are put into context with respect to the generalized approach. Collimated-collimated, collimated-diffuse, and diffuse-diffuse reflectance and transmittance (R&T) spectra of anatase (TiO2) pigments in a polymeric aqueous medium are generated by these four-flux models. To this end, it has been necessary to evaluate previously the scattering and absorption cross sections, average pathlength parameters, and forward scattering ratios that enter these models. The evaluation has been carried out assuming no interaction between neighboring particles, or assuming the presence of particle aggregates in the composite medium. In the first case, the Lorenz-Mie theory has been applied to obtain the expansion of the phase function in Legendre polynomials. For particulate media containing aggregates, the coefficients in this expansion are derived using a transition matrix formalism. Additionally, parameters such as scattering and absorption cross sections, albedo, and asymmetry parameter have been determined. Evaluating synthetic R&T spectra of these materials can be useful to validate inversion procedures using spectral gradient and simulated annealing methods. Quantifying the dependent scattering effect in R&T spectra, both collimated and diffuse, is now possible. This has been done mostly at the level of cross sections, albedos, and asymmetry parameters of particle aggregates, but not for parameters that can be directly correlated with spectrophotometric measurements like R&T. (c) 2025 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Green Chrysina beetles show left-handed circularly polarized light reflections produced by a helicoidal chitin-protein fibre structure, but, unlike reflections of the well-studied golden and silver species, their reflections look diffuse as a result of having an exocuticle with small craters that spread local specular reflections in many directions. At the micrometre scale, the curved reflective structure at the craters behaves as concave reflectors projecting real images above their surfaces, while the convex rims separating the craters form instead virtual images below. The result is an optical illusion where the real topography is inverted: the craters look like hills and the rims like valleys. We have successfully modelled the resulting pattern of multi-coloured reflections, imagining the curved, helicoidal structure as a stack of wavelength-selective micro-reflectors. This spatially projected imaging interpretation becomes relevant for optical characterization studies of such structures at the microscale.

We explore a novel method to obtain scattering and absorption (S&A) coefficients of inhomogeneous materials from measurements of regular transmittance and specular reflectance. We use a Spectral Projected Gradient Method (SPGM) to invert experimental spectra. The SPGM method requires an initial approximation to the S&A coefficients, which is needed to ensure convergence to an optimized solution. We discuss problems associated with multiple solutions and conditions for the obtained optimal solution to be a good approximation to the physical one. We present results for TiO2/polyvinylpyrrolidone (PVP)-water, plasmonic Au/PVP-water and ferromagnetic Fe3O4/PVP-water materials. The obtained S&A coefficients indicate that these materials are dilute suspensions of dense particle clusters in the dependent scattering regime. We argue that the SPGM gives physically realistic solutions for this class of materials. The SPGM has the advantages that it is much less expensive in computational terms and easier to implement than previously used simulated annealing methods.

Extensive radiative transfer calculations to fit the more significant average spectral features displayed in measured reflectance spectra by cuticles of Chrysina resplendes scarabs have been carried out. A simulated annealing approach is the starting point to optimize the values of the spatial average uric acid volume fraction, the thickness of the retarder layer located between the two left-handed chiral structures in the cuticle, and the slope of the parameter describing the variation with depth of the uric acid volume fraction through the retarder layer. Both twisted arrangements are characterized in terms of their photonic behaviors to display the variation with depth of the photonic band gaps, effective refractive index and birefringence, wave numbers and relative densities of states.

Direct transmittance spectra of Dodecane based Pickering emulsions are inverted to obtain the spectral behavior of their intrinsic scattering and absorption coefficients per unit length, gsca and gabs respectively. The spectral range includes ultraviolet, visible, and near infrared wavelengths. Dynamic light scattering has been used to characterize the bimodal lognormal size distribution of the ensemble of particles or micelles. Lorentz-Mie theory is applied to show how the morphology and composition of the particles influence the scattering and absorption cross sections. A spectral projected gradient method is applied to carry out the inversion of transmittance spectra. Three light scattering and absorbing particles in water are considered: those consisting of submicron-sized Dodecane micelles, Dodecane particles capped with self-assembled nano-sized anatase (TiO2) particles, and similar ones anatase capped Dodecane micelles with dissolved Nile Red dye.

Spectrophotometry has been widely used to retrieve the dielectric function of a bulk iridium sample using an extended version of the Drude–Lorentz model. The parameters of the model are optimized using a spectral-projected-gradient-method-assisted acceptance-probability-controlled simulated annealing approach. Furthermore, optimized values of Drude parameters corresponding to the optical response of electrons and holes (scattering frequency of electrons, the ratio between scattering frequencies of holes and electrons, the ratio between effective masses of electrons and holes, the ratio between the number densities of holes and electrons, and electron volume plasma frequency) are used to evaluate charge transport and magnetic properties. These include static and dynamic conductivities, intrinsic mean free paths, the effective mass of charge carriers and their number densities, Fermi velocities and energies, densities of states at Fermi energies, mobilities, specific heats, Hall’s coefficient, thermal conductivities, charge carrier coupling constant, paramagnetic and diamagnetic susceptibilities, and the number of Bohr magnetons. In addition, optimized resonance energy values of the Lorentz contribution to the dielectric function were compared with the background information provided by density-functional-theory calculations for iridium. A decomposition of the energy loss function was used as the starting point to calculate the effective numbers of bound electrons involved in interband transitions, as well as the densities of states at the final energies of the sets of transitions considered. The Drude–Lorentz model involves charge carrier parameters for both electrons and holes, as well as the resonance energies correlating with the energies associated with quantum transitions. To a large extent, several physical quantities calculated from optimized parameters exhibit values close to those obtained from measurements or by applying other models, including quantum mechanics formulations.

The formalism of the Lorenz-Mie (LM) theory is used to estimate the contributions of radiation being scattered into the detector acceptance angle of spectrophotometers (with or without an integrating sphere) to the apparent direct transmittance and specular reflectance measurements. These contributions are subtracted from the measurements before doing the inversion to obtain the spectral variation of the scattering and absorption (S&A) coefficients of optically particulate liquid samples, decoupling the contribution to extinction from these two mechanisms. The method is applied to transmittance spectra of Pickering emulsions of dodecane oil (DO) and DO micelles with saturated dissolved Nile red dye, stabilized with anatase (T i O 2) in water. What we believe to be novel expressions are given to estimate the forward and backward average path-length parameters of propagating diffuse radiation from its equivalent quantities defined within the formalism of the LM theory and to relate these single particle average path-length parameters with the forward scattering ratio, asymmetry parameter, and higher-order coefficients in the expansion of the LM phase function. This novel approach makes it possible to avoid the underestimation of the S&A coefficients, which arises when the correction is not made by subtracting that contribution due to scattered radiation that is reaching the detectors from the directional reflectance and transmittance measurements.

Transmittance or absorbance spectra of liquid samples are usually obtained by illuminating a quartz cuvette container with non-polarized light.A direct inversion of the absorbance allows to obtain the optical thickness and the absorption coefficient, as well as the intrinsic absorptivity of the medium when the concentration of the light absorbing molecules is known.A numerical inversion of transmittance spectra, based on a spectral projected gradient method, is carried out to obtain other optical parameters: optical constants (refractive index and extinction coefficient) and electronic polarizability of the molecules.This inversion approach is applied to transmittance spectra by samples of transparent Dodecane oil and light absorbing Nile Red dye, with an automated determination of the HOMO-LUMO energy gap of saturated Nile Red dissolved in Dodecane oil.The spectral variation of the optical constants of these two materials is reported for the first time, for wavelengths between 300 and 800 nm.

A theoretical treatment of the optical properties of chiral materials exhibiting a photonic crystal behavior is summarized and extended. It can be applied to interpret the photonic characteristics of synthetic cholesteric liquid crystals and natural elytra of jewel scarabs showing graded spatial periods or pitches. The model is applied here to analyze the photonic properties of the cuticle of a C. resplendens scarab which reflects both components of circular polarization when illuminated with non-polarized light. The structural composition of the elytra has shown to consist of an optically homogenous epicuticle, an upper left-handed twisted arrangement of chitin fibrils embedding uric acid crystallites, followed by a unidirectional layer, and a second left-handed thicker helicoid. The basic information about this morphological structure is taken from the pioneering work of Caveney [Proc. Roy. Soc. Lond. B178, 205 (1971)10.1098/rspb.1971.0062]. Some of his findings are also obtained within the context of our theory (birefringence of the unidirectional layer, average volume fraction for the whole structure, and typical spectral composition of left- and right-handed reflectance spectra), and new ones are reported. Novel expressions for the photonic band gap and its width, the ordinary and extraordinary effective refractive indices and birefringence are also reported.

Chlorophyll pigmented grains were extracted from leaves of palm trees and maintained in aqueous solutions containing an organic solvent. Their transmittance spectra were measured for visible wavelengths, from 400 to 700 nm. Four dissolutions were prepared to consider different pigment concentrations. The transmittance measurements show a somewhat spectrally structured band between 474 and 650 nm, limited by the light extinction (scattering + absorption) due to the pigments.

Optical, charge carriers transport, quantum mechanics, magnetic, thermal, and plasmonic properties of the transition metal rhodium are considered. An extended Drude-Lorentz (DL) model is applied to describe the dielectric function (DF) of rhodium in a spectral range going from the mid-infrared (12.4 μm) to the vacuum ultraviolet (32 nm). The Drude term of the DF includes, as optimization parameters, the inverse of the high frequency dielectric constant, the volume plasma frequency and scattering frequency of the electrons, the scattering frequency of holes relative to that of electrons, the ratio between the effective masses of electrons and holes, the number of holes per atom relative to that of electrons, and the renormalized times between grain boundary scattering events for electrons and holes. The Lorentz contribution to the DF includes the number of conduction electrons per atom, the oscillator strengths, the resonance energies, and the Lorentzian widths. Values of the parameters involved in the DF are optimized by an acceptance-probability-controlled simulated annealing method that minimizes spectral differences between the real and imaginary parts of the DF values obtained from the literature and those evaluated from the DL parametric formulation, accounting for the presence of electrons and holes as charge carriers. Once an optimized spectral description of the DF of rhodium is obtained, a large set of charge-transport, magnetic, thermal, plasmonic, and quantum mechanics derived quantities are evaluated: mobilities, relaxation times, Fermi velocities, effective masses, electrical and thermal conductivities, heat capacity coefficients, Hall coefficient, diamagnetic and paramagnetic susceptibilities, effective number of Bohr magnetons, Fermi energies and corresponding densities of states, energy loss functions, effective number of charge carriers participating in conduction, and effective number of electrons involved in inter-band transitions.

Solutions of the differential equations for the diffuse components of the four-flux model are obtained, including explicit expressions for the collimated-diffuse and diffuse-diffuse reflectance and transmittance for an optical three-layer system. We establish a method to invert collimated-diffuse reflectance and transmittance in order to obtain the spectral variation of all average pathlength parameters and forward scattering ratios, by applying a spectral simulated annealing method. The inversion procedure was tested with synthetic collimated-diffuse reflectance and transmittance spectra and showed very good convergence. Subsequently, the method was applied to measured spectra of a light scattering and absorbing medium containing TiO2 particles in a low-absorbing matrix, which was enclosed between two glass slides. Hence the present paper, together with our previous one, wherein we inverted collimated-collimated reflectance and transmittance spectra to obtain scattering and absorption coefficients, establishes a reliable inversion method for obtaining all parameters in the most general version of the four-flux theory from experimental data.

Measured reflectance spectra by cuticles of C. resplendens scarabs, in the spectral range 300 to 1000 nm, show high variability depending on the magnitude of the illuminated area, the position of the illuminated section, the elytron, a single specimen, and on the different specimens considered. This fact suggests variability in the depth dependence of the pitch and of the average uric acid concentration, and on the thickness of the retarder layer located between the two helicoidal structures characterizing the cuticle of these beetles.

Four-flux radiative transfer models have been extensively used to describe reflectance and transmittance (R&T) spectra of light scattering and absorbing (S&A) media. Solutions to the differential equations corresponding to the collimated fluxes are obtained by subsequent application of boundary conditions. Explicit expressions for the collimated R&T of light are reported, when considering a light S&A medium contained between two glass slides, an experimental arrangement which is appropriate for liquid suspensions and viscous matrices containing solid particles. A spectral simulated annealing method is applied to retrieve, from measured R&T spectra of collimated light under normal incident radiation, the scattering and absorption coefficients of the composite medium. First, the accuracy of the method is established by applying it to synthetic collimated R&T data. Secondly, we apply the method to experimental data and use it to determine the S&A coefficients of a layer of TiO2 particles dispersed in a PVP/water matrix.

To facilitate optical design of energy-efficient materials and devices, a detailed knowledge of their basic optical parameters is necessary. In this paper we present a novel method for determining scattering (S) and absorption (K) coefficients from total transmittance and reflectance measurements by inversion of the Kubelka-Munk theory. The reflectance parameters appearing in this theory depend on the angular distribution of scattered light inside the material. The versatility of our method is demonstrated by a reanalysis of experimental data for several materials of interest in energy-related applications. Specifically, we report spectra of S and K for: (a) pigmented polymer foils for radiative cooling applications; (b) suspended particle devices for smart windows; (c) solar reflecting TiO2-pigmented paints and (d) selective solar absorbing paints for solar collectors.

A parametric Drude–Lorentz (DL) model is used to describe the spectral variation of the dielectric functions of bulk palladium samples at low and room temperature. In addition to the contribution of conduction electrons, the contribution of holes is also explicitly accounted for in the model. A simulated annealing method is applied to obtain the optimized values of the parameters involved in the model: volume plasma frequency of conduction electrons, high frequency dielectric constant, collision frequency of holes and corresponding relaxation time, and two additional parameters from which the effective mass of holes and collision frequency of conduction electrons are evaluated. Oscillatior strengths, resonance frequencies, and widths entering in the Lorentz contribution to the dielectric function are also optimized. Renormalization of the oscillator strengths requires the introduction of a new parameter in the context of the DL model: the ratio between number density of conduction electrons and number density of metal atoms, whose optimized value fits very well with its evaluation from band structure calculations and from independent measurements. Inclusion of this parameter in the framework allows us to evaluate additional quantities related to the charge-carrier transport: average effective masses, Fermi energies and electronic densities of states at the corresponding Fermi energies, intrinsic electrical resistivity, intrinsic mean free paths, heat capacities, mobilities, as well as paramagnetic and diamagnetic susceptibilities, for both electrons and holes. The optimized resonance frequencies are compared with energy differences between plausible interband transitions, in accordance with reported band structure diagrams and with our own band structure obtained from density functional theory calculations.

Many smart materials with applications in the fields of solar energy and energy-efficient buildings are structured and exhibit significant light scattering. To facilitate optical design of material ...