David Barrios

The effective optical constants (EOCs) of a 480 nm tungsten trioxide (WO3 ) thin film deposited on fluorine-doped tin oxide (FTO)–coated glass were determined from collimated transmittance and reflectance measurements over the 250–2500 nm solar spectral range. The WO3 EOCs were obtained by fitting the experimental spectra with the four-flux model (4FM), using previously determined optical constants of the FTO and glass layers as input parameters. This work extends previous analyses of FTO–glass bilayers to a more complex WO3–FTO–glass trilayer configuration, enabling validation of the proposed framework under multilayer boundary conditions. A three-extinction matching requirement was identified from the agreement between extinction coefficients calculated from forward and backward 4FM collimated differential equations and from the EOCs under wavelength compression. In addition, a spectral limit of the optical behavior between film and substrate was consistently observed at wavelengths approximately three times the layer thickness for both FTO and WO3 layers, confirming its applicability beyond bilayer systems. Thickness gradient plots of collimated light fluxes and complex electric fields revealed the transition from substrate-like incoherent behavior to film-like coherent behavior. In the coherent regime, complex electric fields were represented through wavelength-dependent Bode and Nyquist diagrams, allowing analysis of their magnitude–phase evolution within the multilayer structure. The results provide a consistent framework for determining EOCs and identifying optical regime transitions in multilayer systems, demonstrating applicability to trilayer electrochromic structures. Received: 16 January 2026 | Revised: 30 April 2026 | Accepted: 27 May 2026 Conflicts of Interest The author declares that he has no conflicts of interest to this work. Data Availability Statement Data sharing is not applicable to this article as no new data were created or analyzed in this study. Author Contribution Statement David Barrios-Puerto: Conceptualization, Methodology, Validation, Investigation, Resources, Data curation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration.

Intrinsic and extrinsic scattering and absorption coefficients (S&A-C) of four inorganic electrochromic device (ECD) samples were determined at their bleached-off optical state (BOOS, 0 V) and at their colored-on optical state (COOS, +3 V DC). The same inversion procedure previously applied to suspended particle device (SPD) and polymer-dispersed liquid crystal (PDLC) samples, approximated for a single layer sandwich structure, was used here for a third smart window technology, based on redox reactions. The four studied ECDs incorporated cathodic WO3 and anodic NiO layers corresponding to the four possible combinations of two thicknesses for each material —single (120 nm) and double (240 nm)— within their sandwich structures. The analysis of their S&A-C was performed in three stages. First, determination of optical constants using the collimated solutions of the four-flux model (4FM) from collimated transmittance and reflectance (T&R) measurements. Second, estimation of extrinsic S&A-C from the two-flux model (2FM) using total (collimated + diffuse) T&R data. Third, extraction of intrinsic S&A-C from the diffuse 4FM solutions by means of diffuse T&R data, employing the diffuse interface reflectance, an approximate average crossing parameter, and the forward scattering ratio. Although light scattering was found to be weak, absorption for the COOS showed a clear dependence on the WO₃–NiO active layers thicknesses: neutral gray ECD11 (single WO3–single NiO), brownish gray ECD12 (single WO3–double NiO), bluish gray ECD21 (double WO3–single NiO), and dark gray ECD22 (double WO3–double NiO). As expected, similar optical appearances were observed in the BOOS for the four ECDs. The values weighted by the CIE 1931 standard observer color matching function (ȳ) indicate that increasing the WO3 thickness twofold leads to a larger absorption coefficient increment than a twofold increase in the NiO thickness. The single-layer 4FM implementation adopted in this study provides reliable agreement in the 250–1500 nm wavelength range, while fitting inaccuracies appear beyond 1500 nm (i.e., in the 1500–2500 nm region) between the calculated and the measured diffuse transmittance at BOOS, for ECD11 and ECD22, and at COOS, for ECD11. No fitting inaccuracies were observed between the calculated and the measured diffuse reflectance. Concerning the 2FM study, an excellent agreement in fitting was observed between the calculated and the measured total T&R for the four ECD samples.

The optical constants of a fluorine-doped tin oxide (FTO) transparent conductive layer (of 400 nm thickness), deposited over a glass (G) substrate, were determined from the collimated transmittance (T) and reflectance (R) optical measurements of a bilayer FTOG sample in the solar wavelength range (from 250 nm to 2500 nm). The three-extinction matching requirement (3εMR) phenomenon was used as a feedback system to validate the accuracy of the optical constants results, obtained by fitting to the collimated T and R solutions of the four-flux model (4FM). The same extinction coefficients (ε) were observed than the obtained when computing light fluxes at the interfaces and substituting into the 4FM collimated differential equations (CDE). The compression of the wavelength of the incident light inside G and FTO layers was validated by computing ε from optical constants. Despite a layer is considered a thin film when its thickness is lower than the wavelength of incident light beam, and a substrate when higher, the 3εMR allowed to determine the spectral limit of the FTO layer behavior, as a substrate and as a film, at 1200 nm (instead of at 400 nm, i.e., three times higher). FTO layer behaves as a substrate, for the shorter wavelength range of the solar spectrum (SWRSS), and as a film, for the longer wavelength range of the solar spectrum (LWRSS). The thickness gradients of the forward and backward (for&back) collimated light fluxes were determined at the glass substrate of the G and FTOG samples and for the SWRSS of the FTO as a substrate layer. Then, for the LWRSS, the thickness gradients for the complex magnitude and phase (Bode plots) of for&back electric fields of the FTO film layer have been spectrally determined and represented by means of imaginary versus real parts (Nyquist plots) depending on wavelength. Received: 12 February 2025 | Revised: 9 May 2025 | Accepted: 29 May 2025 Conflicts of Interest The author declares that he has no conflicts of interest to this work. Data Availability Statement Data sharing is not applicable to this article as no new data were created or analyzed in this study. Author Contribution Statement David Barrios-Puerto: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration.

The optical constants of a thin film layer of transparent conductive indium tin oxide (ITO), deposited on a thick glass substrate (G), were obtained for an ITO-G sample using a multilayer matrix method by fitting the regular transmittance and specular reflectance measurements to the collimated equations of the four-flux model, once the optical constants of the glass substrate layer had previously been obtained from a G sample of a single glass substrate layer. The accuracy of the results was validated using a feedback system called the three-extinction matching requirement, that is, obtaining the same extinction coefficients from the optical constants and from the forward and backward collimated differential equations of the four-flux model. To calculate the extinction coefficients of the glass from optical constants, the compression of the wavelength of the incident light in the glass substrate and in the thin ITO film layer was demonstrated, with a thickness less than the wavelength of incident light. The thickness gradients of the forward and backward collimated light intensities, on the glass substrate of the ITOG sample, were compared with those obtained for the single-layer G sample. Then, the thickness gradients for the complex forward and backward electric fields of the ITO thin film layer were obtained and plotted using spectral magnitude and phase Bode plots and new Nyquist plots, i.e. imaginary versus real parts, dependent on wavelength.

The intrinsic and extrinsic scattering and absorption coefficients of a thin black paint substrate used as a solar thermal collector, deposited on a thick glass substrate, were obtained in an approximated single-layer method, using the new solved equations recently determined from the collimated and diffuse differential equations of the four-flux model, for the intrinsic coefficients, and considering the differential equations of the two-flux model with total light intensities, for the extrinsic coefficients. In a first step, the optical constants were determined from the measurements of regular transmittance and specular reflectance, fitting them to the collimated transmittance and reflectance solutions of the four-flux model. In a second step, the intrinsic coefficients were obtained from the diffuse transmittance and reflectance measurements, considering four forward scattering ratios (for collimated and diffuse light intensities and for forward and backward light directions) and two average crossing parameters (for forward and backward light directions). In a third step, the extrinsic coefficients were obtained, using the new solved equations recently determined considering the total differential equations as the sum of the collimated and diffuse differential equations. Due to the approximated single-layer method, a simple rule of three was applied to the intrinsic and extrinsic results, relating the extinction coefficients calculated from the optical constants to those calculated from the collimated differential equations. The same external and internal diffuse interface reflectance was observed by integrating the collimated interface reflectance up to the critical angle of total internal reflection.

Collimated, diffuse, and total light intensity gradients, for forward and backward light senses, were determined in a three substrate layers – glass/electrolyte/glass – almost transparent sample, using optical constants and new equations for intrinsic and extrinsic scattering and absorption coefficients. These new equations were obtained for the inner electrolyte layer from the systems of differential equations of the four-flux and two-flux radiative transfer models, used for determining intrinsic and extrinsic coefficients, respectively, once knowing the optical constants of outer glass layers, from a single glass substrate sample measured in advance. Extinction coefficients were determined from optical constants and considering the wavelength compression of light when it enters into a material, decreasing its speed with respect to the vacuum. The same extinction coefficients for glass and electrolyte layers were computed in three different ways. First, from optical constants, they were determined using collimated transmittance and reflectance solutions of four-flux model. From them, collimated interface reflectance and attenuation due to extinction were computed. These intermediate parameters for glass and electrolyte layers were required for determining inner collimated light intensities at the interfaces, which were used at the collimated forward and backward differential equations, solving for the forward and backward extinction coefficients. The three-extinction matching requirement was successfully satisfied for the glass and electrolyte layers. Two average crossing parameters equations, for each sense, and four forward scattering ratios equations, for collimated and for diffuse light intensities for each sense, were used in the system of diffuse differential equations (DDE) for intrinsic coefficients. For them, intuitive equations were proposed based on collimated and diffuse light intensities at each interface. New equations for extrinsic parameters were determined by equalizing the system of total differential equations of the two-flux model to the sum of the systems of collimated and DDE of the four-flux model. Received: 9 December 2023 | Revised: 4 March 2024 | Accepted: 25 April 2024 Conflicts of Interest The author declares that he has no conflicts of interest to this work. Data Availability Statement Data are available from the corresponding author upon reasonable request.

Los coeficientes intrínsecos y extrínsecos de dispersión y absorción de ocho muestras de cristales líquidos dispersos en polímero no absorbedoras de luz, con dos diferentes tamaños del área activa y con cuatro diferentes espesores de la capa interna active, en sus estados de apariencia óptica translucidos apagados y transparentes encendidos –sin y con voltaje eléctrico aplicado respectivamente– fueron satisfactoriamente determinados siguiendo el mismo procedimiento descrito en trabajos anteriores con un dispositivo de partículas suspendidas –una muestra de ventana inteligente absorbedora de luz– y con otra muestra de ventana inteligente de cristales líquidos dispersos en polímero. Este procedimiento, basado en los modelos de transferencia radiativa de dos flujos y de cuatro flujos, considera el ángulo crítico de reflectancia interna total para determinar la reflectancia difusa interna de interface, y utiliza la misma ecuación propuesta previamente para el parámetro de camino promedio, basada en las intensidades difusas y totales de los haces de luz hacia adelante y hacia atrás, para resolver el modelo de cuatro flujos a fin de determinar los coeficientes intrínsecos. La apariencia óptica simulada resultó ser un estado apagado translucido blanco lechoso y un estado encendido transparente incoloro que llega a ser más transparente para las muestras de cristales líquidos dispersos en polímero con espesores más finos, requiriendo mayor voltaje aplicado para los estados ópticos encendidos transparente de las muestras caracterizadas de cristales líquidos dispersos en polímero.

Smart windows based on suspended particle devices (SPDs) are able to switch optically from dark to clear visual appearance when applying an AC electrical signal. This effect is due to light absorbing nanoparticles that get aligned by the applied voltage. The sandwich structure of an SPD consists of several layers and includes two outer glass substrates, each one covered on its inwards-facing side with a transparent conducting thin layer surrounding the centrally positioned SPD active layer. A knowledge of the optical constants of each layer—i.e., the complex refractive index, including its real and imaginary (absorption) parts—is a key in the design of the visual appearance of the SPD window and is a useful tool to determine the optimum thickness of the active layer.