{"id":1018,"date":"2023-10-24T09:01:13","date_gmt":"2023-10-24T09:01:13","guid":{"rendered":"https:\/\/uneedtalk.com\/config\/tredword\/?page_id=1018"},"modified":"2023-12-01T13:12:12","modified_gmt":"2023-12-01T13:12:12","slug":"inversion-method-to-derive-complex-valued-refractive-index-functions-of-transparent-materials","status":"publish","type":"page","link":"https:\/\/radi-cal.org\/method\/inversion-method-to-derive-complex-valued-refractive-index-functions-of-transparent-materials\/","title":{"rendered":"Inversion method to derive complex-valued refractive index functions of transparent materials"},"content":{"rendered":"<div class=\"row\"  id=\"row-1644105725\">\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<h1>4.12. Inversion method to derive complex-valued refractive index functions of transparent materials<\/h1>\n<h2>4.12.1. Background and theory<\/h2>\n<p>As stated above, it is essential to have knowledge of the complex-valued refractive index (cRI) functions of all materials used in physically-based raytracing. While it is possible to retrieve models or empirical data for some \u201cbulk\u201d materials, e.g. of the refractiveIndex database (Polyanskiy, 2022), this data is unfortunately currently not available for many other materials. For building science applications, it is of particular importance to have models for coated and uncoated glass panes, to be able to model glazings accurately. While the necessary refractive index information is not available, Lawrence Berkley National Laboratories kindly maintains the freely accessible and continuously updated glazing database IGDB (LBNL Windows and Daylighting Group, 2021), containing spectral measurements for almost all relevant types of glass. The database provides transmittance and reflectance spectra for the global radiation range. The datasets typically include measured spectral functions for the frontside reflectance \ud835\udc45\ud835\udc53, backside reflectance \ud835\udc45\ud835\udc4f and transmittance \ud835\udc47. These measurements are generally provided for normal, i.e. near-normal, incidence only. The data is provided for uncoated as well as for coated glass panes.<br \/>\nA method has been developed to generate the required cRI functions based on these measured spectral data by inversion, i.e. appropriate parametric models for the cRI functions are varied until the modelled reflectance and transmittance spectra match empirical spectra provided by IGDB. The two real-valued components forming the complex-valued refractive index function, the refractive index \ud835\udc5b and the extinction coefficient , have to be determined for the entire global radiation spectrum. Due to the many resulting degrees of freedom, the inversion can only be performed using a multi-parameter optimisation algorithm.<\/p>\n<\/div><\/div>\n<div class=\"col medium-6 small-12 large-6\"  ><div class=\"col-inner\"  >\n<h2>4.12.1.1. Thick slab mode<\/h2>\n<p>While the MC-raytracing approach could be used to determine the total reflectance and transmittance of the uncoated or coated glass panes, the optimisation algorithm to fit the refractive indices relies on fast computation. Since the total reflectance and transmittance of the glass panes, being the result of interreflections, can be solved analytically, there is no reason to apply the \u2013 for this application slower &#8211; MonteCarlo method. Instead, the thick<\/p>\n<\/div><\/div>\n<div class=\"col medium-6 small-12 large-6\"  ><div class=\"col-inner\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1082717277\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"579\" height=\"385\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a8-5.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1082717277 {\n  width: 100%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>slab model is applied to efficiently model a coated or uncoated glass pane&#8217;s front- and backsidereflectances and transmittance. As discussed in section 4.3.7, this model assumes a non-coherentsuperposition of many interreflections within the glass pane. Figure 53 provides a schematic overviewof the model. Since the measurements are performed with air contact, the media 0 and 2 are modelledas air (approximated as vacuum). Medium 1 is the actual glass material. Interfaces 01 and 12 aremodelled using the generalized Fresnel approach (see section 4.7). At interface 01, a thin-film stack consisting of N layers can additionally be considered. The interfaces&#8217; reflectivity and transmissivityvalues are denoted with ordered indices to indicate the direction, even though most are identical forboth directions. All coefficients are determined using equations (53). Note that for the case of a thinfilm stack present at interface 01, any absorption within the thin-film will also be included in themodel, as for this case \ud835\udc4701 + \ud835\udc4501 &lt; 1 will apply.<br \/>\nBeyond the Fresnel coefficients, relevant for the scattering processes at the interfaces, the transmittance \ud835\udc471 of the glass pane is required. It is obtained by applying Lambert-Beer\u2019s law (see section 4.3.6):<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_204759961\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"622\" height=\"187\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-5.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_204759961 {\n  width: 70%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Whereas the attenuation coefficient \uf061 is calculated using equation (40), and the angle \ud835\udf031 is calculated based on Snell\u2019s law (46).<br \/>\nThe derivation of the total reflectance and transmittance, being the superposition of all interreflections, can easily be comprehended if the coefficients R and T are understood as probabilities. In this view, the probabilities for each path have to be added, while the required probabilities forming a path have to be multiplied. A similar derivation, but with slightly different components, is, e.g., provided by Stenzel (2016, p. 131).<br \/>\nThe infinite summation of all reflection paths (see Figure 53) then leads to a geometric series, which can easily be evaluated into a simple fractional term:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_520365875\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"796\" height=\"224\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-6.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-6.jpg 796w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-6-768x216.jpg 768w\" sizes=\"(max-width: 796px) 100vw, 796px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_520365875 {\n  width: 81%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>In complete analogy, the total reflectance for the front side and, by flipping the direction and exchanging the interface indices correspondingly, the backside of the glass pane can be derived. The frontside reflectance \ud835\udc45\ud835\udc53,\ud835\udc5d\ud835\udc4e\ud835\udc5b\ud835\udc52 and the backside reflectance \ud835\udc45\ud835\udc4f,\ud835\udc5d\ud835\udc4e\ud835\udc5b\ud835\udc52 are:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_749347772\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"313\" height=\"164\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a11-5.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_749347772 {\n  width: 38%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Note that the expressions could further be simplified by expressing transmittance values by reflectivities and applying identities for transmittance values in reverse directions. However, since the coefficients are readily provided by the algorithms and in order to average out numerical inaccuracies, the expressions were implemented as stated in equations (76) and (77). Note also that all quantities Light, sun and optics &#8211; applied principles, models and methods RadiCal, D. R\u00fcdisser 95 are functions of wavelength and incidence angle; for reasons of clarity, the corresponding arguments were omitted in the above equations.<\/p>\n<h2>4.12.1.2. A parametric model for coated and uncoated glass panes<\/h2>\n<p>After some testing, the following models were identified as best suited to determine the cRI functions by inversion. As is always the case for such optimisation problems, it is critical to provide a physically appropriate model with as many variable parameters as required &#8211; but only as few as necessary, to avoid overtraining the model. In the implemented method, mainly natural cubic spline functions (G., Press et al., 1991) with N knots are used to individually approximate the real-valued functions \ud835\udc5b and that constitute the complex-valued refractive index function. Since often ranges over several orders of magnitude, a logarithmic mapping of the spline was tested but was found not to perform better than the linear approach.<br \/>\nThe following constraints were applied in the optimisation process:<\/p>\n<ul>\n<li>the wavelength (\u201cx-coordinate\u201d) of the first knot of the spline was constrained to range within 0 nm and 300 nm.<\/li>\n<li>the wavelength (\u201cx-coordinate\u201d) of the last knot of the spline was constrained to range within 2500 nm and 3000 nm.<\/li>\n<li>all values for glazings were constrained in the range of [1E-8, 10].<\/li>\n<li>all values for coatings were constrained in the range of [1E-8, 100].<\/li>\n<li>all \ud835\udc5b values for glazings were constrained in the range of [1E-5, 4].<\/li>\n<li>all \ud835\udc5b values for coatings were constrained in the range of [1E-5, 6].<\/li>\n<\/ul>\n<p>Note that the actual \ud835\udc5b and functions for the coatings do not have to reflect the true physical properties if taken individually and, therefore, can seem entirely out of range. A two-layer substitute model is applied to approximate the behaviour of the thin-film stack, potentially consisting of a double-digit number of different materials, e.g. for a low-E coating (Ding and Clavero, 2017). The actual optical behaviour of the thin-film assembly is determined as the result of the multiplications of the individual characteristic matrices of each layer (see equations (50) &#8211; (52)). This matrix multiplication relies on the cRI functions of the individual layers, as well as on their thicknesses \ud835\udc51\ud835\udc56 . Therefore, physically not meaningful values for \ud835\udc5b, and d achieved for the single layers of the twolayer model can still be suitable to resemble the correct physical behaviour of a multi-layer assembly. While spline functions are used to model the refractive index and extinction coefficient functions of all coating layers, the refractive index of the glass can be modelled based on a less complex model. The following dispersion formula (see TRIsimpleDispersionAndKspline of Table 5) has proven to be suitable to model the relevant uncoated glass pane types:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_91451197\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"192\" height=\"59\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a12-5.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_91451197 {\n  width: 20%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>As can be seen, this function relies on three parameters only, while the natural cubic spline function has N degrees of freedom.<br \/>\nSince IGDB contains data for coated and uncoated glass panes, the uncoated panes\u2019 spectra are always used to determine the model parameters for the plain glass material. The given thickness of the glass pane is provided as a constant model parameter.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1845889692\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"777\" height=\"441\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-5.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-5.jpg 777w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-5-705x400.jpg 705w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-5-768x436.jpg 768w\" sizes=\"(max-width: 777px) 100vw, 777px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1845889692 {\n  width: 77%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>In order to determine the model for coated glass panes, the model parameters of the uncoated glass, comprising three constants for the refractive index (eq. (79)) and a series of N spline knots for the extinction coefficient, are used as constant values. Only the parameters relevant for the two coating layers, comprising the thickness and the two series of knots coordinates for each coating layer, are the free parameters in the optimisation process to replicate the measured spectra of the coated glass pane.<\/p>\n<h2>4.12.1.3. An ideal optimisation cost function<\/h2>\n<p>Ideally, the optimisation should be controlled by applying transmittance and reflectance spectra for multiple incidence angles as the target quantity of the optimisation. Based on the experience gained when working with the optimisations, it seems likely that a number of only two to four different incident angles should be sufficient for this purpose.<br \/>\nThe objective of the optimisation process is defined by providing a cost function that rates the optimality of a solution. For the present case, the following definitions are proposed: first, the deviations (\u2206\ud835\udc47, \u2206\ud835\udc45\ud835\udc53 and \u2206\ud835\udc45\ud835\udc4f) of the model vs. the measurement for a specific incidence angle \ud835\udf03\ud835\udc5b and the wavelength are defined as:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1732240230\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"390\" height=\"98\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a1-7.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1732240230 {\n  width: 40%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Where \ud835\udc47, \ud835\udc45\ud835\udc53 and \ud835\udc45\ud835\udc4f are the model results, according to (76) &#8211; (78), and the corresponding values marked with the hat accent represent measurement values provided by the IGDB spectrum files. The global radiation spectrum is sampled in two dimensions. First, by using \ud835\udc40 distinct, energetically equidistant wavelengths (see section 4.4.2), and second, for \ud835\udc41 available incidence angles \ud835\udf03\ud835\udc5b. The loss functions to assess the quality of the fitting are defined as:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_118044440\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"745\" height=\"210\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a2-8.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_118044440 {\n  width: 69%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>The weighted loss functions finally determine the cost function for the optimisation process:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1328932995\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"471\" height=\"38\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a3-6.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1328932995 {\n  width: 55%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>The coefficients \ud835\udc84\ud835\udc8a can be used to balance the three components of the cost function. Unfortunately, this approach could not be applied and tested fully, as only spectra for one angle (near-normal incidence) were available. However, based on the presented definitions, an extended cost function is introduced below.<\/p>\n<h2>4.12.1.4. The pragmatic optimisation cost function<\/h2>\n<p>It turned out that applying spectral data only for one angle (near-normal incidence) was insufficient to perform a meaningful determination of the model parameters for the refractive index functions. As the model offers a relatively large number of free parameters, overfitting occurs if insufficient fitting data is provided. Therefore, additional data or constraints are needed. Since no measurements for varying angles are available, two additional constraints are imposed:<\/p>\n<ul>\n<li>1. the normal-incidence reflectance and transmittance spectra should also be valid for a specified low, but non-zero, incidence angle \ud835\udf031.<\/li>\n<li>2. the ratio of the total transmittance for a specified higher oblique angle \ud835\udf032 to the total transmittance at normal incidence should match a specified value.<\/li>\n<\/ul>\n<p>Both constraints are physically reasonable. The first condition helps to avoid spectrally sensitive thin-film solutions that frequently occur if the constraint is not applied. Such solutions will provide the correct reflectance behaviour for the normal incidence but will show strong spectral and angular dependences for oblique angles, e.g. as the colourful \u201crainbow\u201d interference known from thin oil or soap films. Such effects are, in fact, avoided in the production of glazings, as this would impair the visual quality of the glazing. Since the angular characteristics of glazing generally show very little change at incidence angles less than 10 to 20 degrees (Rubin et al., 1999; Roos et al., 2001), an angle \ud835\udf031 within this range can be chosen.<br \/>\nTo further restrict the optimisation results to physically plausible solutions, a target value for the decrease of the transmittance with higher angles is specified in the second constraint. While the entire spectrally resolved reflectance and transmittance data is provided in the first constraint, here, only the total (=spectrally integrated) value of the transmittance is considered. Since the sampling wavelengths \uf06c\ud835\udc5a are provided for energetically equidistant intervals, the total transmittance can simply be calculated by averaging the values. Hence, the additional loss function can be written as:<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1504041967\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"418\" height=\"93\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a4-7.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1504041967 {\n  width: 46%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Where \ud835\udc53\ud835\udc47 represents the target value for the decay of the total transmittance. The other loss functions remain the same as stated above, however two angles are considered there now: \ud835\udf030 representing near-normal incidence, and \ud835\udf031 as low angle for which the normal-incidence spectra should still be valid.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_783463526\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"713\" height=\"39\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a6-6.jpg\" class=\"attachment-large size-large\" alt=\"\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_783463526 {\n  width: 66%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>In the practical implementation, the relevant angles and weighting-factors were defined as: \ud835\udf030 = 10\u22123 \u00b0 (nonzero for numerical stability), \ud835\udf031 = 15 \u00b0 , \ud835\udf032 = 70 \u00b0 and \ud835\udc501 = \ud835\udc502 = \ud835\udc504 = 1, while \ud835\udc503 = 0.5.<\/p>\n<h2>4.12.2. Implementation<\/h2>\n<p>The implementation is contained in the module rc_fittrrspectrum3, whereby the version number indicates the evolution of the module. The class TfitTRRSpectrum contains all functions necessary for the optimisation process and additional tools used to import, export and report the optimisation results.<br \/>\nThe progress of the potentially multi-hour-long optimisation can be monitored in text form, as well as in the form of an updating chart.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_2102908255\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"923\" height=\"559\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-7.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-7.jpg 923w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-7-660x400.jpg 660w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-7-768x465.jpg 768w\" sizes=\"(max-width: 923px) 100vw, 923px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_2102908255 {\n  width: 89%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>The implementation is an application of the simulated annealing algorithm (see section 5.2), where equation (84) is used as a cost function. The relations for the thick slab model derived above, see (76) &#8211; (78), are used to compute the analytic transmittance and reflectance values used for the fitting to the empirical data during the optimisation process.<br \/>\nIn the implementation, the ipfisn and ipfspn of the NumLib library (Free Pascal Team, 2021) are used to solve and evaluate the linear equation systems for the applied natural cubic spline fitting.<\/p>\n<h2>4.12.3. Testing and validation<\/h2>\n<p>In order to demonstrate the potential of the method, the complex-valued refractive index (cRI) functions of three different materials are determined from their transmission and reflection spectra. The objective of the test is two-fold:<\/p>\n<ul>\n<li>1.) If applied in the raytracing (or thick-slab model), the determined cRI function should, with good accuracy, result in transmission and reflection spectra matching the measured data they were generated on. The match is quantified with three different parameters that are also applied during the optimisation process:\n<ul>\n<li>a. Mean error of samples: as the wavelength samples are distributed according to the global radiation spectral density, the total mean error also coincides with the integrated total error considering the entire spectrum.<\/li>\n<li>b. Root-mean-square error of samples: this value is the best indicator for the average spectral deviation of the three spectra.<\/li>\n<li>c. Max. error of samples: the maximum deviation for any sample in any of the three spectra (transmission, reflection-frontside, reflection-backside) is determined.<\/li>\n<\/ul>\n<\/li>\n<li>2.) The determined cRI functions should be physically correct, i.e. represent the \u201ctrue\u201d cRI functions.<\/li>\n<\/ul>\n<h2>4.12.3.1. Case 1 &#8211; determination of cRI functions for uncoated window pane<\/h2>\n<p>In this first case, the cRI of an uncoated pane used for insulated glazing units (IGU) is determined. The result is later used in the full-system validation (chapter 8). The pane is of type Planibel Clearlite by the manufacturer AGC Glass Europe. The nominal thickness is 4 mm, while the actual thickness is 3,85 mm. The related spectra were exported from the IGDB database (LBNL Windows and Daylighting Group, 2021).<br \/>\nFor the inversion process to determine the cRI of the uncoated glass, the dispersion relation stated in equation (79) was used to model the refractive index function (\ud835\udc5b), while a spline function with 10 knots was used to model the extinction coefficient function. The knots are moved in both dimensions during the optimisation process. Only the wavelength coordinates of the first and last knot are held constant at the boundaries of the global radiation spectrum. The optimisation process comprised 35 million iteration steps and lasted a few hours. The finally determined cRI is depicted in Figure 55. The dots on the blue line indicate the knots of the spline function.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_216497486\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"965\" height=\"524\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-6.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-6.jpg 965w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-6-737x400.jpg 737w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-6-768x417.jpg 768w\" sizes=\"(max-width: 965px) 100vw, 965px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_216497486 {\n  width: 100%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Figure 56 shows the cRI as determined by Rubin (1985) for unspecified \u201cclear glass\u201d. As Rubin notes in his publication, the absorptance and reflectance of the window glass are affected by the FeO concentration and further additives. This means that significant variations can be expected among different clear glasses. However, as can be seen by comparison of Figure 55 and Figure 56, the match between the values for a modern glass, determined with the novel method presented here and the values determined by Rubin in 1985, is remarkably high.<br \/>\nThe match of the transmission and reflection spectra is depicted in Figure 57. The precision of the reconstruction is very high, as the only visible deviation occurs in the transmission spectrum at approx. 2.2 \u00b5m. However, the deviation is still small, and the energetic impact of the deviation is even less significant, as this part of the global radiation spectrum exhibits a low spectral flux density.<br \/>\nThe maximum sample error is 1.1% for the transmittance and less than 0.08% for the reflectance, and the root mean squared error is 0,4% for all transmittance and reflectance samples. The mean sample error is 0.09% for the transmittance and + 0.001% for the reflectance samples. Since mean sample errors are spectrally weighted, the errors are consistent with the total error for the entire global radiation spectrum.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_868718028\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"955\" height=\"417\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-7.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-7.jpg 955w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-7-800x349.jpg 800w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-7-768x335.jpg 768w\" sizes=\"(max-width: 955px) 100vw, 955px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_868718028 {\n  width: 90%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Figure 58 finally shows the angular dependence of the transmittance and reflectance of the glass pane for global radiation. As discussed in the state-of-the-art analysis (section 3.2), even the more advanced, commonly used modelling approaches apply simplified models in order to determine the angular dependence of transmittance and reflectance. The depicted angular dependencies are relevant for global radiation determined by spectrally weighted integration.<\/p>\n<h2>4.12.3.2. Case 2 &#8211; determination of cRI functions for low-E coated window pane<\/h2>\n<p>In this example, a low emissivity (low-E) coating (of type iplus Top 1.1) is applied to the previously considered uncoated glass pane. As discussed in the theory section, the coating, consisting of multiple layers, is approximated using a two-layer substitute model. The already determined cRI for the 3.85 mm thick glass pane is held constant in the optimisation process. The optimisation involves the \ud835\udc5b and functions of both layers, where each function is modelled using spline functions with 10 knots. Beyond that, the two thicknesses d of the layers are variables during the optimisation process. As discussed above, an additional parameter for the total angular weakening of transmittance is required due to the lack of angular spectral measurements. The value was taken from measurement data contained in a publication by Rubin et al. (1999). The transmission of the specimen \u2018double Ag 3mm clear\u2019 was found to decrease to 63.8% of its near-normal value at 70\u00b0. This value was used as a target for the optimisation process. The optimisation terminated after 12.3 million iterations. The layer thicknesses for the substation model were determined as 9.0 nm for the first (=glass-sided) layer and 37.6 nm for the second (=top) layer, respectively.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_2053373727\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"753\" height=\"552\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a12-6.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a12-6.jpg 753w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a12-6-546x400.jpg 546w\" sizes=\"(max-width: 753px) 100vw, 753px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_2053373727 {\n  width: 79%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col medium-6 small-12 large-6\"  ><div class=\"col-inner\"  >\n<p>As mentioned in the theory section, the two individual coating layers do not necessarily need to have physically meaningful cRI functions or thicknesses, as their effect is determined by applying the matrix method to the entire stack. However, it can clearly be seen that the method again captures relevant physical properties of the materials, as the cRI of layer 1 (n1, k1 in Figure 59) shows a clear qualitative and quantitative resemblance to the measured values (Ciesielski et al., 2017; Polyanskiy, 2022) for a 20 nm thick silver coating deposited on a SiO2 substrate (see Figure 60). This is remarkable, as the optimisation process has only a few restrictions and is offered a large number of degrees of freedom available. While the actual layering of the lowE coating is not available to the author, it can<\/p>\n<\/div><\/div>\n<div class=\"col medium-6 small-12 large-6\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1416122670\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"478\" height=\"529\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-6.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-6.jpg 478w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-6-361x400.jpg 361w\" sizes=\"(max-width: 478px) 100vw, 478px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1416122670 {\n  width: 90%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>be assumed that it comprises several different materials, including one to three silver layers. Due to production and durability aspects, the actual layering is usually complex and involves additional materials (Ding et al., 2017).<br \/>\nThe comparison of the measured vs. the modelled spectra (Figure 61) shows that the applied twolayer model is able to replicate the transmittance and reflectance behaviour of the more complex layering very well. This is also supported by the error measures: transmission: max. sample error: 2.8%, mean sample error: &lt;0.01%; reflection-frontside: max. sample error: 0.85%; mean sample-error: &lt;0.01%; reflection-backside: max. sample error: 3.2%; mean sample-error: &lt;0.01%;).<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1949137018\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"963\" height=\"514\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-5.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-5.jpg 963w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-5-749x400.jpg 749w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-5-768x410.jpg 768w\" sizes=\"(max-width: 963px) 100vw, 963px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1949137018 {\n  width: 90%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Figure 62 shows the angular dependencies for transmittance, frontside- and back-side reflectance as well as absorptance. Additionally, two references for the angular transmittance are depicted. The first is taken from Rubin et al. (1999) and represents measured data for a silver double-coated pane with a nominal thickness of 3 mm. Since the transmission ratio at 70\u00b0 was an optimisation target, it shows a perfect match at this angle. However, a good match is also visible regarding the general shape of the curve. The second reference was taken from Roos et al. (2001). Figure 10 of the cited publication provides a model fit for a silver-coated Planitherm pane with unspecified thickness and few angular measurements. The angular dependence modelled by Roos et al. shows slightly higher values than are determined here.<\/p>\n<h2>4.12.3.3. Case 3 &#8211; determination of cRI functions for ETFE membrane<\/h2>\n<p>In order to include a non-glass example of practical relevance, the cRI function of a clear ETFE membrane, as used for skylights in buildings, is determined. Transmission and reflection spectra for the relevant global radiation range were extracted from a chart provided by a manufacturer (Buitink Technology, 2022) (\u201cstandard foil\u201d curves). The thickness of the sample that was used to determine the spectra is specified as 200 \u00b5m.<br \/>\nWhile an appropriate dispersion relation was used to model the \ud835\udc5b function for the uncoated glass, the cRI of the ETFE membrane is now determined by optimising two spline functions. The number of available knots was set to 9 for the \ud835\udc5b-component and 19 for the \uf06b-component, as more fluctuation in the absorption can be expected. Even though the optimisation is offered a very high number of degrees of freedom for the spline functions, remarkably smooth and plausible \ud835\udc5b and \uf06b functions are determined in the optimisation process (see Figure 63). As can be seen in the \ud835\udc5b function, the optimisation process concentrated most knots below 1 \u00b5m to capture the rise towards shorter wavelengths accurately.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_412431214\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"746\" height=\"481\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a15-4.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a15-4.jpg 746w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a15-4-620x400.jpg 620w\" sizes=\"(max-width: 746px) 100vw, 746px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_412431214 {\n  width: 70%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>Again literature was reviewed to assess the validity of the inversion result. A reference containing measurement data has been found in a publication on Optical properties of materials for concentrator photovoltaic systems (French et al., 2009). The publication includes results of spectroscopic ellipsometer and spectrophotometer measurements performed on a clear ETFE membrane.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1547793583\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"952\" height=\"368\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-5.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-5.jpg 952w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-5-800x309.jpg 800w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-5-768x297.jpg 768w\" sizes=\"(max-width: 952px) 100vw, 952px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1547793583 {\n  width: 80%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<p>The comparison of the measured \ud835\udc5b-function (blue curve in Figure 64 left) vs. the function determined by inversion (Figure 63) shows that for wavelengths longer than 1,3 \u00b5m, both functions quickly converge to an almost constant value around 1.40. In the UV range, a sharp rise towards shorter wavelengths, typical for polymers, is visible. While the shape of the rise is very similar, a quantitative deviation is clearly visible in this region. However, the products are of different manufacturers, and limitations of the measurement devices in the UV region can distort the measurement results at the end of the measurement range.<br \/>\nThe limited vertical resolution of the absorption coefficient chart (blue curve in Figure 64 right) allows only a crude assessment of the validity of the \uf06b-function, containing the extinction coefficients. As can be seen in the chart, three main absorption regions can be identified (indicated with blue arrows). These correspond to three absorption regions determined in the inversion, see Figure 63. Note that Figure 64 uses a linear scale, while the extinction coefficient in Figure 63 is depicted using a logarithmic scale. By converting the estimated absorption values using Beer-Lamberts law, eq. Figure 64 Measured index of refraction and absorption coefficients of ETFE membranes (blue lines) source: French et al. (2009) Light, sun and optics &#8211; applied principles, models and methods RadiCal, D. R\u00fcdisser 104 (26), it can be shown that the values match, at least in order of magnitude, being the relevant scope here.<br \/>\nFigure 64 Measured index of refraction and absorption coefficients of ETFE membranes (blue lines) source: French et al. (2009) Light, sun and optics &#8211; applied principles, models and methods RadiCal, D. R\u00fcdisser 104 (26), it can be shown that the values match, at least in order of magnitude, being the relevant scope here. Figure 65 demonstrates that the match between the measured and modelled transmission and absorption spectra (the optimisation process&#8217;s target) is excellent: transmission: max. sample error: 1.8%, mean sample error: &lt;0.01%; reflection: max. sample error: 0.4%; mean sample-error: &lt;0.01%. The derived angular transmittance and reflectance for global radiation are depicted in Figure 66.<\/p>\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_1083766463\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"973\" height=\"527\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a17-5.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a17-5.jpg 973w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a17-5-739x400.jpg 739w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a17-5-768x416.jpg 768w\" sizes=\"(max-width: 973px) 100vw, 973px\" \/>\t\t\t\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\n<style scope=\"scope\">\n\n#image_1083766463 {\n  width: 90%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<\/div>\n<style>\n#menu-main li:nth-child(4) .sub-menu{display:block !important;}\n<\/style>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-right-sidebar.php","meta":{"footnotes":""},"_links":{"self":[{"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages\/1018"}],"collection":[{"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/comments?post=1018"}],"version-history":[{"count":3,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages\/1018\/revisions"}],"predecessor-version":[{"id":1966,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages\/1018\/revisions\/1966"}],"wp:attachment":[{"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/media?parent=1018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}