{"id":667,"date":"2023-10-20T10:52:54","date_gmt":"2023-10-20T10:52:54","guid":{"rendered":"https:\/\/uneedtalk.com\/config\/tredword\/?page_id=667"},"modified":"2023-12-01T13:09:04","modified_gmt":"2023-12-01T13:09:04","slug":"generalized-fresnel-coefficients","status":"publish","type":"page","link":"https:\/\/radi-cal.org\/method\/generalized-fresnel-coefficients\/","title":{"rendered":"Generalized Fresnel coefficients"},"content":{"rendered":"<div class=\"row\"  id=\"row-1952863031\">\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner\"  >\n<h1>Generalized Fresnel coefficients<\/h1>\n<h2>4.7.1. Background and theory<\/h2>\n<p>The derivations and equations presented below are provided in different forms in various publications. A comprehensive derivation that starts with the Maxwell equations in vector form can be found in the book Thin-film Optical filters (Macleod, 2001). For the practical implementation, the publications by Craig (1987) and Keller (2001) were helpful.<br \/>The commonly known Fresnel\u2019s equations are used as a starting point for reference. They can be derived by solving Maxwell\u2019s equations for non-absorbing media. They are commonly stated 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_124069789\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"1002\" height=\"245\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-2.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-2.jpg 1002w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-2-800x196.jpg 800w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a7-2-768x188.jpg 768w\" sizes=\"(max-width: 1002px) 100vw, 1002px\" \/>\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_124069789 {\n  width: 79%;\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 ratio of the incidence angle \ud835\udf031 and transmittance angle \ud835\udf032 is determined by Snell\u2019s law, see eq. (18). The Fresnel coefficients \ud835\udc5f\ud835\udc60,\ud835\udc5d and \ud835\udc61\ud835\udc60,\ud835\udc5d determine the amplitudes of the reflected and transmitted radiation for each polarisation state. The coefficients for the reflected and transmitted power, referred to as reflectance \ud835\udc45\ud835\udc60,\ud835\udc5d and transmittance \ud835\udc47\ud835\udc60,\ud835\udc5d are given by:<\/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_866503680\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"720\" height=\"74\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a8-2.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_866503680 {\n  width: 71%;\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>This takes into account that the power is proportional to the square of the amplitude and that the angle as well as the impedance of the medium will change for the transmitted beam. For normal incidence (\ud835\udf031 \u2261 \ud835\udf032 \u2261 0) the amplitude coefficients (\ud835\udc5f,\ud835\udc61) and power coefficients (\ud835\udc45, \ud835\udc47) simplify to the elegant form:<\/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_553235355\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"716\" height=\"184\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a9-2.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_553235355 {\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>By using complex-valued refraction indices and introducing the concept of effective refractive indices, more general expressions valid for absorbing media and oblique (=non-normal) incidence angles will be derived now. The finally obtained relations show a striking similarity to equations (43). <br \/>The derivation is carried out by solving Maxwell\u2019s equation for interfaces of materials and demanding continuity of the electric and magnetic field at the boundary. While this can be done by applying vector calculus in three dimensions, a more straightforward and more efficient approach is possible when the energy flow normal to the interface is considered. In this case, the relevant electric field vector \ud835\udc38\u20d7 and the magnetic field vector \ud835\udc3b\u20d7 are oriented parallel (tangential) to the boundary. It is, therefore, sufficient to consider the amplitudes only, denoted as \ud835\udc38\u0303 and \ud835\udc3b\u0303. In order to include phase information, the amplitudes are represented by complex-valued numbers. The tilde in superscript is further used to indicate all quantities that are complex numbers. s and p subscripts are used in the following to indicate the direction of polarisation for which the relation is valid. If the letters for Light, sun and optics &#8211; applied principles, models and methods RadiCal, D. R\u00fcdisser 63 both directions are indicated in an equation (separated by a comma), it will mean that the equation is separately valid for either polarisation state (using only s or only p polarised quantities throughout the equation). <br \/>Following the approach of considering the energy flows perpendicular to the boundary, it is helpful to introduce an effective index of refraction, which links the two tangential electromagnetic field components \ud835\udc38\u0303 and \ud835\udc3b\u0303:<\/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_1840965207\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"144\" height=\"87\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a10-3.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_1840965207 {\n  width: 16%;\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 applied approach considering the flow of energy perpendicular to the boundary, it can be shown that for the s-polarised and p-polarised components, the effective indices of refraction can be derived of the complex-valued index of refraction \ud835\udc5b\u0303 and the propagation angle \ud835\udf03\u0303 by:<\/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_1600767416\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"671\" height=\"76\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a11-2.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_1600767416 {\n  width: 67%;\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>It is important to note that the general propagation angle \ud835\udf03\u0303 is now a complex quantity. Consequently, the angle of refraction \ud835\udf03\u0303 2 for the transmitted beam has to be derived from the angle of incidence \ud835\udf03\u0303 1 by solving Snell\u2019s law in the complex-valued form:<\/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_1698597032\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"226\" height=\"51\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a12-2.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_1698597032 {\n  width: 25%;\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>Unlike its real-valued counterpart, the complex-valued effective index of refraction \uf068\u0303 contains angular information, as well as information on the attenuation (through the complex-valued index of refraction \ud835\udc5b\u0303). The amplitude transmission coefficients for reflection (\ud835\udc5f\u0303) and transmission (\ud835\udc61\u0303) and the resulting reflectance (\ud835\udc45) and transmittance (\ud835\udc47), at the boundary of two absorbing or non-absorbing media 1 and 2, can for any incidence angle be stated 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_289862439\">\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=\"175\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a13-2.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_289862439 {\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>It can be seen how this very elegant and simple form of the generalized form of Fresnel equations results only through the introduction of the complex-valued effective index of refraction. The similarity to the earlier, non-complex-valued form is remarkable, see equation (43). However, taking the generalization a step further, a different definition for the amplitude coefficients is chosen to additionally consider the effect of potentially present thin-films in between the two media. It can be shown (Macleod, 2001) that they can be calculated by the tangential electric and magnetic field amplitudes in the incident medium:<\/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_316223992\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"803\" height=\"86\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-2.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-2.jpg 803w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-2-800x86.jpg 800w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a14-2-768x82.jpg 768w\" sizes=\"(max-width: 803px) 100vw, 803px\" \/>\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_316223992 {\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>If the incident field amplitudes \ud835\udc38\u0303 1 \ud835\udc60,\ud835\udc5d and \ud835\udc3b\u0303 1 \ud835\udc60,\ud835\udc5d are determined by:<\/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_1924413326\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"244\" height=\"83\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a15-1.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_1924413326 {\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>Where \ud835\udc36\u0305\ud835\udc60,\ud835\udc5d is the characteristic matrix of a thin-film assembly consisting of N thin-films.<\/p>\n<\/div><\/div>\n<div class=\"col medium-7 small-12 large-7\"  ><div class=\"col-inner\"  >\n<p>In the absence of thin-films, the matrix \ud835\udc6a\u0305\ud835\udc60,\ud835\udc5d can be omitted, i.e. replaced with the unity matrix. For this case, it is evident that equations (48) will turn into equations (47) again. Hence, the definition for the reflection and transmission coefficients provided by (48) is the most general, as it allows consideration of 0 to N thin-films at the boundary of two materials. For this reason, this definition has been chosen for the actual implementation. In order to get the characteristic matrix of the thin-film the matrix multiplication is repeatedly applied using the characteristic matrix for each layer \ud835\udc50\ud835\udc57 :<\/p>\n<\/div><\/div>\n<div class=\"col medium-5 small-12 large-5\"  ><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_967768426\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"402\" height=\"417\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-2.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-2.jpg 402w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a16-2-386x400.jpg 386w\" sizes=\"(max-width: 402px) 100vw, 402px\" \/>\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_967768426 {\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 text-center\"  >\n\t<div class=\"img has-hover x md-x lg-x y md-y lg-y\" id=\"image_234311953\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"294\" height=\"47\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a17-2.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_234311953 {\n  width: 32%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-left\"  >\n<p>It is crucial, of course, to keep the correct order of the layers, as matrix multiplication is not commutative. The coatings are counted starting at the 2nd medium (see Figure 33). The characteristic<br \/>\nmatrix for every layer \ud835\udc57 is given by:<\/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_171660098\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"387\" height=\"130\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a18-2.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_171660098 {\n  width: 35%;\n}\n<\/style>\n\t<\/div>\n\t\n<\/div><\/div>\n<div class=\"col small-12 large-12\"  ><div class=\"col-inner text-left\"  >\n<p>The characteristic matrix transforms the complex amplitudes of the electric and magnetic field according to the layer&#8217;s electrodynamic properties and thickness \ud835\udc51\ud835\udc57 . The phase factor \ud835\udeff\u0303 \ud835\udc57 results of the ratio of the effective optical thickness \u2206\u0303 \ud835\udc57 and the wavelength \uf06c:<\/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_772336005\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"361\" height=\"75\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a19-2.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_772336005 {\n  width: 35%;\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 specific propagation angles \ud835\udf03\u0303 \ud835\udc57 and the corresponding effective refractive indices for each layer j can again be found by applying Snell\u2019s law, eq. (46).<br \/>Finally, the reflectance coefficient \ud835\udc45 and transmittance coefficient \ud835\udc47 to calculate the power of the reflected and the transmitted beam for each polarisation state can again be found by:<\/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_1389602881\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"661\" height=\"57\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a20-1.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_1389602881 {\n  width: 60%;\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 additional factor required to calculate the transmittance coefficient considers that the impedance, as well as the propagation direction, is generally different in the two media. In the further text, these coefficients will be referred to as generalized Fresnel coefficients. In the following application, it is essential to additionally consider the phase change \ud835\udf16 of the reflected and transmitted beam. They are simply represented by the arguments of the complex coefficients \ud835\udc5f\u0303and \ud835\udc61\u0303:<\/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_1387282263\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"654\" height=\"34\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a22-1.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_1387282263 {\n  width: 65%;\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 corresponding coefficient for absorptance \ud835\udc34 can be found by applying the law of energy conservation:<\/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_1846787145\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"170\" height=\"39\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a23-1.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_1846787145 {\n  width: 18%;\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, again, all relations in (53),(54) and (55) are valid for each polarisation state separately;  for the purpose of clarity, the indices s and p were omitted here.<\/p>\n<h2>4.7.2. Implementation<\/h2>\n<p>The implementation of above formulae is performed in the module TFresnel of the library rc_fresnel:<\/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_1114082912\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"1020\" height=\"668\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a1-3.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a1-3.jpg 1082w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a1-3-610x400.jpg 610w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a1-3-768x503.jpg 768w\" sizes=\"(max-width: 1020px) 100vw, 1020px\" \/>\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_1114082912 {\n  width: 72%;\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 module contains, on the one hand, private functions which perform the actual calculation in line  with the relations given above. The operator overloading features of the Pascal language allows a  convenient and transparent implementation of the relevant equations, as the complex-number algebra can be converted into programming code straightforwardly. On the other hand, the public functions of the module allow the initialisation of the materials and the direct calculation of the M\u00fcller  matrices for reflection and transmission (see section 4.9). Alternatively, only the parameters required  to derive the M\u00fcller matrices can be generated. These essential parameters are contained in the type  definition TTRpolRecord and reflect the results of equations (48) and (53). In the actual implementation of the raytracing (see chapter 7), the function getRTandThetaS is called to perform the Fresnel  calculations, while the function getRandTMuMat is called to establish the M\u00fcller matrices of the calculated parameters.<\/p>\n<h2>4.7.3. Testing and validation<\/h2>\n<p>The full validation is implicitly included in the M\u00fcller matrix section (4.9.3), as the M\u00fcller matrix elements rely on the Fresnel calculations. During the implementation, the Fresnel module was additionally tested based on commonly known material behaviours.<br \/>\nExemplary, Figure 34 illustrates the angular reflectivity for three different materials, resolved for three different wavelengths and the main polarisation states. As can be seen, the dielectric (i.e. nonconducting) glass exhibits limited wavelength dependence and a pronounced Brewster angle at 57\u00b0 where only s-polarised light is reflected off the surface. Gold shows high reflectivity in the visual and infrared spectral range; however, less than half of the incident radiation is reflected in the UV range. Aluminium generally exhibits a high reflectivity for all wavelengths considered.<\/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_1283393270\">\n\t\t\t\t\t\t\t\t<div class=\"img-inner dark\" >\n\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"1020\" height=\"370\" src=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a2-3.jpg\" class=\"attachment-large size-large\" alt=\"\" srcset=\"https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a2-3.jpg 1115w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a2-3-800x290.jpg 800w, https:\/\/radi-cal.org\/method\/wp-content\/uploads\/2023\/10\/a2-3-768x278.jpg 768w\" sizes=\"(max-width: 1020px) 100vw, 1020px\" \/>\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_1283393270 {\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\/667"}],"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=667"}],"version-history":[{"count":56,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages\/667\/revisions"}],"predecessor-version":[{"id":1961,"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/pages\/667\/revisions\/1961"}],"wp:attachment":[{"href":"https:\/\/radi-cal.org\/method\/wp-json\/wp\/v2\/media?parent=667"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}