Optical models for material appearance 9782759826483

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Institut d’Optique Graduate School Textbook

Mathieu HÉBERT

Optical Models for Material Appearance

Institut d’Optique Graduate School Textbook Series: The IOGS Textbook series is a collection of books based on the training provided to the school’s engineering students. Focusing on the main field of modern photonics, these books present in a comprehensive manner the knowledge essential to the professions of engineers and researchers in this discipline. Written in a clear and pedagogical manner, in English or in French, these books are intended to support experts, during their training at the school, but also in their daily professional life. Institut d’Optique Graduate School Advanced Series: Based on seminars, conferences or the long-standing experience of the school’s researchers or collaborators, these books aim at presenting recent developments in one of the many fields of this discipline.

Printed in France

EDP Sciences – ISBN(print): 978-2-7598-2647-6 – ISBN(ebook): 978-2-7598-2648-3 DOI: 10.1051/978-2-7598-2647-6

All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. Ó Science Press, EDP Sciences, 2022

The Author

Mathieu Hébert is professor assistant since 2010 at Institut d'Optique Graduate School and in Hubert Curien Laboratory in Saint-Etienne, France. He is also head of program of a master dedicated to advanced imaging and material appearance. After engineering studies at CPE-Lyon and a master in image processing at the University Jean Monnet of Saint-Etienne, he spent 8 years at the Ecole Polytechnique Fédérale de Lausanne (EPFL) in Switzerland) for his PhD (graduated in 2006) and post-doctoral research dedicated to optical models for printed surfaces and colored materials. For more than ten years, his teaching is focussed on color sciences, radiometry, and optical models for the apparences of surfaces. His researches in these domains adress various applications fields including visual appearance of products, security printing, and medical imaging. Since 2019, he is the director of a grouping of research ‘Appamat’ of the French National Centre for Scientific Research (CNRS) dedicated to material appearance sciences.

Preface

Of our five senses, it is estimated that vision provides almost 80% of the total input to the brain related to our environment. This means that the amount of data passing through our small pupils (a few millimeters in diameter) is truly enormous and that it is important to know how this information is received and processed. This book explains why a very large amount of research effort has gone into discovering more about man’s visual system over a large span of years, and a lot of work is currently being conducted about what is going on beyond the numerous detectors of the retina at the back of the eye and up to the cortex through our complicated array of neural networks. Viewing an object, material, or product provides the observer with the feeling of its “appearance”, which results from the physical and psychological entanglement between the illuminating source of light, the scene of interest and the human observer. The viewer’s retina is the physical component that links these three partners together and hence plays a key role in the appearance of the observed scene since it is the organ which detects the image of the objects of interest and transfers it to the cortex for processing by the brain and converting into the apparent image. One will notice that the visual appearance of an object is distinct from its visual recognition. Whereas a familiar object remains easily recognized under different lighting conditions, its appearance will usually differ strongly if one looks at it under a directional light source such as the sun or under a diffuse, uniform illumination from an overcast sky. Also, if different individuals (male/female; young/old; naïve/experts,…) are asked to look at the same object under identical illuminating and viewing configurations, their evaluations of its appearance will certainly differ, which may occur in the case of a given observer, depending upon his or her state of mind or health. This dispersion is not surprising because the sensation of appearance arises from a very large number of both visual and optical contributors or attributes such as, in a first approach, luminance (lightness), color (chroma, hue), gloss (contrast, haze), sparkle, texture, graininess, translucency (blur, transparency, DOI: 10.1051/978-2-7598-2647-6.c901 Ó Science Press, EDP Sciences, 2022

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clarity), iridescence, and because some of these parameters interfere with one another. As a consequence, the research work of the scientific community dealing with visual appearance is progressing along two axes. Firstly, there is a need for correctly evaluating the optical input, i.e., to adequately define the optical behavior of materials and surfaces in order to qualify the retinal image of the human viewer. As was done almost a century ago in the fields of photometry and colorimetry, the second task consists in defining and setting up psychophysical experiments in which panels of observers are asked to grade some of the appearance attributes of the above mentioned sets of physically and optically characterized samples (materials and surfaces) in a controlled lighting environment (light booth). This book “optical models for material appearance” by Mathieu Hébert originates for the most part from his research and teaching experience at the Institut d’Optique graduate school. It presents the optical tools that one should master in order to compute the spectral, spatial and angular behavior of the visible radiation which, after reflection or scattering by the object of interest converges towards the observer’s eyes and forms the retinal image. It reviews and specifies the basic parameters of the scene: the geometrical configuration (position and orientation of the light sources, of the object and of the observer), the spectral absorption, transmission and reflection of materials, texture and shape of the boundaries. Emphasis is placed on the surface reflectance properties such as the bidirectional reflectance/transmittance distribution functions, and the associated metrological procedures and instruments. A very wide selection of interfaces is analyzed, ranging from the rough or diffuse ones, the more or less specular or glossy boundaries, the highly specular or mirror-like surfaces. One will particularly appreciate the didactic approach of the author: the 3 first chapters review the necessary basic quantities in optics, photometry and colorimetry, and the subsequent chapters analyze the optical properties of several types of interfaces of increasing complexity, i.e., going from flat optical interfaces and transparent layers, to diffusing single and multi-layers, and to non-scattering layers on a diffusing background. There is no doubt that this book is of prime interest to students and researchers in optics who are involved in the interactions of light with matter. It will be of great help also to the growing population of professionals dealing with the visual appearance of materials and surfaces. This is particularly true for engineers in the manufacturing industry, the appearance of a product being an important design parameter and essential for its acceptance by the public. This book addresses numerous business activity sectors: automotive, printing, luxury, cosmetics, furniture, museums, video games, virtual reality and notably those which are in the search for coatings or pigments capable of special visual effects. Jean-Louis Meyzonnette Former professor at Institut d’Optique Graduate School Independent Expert Assessor in Radio-Photometry for COFRAC Orsay, February 2022

Foreword

The human visual system is a fascinating calculator. It is difficult to imagine the intense, continuous processing the brain performs on the light signal entering into the eyes. This huge, mainly unconscious activity brings a multitude of details to the threshold of our consciousness about the appearance of things and about the things themselves, their size, shape, the material they are made of, their condition, etc., that we perceive in a selective and hierarchical way, by often prioritizing the more global characteristics. It is only when we take a closer look that some more details become perceptible. Let us consider, for example, the object shown in the picture below. Everyone recognizes at first sight a vase, made of glass, locally opacified, with a dragonfly and flowers in relief, and with green opaque and metallic patterns at its base; art specialists will also recognize the Art Nouveau style by its characteristic aesthetics, and experts in materials will maybe think of the inclusions that give the glass its whitish appearance. The approach of other categories of people will be to give a subjective judgment of quality, preciousness, originality, or beauty. All this is a global, high-level analysis of the object – the most difficult to characterize objectively, although it is of primary interest to manufacturers of industrial products. A scientist of material appearance will be most interested in the various visual attributes that characterize each tiny portion of the object: the nuances of color, shine, metallic luster, translucency, haze, and other attributes that are all found in this example with infinite variations. These details, to which we usually do not pay particular attention although we perceive them, underlie our global perception of the object, and even become of crucial interest in some specific situations. They are necessary, for example, when one wants to accurately document the degradation of the piece over time, or to make a close copy. The same visual attributes can be found in most manufactured objects from the cosmetics, automotive, architecture, fashion, watchmaking, jewelry or design industries, where the aim is to control the appearance of products produced in large series. The question for the scientist is then the DOI: 10.1051/978-2-7598-2647-6.c902 Ó Science Press, EDP Sciences, 2022

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following: do we have the qualifying terms to describe these appearance attributes? Are we able to quantify all of them on a relevant scale, to measure them on a particular object? The answer today is: “definitely not”, but it would probably be wiser to say: “not yet”…

Vase decorated with dragonflies and butter cup ranunculus [Vase à décor de libellules et de renoncules, Daum Frères Manufacture, 1904, Musée des Beaux-Arts de Nancy, France].

Indeed, the newly formed scientific community around material appearance is progressing towards this objective, with significant advances on three fronts: the mathematical representation of visual attributes, their physical measurement and their reproduction. If the scientific works in this domain necessarily relate to the sensory and cognitive dimension of appearance, it actually relates much more to light, since it is light that conveys the signal allowing the brain to perceive and analyze things. It is essential to point out that light is the only a physical quantity that can be measured with instruments today – the physical measurement of sensations in the human brain using neuroscience tools is still in its infancy, and the only alternative that we have for the moment are psycho-physical experiments based on questionnaires, crucial but less precise than those given by a detector receiving a certain amount of light.

Foreword

IX

One of the ultimate goals of this material appearance community is to extend this work to global appearance using the methods developed for the color attribute alone in the framework of the so-called color science. The 20th century has seen the development of standard color appearance models created by connecting optical quantities (spectral distribution of light in the visible domain) to visual sensations produced on human observers, giving rise to standardized representations of colors by three coordinates in various color spaces (e.g., CIE 1976 L*a*b*). These colorimetric values can be related to the optical measurements performed on samples with spectrocolorimeters, scanners, or digital cameras, and with colorization machines, e.g., printers, allowing an accurate reproduction of the original colors onto a new support. These methods are satisfying in the case of matte, opaque, plane surfaces, which do not need specific appearance attributes beyond color. However, the appearance of most 3D manufactured products issued the cosmetics, automotive, architecture, fashion, horology, jewelry, or design industries, a fortiori objects like the Daum vase above, cannot be reduced to their color. The approach initiated by color sciences must deal with all other appearance attributes. The task is daunting because most attributes, except color and gloss in some simple situations, have no commonly accepted representation system. Fortunately, acquisition systems are progressing rapidly, in particular imaging systems (color, spectral, 3D) together with portable, multi-angle spectrophotometers. However, the optical data collected are still difficult to interpret in terms of visual sensations, and its interpretation in terms of structural and optical properties of the materials needs optical models specifically dedicated to the type of material under study. Reproduction is also a central concern – our learning of appearance is directly connected to our ability to reproduce a certain form of reality. The spectacular achievements of virtual reproduction with different visual rendering software, developed in particular for image synthesis aimed at cinema and gaming, have reached a very high level. The contribution of optical models to these results was crucial. The physical reproduction of objects has also reached a high degree of realism with laboratory 3D printers combining translucent and colored materials [26]. However, they are still in need of “smart” drivers managing global appearance and the underlying optical phenomena in the way color is managed by the drivers of 2D printers. Besides, there is a huge change in manufacturing processes in most industrial domains: traditional manufacturing based on an empirical (often remarkable) know-how has been gradually replaced by automated digital processes, generating an increasing demand for managing appearance in innovating ways. The time where standard appearance management procedures will exist for every kind of product has not come; we will remain in an intermediate stage for a certain number of years. Meanwhile, ad hoc solutions must be found based on available theories and technologies. This requires a certain level of expertise for engineers in charge of finding these solutions, whereas concepts and techniques involved in appearance management problems, in particular notions related to optics, are not easy to grasp. It is precisely for these persons and these needs, as well as for students or young researchers entering into this domain, that this book focused on optical concepts has been conceived.

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Foreword

As explained above, the definition of appearance attributes and their determination on a given sample requires optical measurements. The way they are performed, their relevance to the material under study, and the understanding of the optical principles underlying them are the main objective of the first part of the book, chapters 1 and 2. One will find there as good recaps about some important properties of light and useful radiometric quantities, and the presentations of some current measurement setups (for example, for measurements of Bidirectional Reflectance Distribution Functions – BRDF, reflectance or transmittance). Then in chapter 3, we present some classical models allowing the conversion of measured light quantities into color and gloss values. It is also important to have some familiarity with light-matter interaction in order to understand the relationship between the material structure and the appearance of an object. In the second part of the book, chapters 4–8, we progressively introduce some optical models allowing to understand how light is attenuated and scattered in some material layers, how it is reflected or refracted multiple times at its boundaries, how thickness influences the coloring or masking power of a layer on a background, how surface roughness impacts the glossiness of a surface, etc. Various material configurations are considered, from plates of transparent materials to diffusing ink-printed surfaces. The scope has been intentionally limited to plane surfaces, which prove easier to study than curved surfaces. It has also been limited to incoherent optics models – interferential phenomena are therefore not addressed – which already cover many types of common materials whether transparent or strongly scattering. The so-called “two-flux” approach originally introduced in 1931 by Kubelka and Munk for paints is extended into a much more general form, the “generalized two-flux model”, valid for stackings of optically thick layers of non-scattering and/or strongly scattering materials. An original mathematical formalism based on flux transfer matrices provides analytical formulae for the spectral reflectance and transmittance of any object of this kind. It allows to predict easily the new characteristics of a similar object induced by the modification of a parameter, for example the thickness of a layer. It has limits, however, that we have tried to formulate as clearly as possible; going beyond these limits requires more advanced models that are only briefly evoked in the last chapter. Finally, although this book is oriented towards the appearance of manufactured materials, the models presented here can be of interest wherever thick layered structures are encountered. One can cite the domains of medicine (e.g., skin analysis in dermatology), agronomy (e.g., photosynthesis in plant leaves), Earth sciences (e.g., remote sensing for canopies or soils), photovoltaics, etc. We hope that this book will also convince you that optical models have a larger scope than their strict domain of use, and will make you want to delve further into this fascinating domain.

Contents The Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII CHAPTER 1 Light and Optical Radiations . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Light Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Natural Light and Partial Polarization . . . . . . . 1.1.3 Wavefronts and Rays . . . . . . . . . . . . . . . . . . . . 1.2 Refractive Index of a Material . . . . . . . . . . . . . . . . . . . 1.3 Reflection and Refraction by a Smooth Interface . . . . . 1.3.1 Snell–Descartes Laws . . . . . . . . . . . . . . . . . . . . 1.3.2 Fresnel’s Formulae . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Reciprocity Properties . . . . . . . . . . . . . . . . . . . 1.4 Radiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Geometrical Concepts . . . . . . . . . . . . . . . . . . . 1.4.2 The Four Basic Radiometric Quantities . . . . . . 1.4.3 Spectral Radiometry . . . . . . . . . . . . . . . . . . . . 1.4.4 Photometry and Visual Units . . . . . . . . . . . . . . 1.5 Perfectly Diffuse Light . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Lambertian Lighting and Lambertian Surface . . 1.5.2 Integrating Spheres . . . . . . . . . . . . . . . . . . . . . 1.6 Light Source Illuminating a Plane Surface . . . . . . . . . . 1.6.1 Illumination by a Point Source: Bouguer’s Law 1.6.2 Illumination by an Extended Source . . . . . . . . .

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CHAPTER 2 Spectral Radiometry of Surfaces . . . . . . . . . . . . . . . . . . . . 2.1 Types of Reflecting Surfaces . . . . . . . . . . . . . . . . . . 2.1.1 Mirrors and Specular Reflectors . . . . . . . . . . 2.1.2 Very Matte Surfaces: Lambertian Reflectors . 2.1.3 Intermediate Cases . . . . . . . . . . . . . . . . . . . . 2.2 Light Transmission Through Objects . . . . . . . . . . . 2.3 Angular Spectral Reflectance and Transmittance . . .

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2.4

Bi-Directional Reflectance/Transmittance Distribution Function (BRDF/BTDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 BRDF of Some Typical Reflectors . . . . . . . . . . . . . . . . . . . . . 2.4.3 Lambert’s Azimuthal Equal-Area Projection . . . . . . . . . . . . . 2.4.4 BRDF Measurement Systems . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Reflectance and Transmittance Factors . . . . . . . . . . . . . . . . 2.5.1 Reflectance, Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Spectral Reflectance/Transmittance Factor . . . . . . . . . . . . . . 2.5.3 Spectral Radiance Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometries and Devices for the Measurement of Reflectance Factors . 2.6.1 Nicodemus’ Nomenclature for Nine Reflectance Factors . . . . . 2.6.2 Geometries Using Integrating Spheres . . . . . . . . . . . . . . . . . . 2.6.3 Bidirectional and Annular Geometries . . . . . . . . . . . . . . . . . . 2.6.4 Effective Measurement Geometry . . . . . . . . . . . . . . . . . . . . . . Surface Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Bidirectional Scattering-Surface Reflectance Distribution Function (BSSRDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Point Spread Function (PSF) . . . . . . . . . . . . . . . . . . . . . . . . .

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Plane Optical Interfaces and Transparent Layers . . . . . . . . . . . . . . . . . . . . . 4.1 Radiance Reflection and Transmission at an Interface . . . . . . . . . . . . .

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CHAPTER 3 Visual Characterization of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Color and Colorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 A Relative Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Color Mixing and Early Color Representation Systems . . . . . . 3.1.3 Trichromacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 CIE 1931 RGB and XYZ Color Spaces . . . . . . . . . . . . . . . . . . 3.1.5 CIE 1931 Chromaticity Diagram . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Color of Light Sources and Illuminants . . . . . . . . . . . . . . . . . . 3.1.7 CIE 1976 L*a*b* Color Space and Color Appearance Models . 3.1.8 Von Kries Chromatic Adaptation . . . . . . . . . . . . . . . . . . . . . . 3.2 Color Measurement and Color Imaging . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Color Characterization of a Surface by Using a Spectrophotometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Metamerism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Color and Spectral Measurement by Imaging Techniques . . . . 3.2.4 Measuring the Spectral Response of an RGB Camera . . . . . . . 3.2.5 Color Calibration of an RGB Imaging System . . . . . . . . . . . . 3.3 Gloss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Definition for Gloss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Gloss Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 4

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4.2 4.3 4.4

4.5

4.6

4.7

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Bi-Hemispherical Reflectance and Transmittance of an Interface . Metals and Strongly Absorbing Media . . . . . . . . . . . . . . . . . . . . Angular Reflectance and Transmittance of a Clear Plate . . . . . . . 4.4.1 Angular Reflectance and Transmittance . . . . . . . . . . . . . . 4.4.2 Bi-Hemispherical Reflectance and Transmittance . . . . . . . Spectral Transmittance of Absorbing Layers . . . . . . . . . . . . . . . . 4.5.1 Bouguer’s Law and Beer’s Law . . . . . . . . . . . . . . . . . . . . . 4.5.2 Piles of Absorbing Layers and Mixing of Absorbing Media Spectral Reflectance and Transmittance of an Absorbing Plate . . 4.6.1 Angular Reflectance and Transmittance . . . . . . . . . . . . . . 4.6.2 Bi-Hemispherical Reflectance and Transmittance . . . . . . . 4.6.3 Obtaining the Intrinsic Parameters of an Absorbing Plate Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 5 Transparent Multilayers: Two-Flux Models for Directional Light . . . . . 5.1 Piles of Transparent Plates Separated by Air . . . . . . . . . . . . . . 5.1.1 Angular Reflectance and Transmittance of Two Plates . . 5.1.2 Generalization to Non-Symmetric Plates . . . . . . . . . . . . 5.1.3 Generalization to N Plates: Iterative Model . . . . . . . . . . 5.1.4 Generalization to N Plates: Flux Transfer Matrix Model . 5.2 Piles of Identical Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Angular Reflectance and Transmittance . . . . . . . . . . . . . 5.2.2 Degree of Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Bi-Hemispherical Reflectance and Transmittance . . . . . . 5.2.4 Generalization to Non-Symmetric Plates . . . . . . . . . . . . 5.2.5 Invariance of Parameter a . . . . . . . . . . . . . . . . . . . . . . . 5.3 Layers of Different Refractive Indices in Optical Contact . . . . . . 5.3.1 Flux Transfer Matrices for Layers and Interfaces . . . . . . 5.3.2 Examples of Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Optical Characterization of Liquids . . . . . . . . . . . . . . . . 5.3.4 Total Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Nonpolarity of Directional Transmittance . . . . . . . . . . . . 5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Piles of Colored Films Separated by Air . . . . . . . . . . . . . 5.4.2 Piles of Colored Films Separated by Different Media . . . 5.5 Piles of Films on Top of a Specular Background . . . . . . . . . . . .

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CHAPTER 6 Diffusing Layers and Multilayers: Two-Flux Models for Diffuse 6.1 The Kubelka–Munk Model . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Kubelka–Munk Differential Equations . . . . . 6.1.2 Reflectance and Transmittance Formulae . . . . . . 6.2 Layers in Optical Contact with a Background . . . . . . . . 6.3 Light Transfers at the Interfaces Bordering the Layer . .

Light . . . . . . 133 . . . . . . . . . . . 134 . . . . . . . . . . . 134 . . . . . . . . . . . 135 . . . . . . . . . . . 137 . . . . . . . . . . . 138

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6.3.1 6.3.2

6.4

6.5 6.6 6.7 6.8

Saunderson Correction and Inverse Formulae . . . . . . . . . . . . Saunderson Correction and Inverse Formulae for a Diffusing Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deducing K and S from Measurements . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Drawdown Card Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Reflectance and Transmittance Method . . . . . . . . . . . . . . . . 6.4.3 Choosing the Appropriate Method . . . . . . . . . . . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixture of Scattering Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validity of the Kubelka–Munk Model . . . . . . . . . . . . . . . . . . . . . . . Diffusing Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Kubelka’s Compositional Formulae and Flux Transfer Matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Example: Piles of Identical Diffusing Sheets . . . . . . . . . . . . . 6.8.3 Kubelka’s Formulae and Kubelka–Munk Model . . . . . . . . . . 6.8.4 Extended Saunderson Correction for Multilayers: The Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 138 . . . . . . . . .

. . . . . . . . .

141 143 143 145 145 146 148 149 150

. . 151 . . 152 . . 153 . . 154

CHAPTER 7 Nonscattering Layers on a Diffusing Background . . . . . . . . . . . . . . . . . . . 7.1 Uniform Layer on Top of a Diffusing Background . . . . . . . . . . . . . 7.1.1 Williams–Clapper Model . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Berns’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Insurface and Subsurface Reflections According to the Lighting Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Transparent Multilayers on Top of a Diffusing Background . . . . . . 7.3 Generalized Two-Flux Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Configurations Where a Two-Flux Model Applies . . . . . . . . 7.3.2 Multiple Reflection Processes and Homogeneous Discrete-Time Markov Chains . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Transition Probability Matrices . . . . . . . . . . . . . . . . . . . . . 7.3.4 Average Number of Transfers . . . . . . . . . . . . . . . . . . . . . . . 7.4 Spectral Reflectance of Printed Surfaces . . . . . . . . . . . . . . . . . . . . 7.4.1 Halftone Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Spectral Neugebauer Model . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Yule–Nielsen Modified Spectral Neugebauer Model . . . . . . . 7.4.4 Clapper–Yule Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Calibration of the Halftone Color Prediction Models . . . . . . . . . . . 7.5.1 Obtaining Spectral Parameters . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Ink Spreading Assessment Methods . . . . . . . . . . . . . . . . . . 7.5.3 Basic Ink Spreading (BIS) Method . . . . . . . . . . . . . . . . . . . 7.5.4 Superimposition-Dependent Ink Spreading (SDIS) Method . 7.5.5 Predicting the Spectral Reflectance of Halftones . . . . . . . . . 7.5.6 Four-Ink Halftone Colors . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

157 158 158 161

. . . .

. . . .

. . . .

162 165 168 168

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

170 172 173 174 174 176 177 179 180 181 182 182 184 184 186

Contents

XV

CHAPTER 8 Angle-Dependent Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Surface Roughness Measurement . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Modelling a Randomly Rough Surface . . . . . . . . . . . . . . . . . . 8.1.3 BRDF and BTDF Optical Models . . . . . . . . . . . . . . . . . . . . . 8.1.4 Microfacet Model and Smith’s Shadowing-Masking Function . 8.1.5 Spherical Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Scattering Description Parameters . . . . . . . . . . . . . . . . . . . . . 8.2.2 Types of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 The Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Scattering in Lambertian Layers . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

187 187 188 188 189 190 193 195 195 198 199 200

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Chapter 1 Light and Optical Radiations The optical or visual characterization of surfaces cannot be separated from the properties of light itself: its wavelength, polarization, angular distribution are all determining factors in the optical signal that is measured or perceived, and therefore in the way it is interpreted. The optical models used in this book will rarely refer to the wave nature of light, except for important properties which are directly related: wavelength, on which many intrinsic characteristics of matter depend, and polarization, which determines the amount of light reflected at the interfaces between different materials. It will therefore prove useful to introduce the wave model of light before exposing the types of light signals that will be considered throughout the following chapters for the description of the light-matter interactions that give materials their appearance.

1.1

Light

According to the Commission Internationale de l’Eclairage (CIE), light is the generic name for the electromagnetic radiations visible to the human eye [39]. This notion is to be extended to infrared (IR) and ultraviolet (UV) radiations which, respectively, have longer and shorter wavelengths but similar physical properties (see figure 1.1). The limit of wavelengths for the spectral range of visible radiation may vary depending on the radiant power falling on the retina and the sensibility of the observer, but they lie generally between 360 and 400 nm for the lower limit, and between 760 and 830 nm for the upper limit. The CIE tabulates most spectral values related to the response of the standard visual system between 380 and 780 nm [42].

1.1.1

Light Waves

Electromagnetic radiations correspond to a simultaneous vibration of the electric field and the magnetic field, oscillating both perpendicularly to each other and perpendicularly to the wave propagation direction. These vibrations are described

DOI: 10.1051/978-2-7598-2647-6.c001 © Science Press, EDP Sciences, 2022

2

Optical Models for Material Appearance

FIG. 1.1 – Electromagnetic spectrum and visible spectrum (light). by time- and position-dependent three-dimensional vectors, often denoted E for the electric field, and B for the magnetic field. Maxwell’s equations describe their variation in time and space according to the electrical properties of the propagation medium. As oscillations of the electric field and magnetic field are proportional to each other, one can describe the wave by the oscillation of the electric field only (vector E). A light wave is said to be monochromatic when the oscillation of the electric field is sinusoidal, with a given wavelength in a vacuum. It is then said to be temporally coherent: if the wave is split into two identical waves shifted in phase, for example by reflection on the two surfaces of a glass plate, the two waves interfere with each other when superimposed and give rise to a wave whose amplitude is increased at some points, and reduced at other points. Most continuous laser beams are, to a good approximation, monochromatic, and therefore temporally coherent. The superimposition of several monochromatic waves of different wavelengths and phases produces a non-periodic wave. The light power associated with different wavelengths is characterized by the spectral power distribution (SPD). If the range of wavelengths is wide, as it is the case for sunlight and white light in general, the light is qualified as temporally incoherent. When an incoherent wave is reflected at the two surfaces of a glass plate, the two reflected waves are not correlated and do not show evident interferences once superimposed. However, it is possible that these two reflected waves are partially correlated, especially if the plate is very thin: they still interfere with each other, thus producing once a superimposed wave whose amplitude is increased or decreased in comparison to the incident wave into proportions that depend on the wavelength. This interference phenomenon is at the origin of the colored irisations on soap bubbles, thin films or thin coatings [90]. The maximal path difference for which interferences can occur, expressed as a distance, is called the coherence length and depends on the spectral bandwidth of the light. As mentioned above, the electric field oscillates perpendicularly to the propagation direction of the wave. If the direction of the propagation of the wave is taken as z-axis of a coordinate system, then the electric field vibrates in the (x, y) plane, inside which electric vector E describes patterns, called polarization states.

Light and Optical Radiations

3

Figure 1.2 shows the field maps at a given time associated with monochromatic waves with same wavelength but different polarization states. The sinusoidal oscillations of Ex and Ey components of E in the (x, y) plane are also shown. Their respective oscillation amplitude and phase shift φ (represented by a pink arrow) determine the oscillation pattern: the polarization is linear when φ is 0 or π, i.e., Ex and Ey oscillate in phase or in phase opposition, as shown in cases (a) and (b); it is circular when Ex and Ey oscillate with equal amplitude and the phase shift is u ¼ p=2, as shown in case (e); and it is elliptic in other cases. Polarization is generally modified when the refractive index of the matter changes, for example when it is reflected or refracted at the interface between media of different refractive indices.

FIG. 1.2 – Maps of electric field E at a given time associated with monochromatic waves of same wavelength λ but different polarizations (red lines). The oscillations of Ex and Ey components of E are drawn in blue and green lines respectively, and their phase shift φ is represented with a pink arrow. The polarization is linear in cases (a) and (b), elliptic in cases (c) and (d), and circular in case (e).

1.1.2

Natural Light and Partial Polarization

The sun and most common light sources emit many short wavepackets independent of each other and with different polarizations. Polarization of the resulting wave therefore varies very rapidly, in a random and irregular manner. Such a light is called natural light, or unpolarized light. The electric field map of a temporally incoherent and unpolarized wave, shown in figure 1.3a, exhibits these changes of polarization along the propagation axis. However, in most configurations, rapid polarization changes do not require a comprehensive description: it is statistically equivalent to considering the wave as the superimposition of two linearly polarized waves (or two elliptically polarized waves in the most general case, not needed in

Optical Models for Material Appearance

4

FIG. 1.3 – Electric field maps (red lines) of (a) an unpolarized wave, and (b) two linearly polarized waves vibrating in perpendicular planes and each carrying the same power: the superimposition of the two linearly polarized waves is a model of an unpolarized wave. The three waves are temporally incoherent with the same spectral power distribution.

our applications), totally independent from each other [22]. These two waves carry the same power. Moreover, their respective vibration planes are perpendicular (see figure 1.3b). One can choose them arbitrarily, provided they contain the propagation direction.

1.1.3

Wavefronts and Rays

As long as the wave issued from a light source propagates in air, water, glass or any other homogenous and isotropic medium, whose refractive index is constant in every point and every direction, the loci of identical phase form a continuous surface called wavefront. This surface is generally curved, but it can also be spherical or planar in special situations. The theorem of Malus asserts that in these isotropic media, the propagation direction of the wave is always perpendicular to the wavefront. The mathematical lines perpendicular to the wavefronts, corresponding to the local propagation directions of the wave, are called rays. The ray concept in optics as a mathematical line should not be confused with the ray as thin light pencil often used in everyday language: actually, a thin light pencil can be represented by a set of rays. Figure 1.4 shows sectional views of two waves, a planar wave and a spherical wave, where some wavefronts and some rays are represented. Fermat’s principle, also known as the principle of least time, asserts that light follows the quickest optical path between two given points. In a medium of constant refractive index, light propagates at constant speed and the quickest path is a straight line. This is consistent with our everyday experience that light pencils in air propagate along straight lines.

Light and Optical Radiations

5

FIG. 1.4 – Representations of wavefronts and rays in (a) a plane wave, (b) a spherical wave.

1.2

Refractive Index of a Material

In vacuum, light waves propagate at speed c  2:998  108 m.s−1. When the propagation medium is not the vacuum, they advance more slowly, with speed v. Depending upon the medium of propagation, the wavelengths are modified, but their values are most usually expressed in vacuum (or in air, which makes almost no difference for the level of accuracy expected in our application domain). The ratio c/v, larger than 1 and called refractive index, characterizes the optical properties of the medium. When the propagation medium is absorbing, the amplitude of the light wave is attenuated along its path. The refractive index is often represented by a complex number, which generally depends on wavelength: ^ ðkÞ ¼ n ðkÞ þ ijðkÞ n

ð1:1Þ

The real part n ðkÞ, called real refractive index, is related to the phase propagation speed in the medium with respect to the speed of light in vacuum. The imaginary part jðkÞ, called extinction index, characterizes attenuation due to absorption by the medium. The real and imaginary parts, functions of wavelength, are related to each other by the Kramers–Kronig relations [130, 165, 182]. Thus, knowing either the real index or the extinction index over the whole spectrum (i.e., in theory, from the zero to infinity) enables us to obtain the other index for any wavelength. Table 1.1 gives the refractive indices of a few common materials. Ellipsometry is one of the favorite techniques used for refractive index measurements based on polarization analysis of reflected waves. It is suitable for materials shaped under the form of a plate or a thin coating. The constraint is that the sample must be homogenous, non-scattering and very flat, which makes this technique hardly usable with scattering materials or rough surfaces. For dielectric materials such as glass, plastic or cellulose, the attenuation index is low compared to the real index. The refractive index may be considered as being real and absorption is modelled independently by an attenuation factor applied to the

Optical Models for Material Appearance

6

TAB. 1.1 – Refractive indices of materials measured at λ = 589 nm (Sodium D line). Air [103] Water (at 20 °C) [103] Ethanol Sucrose 25 g/100 ml (at 20 °C) [153] Sucrose 50 g/100 ml (at 20 °C) [153] Fused quartz SiO2 Cellulose Polypropylene Acrylic Sucrose 85 g/100 ml (at 20 °C) Polyvinyl alcohol Plexiglass Crown glass [103] Sodium Chloride (NaCl) [103] Amber [103] Polycarbonate [154] Polystyrene [103] Zircon (ZrO2  SiO2 ) [103] Diamond [103] Rutile (TiO2) [103] Gold [22] Silver [22] Copper [22] Platinum [22] Aluminum [22]

1.0003 1.333 1.36 1.37 1.42 1.45 1.47 1.49 1.49 1.50 1.50 1.51 1.52 1.544 1.55 1.58 1.59 1.923 2.417 2.907 0.27 + 2.95 0.20 + 3.44 0.62 + 2.57 2.63 + 3.54 1.44 + 5.23

i i i i i

light beam (see §4.5). The dependence of the real index on wavelength, also called dispersion [165], is at the origin of the chromatic aberrations in optical systems [154]. It is empirically modelled in the visible wavelength domain by Cauchy’s law [22, 73]: a2 n ðkÞ ¼ a1 þ 2 ð1:2Þ k where dimensionless factor a1 and coefficient a2 in m−2 are to be determined for each medium. As the real index varies with respect to wavelength, rays are refracted at different angles and split white light pencils into diverging pencils, commonly called rainbows in the case of rain drops [195]. However, dispersion has no significant effect when the incident light is diffuse or when the medium is diffusing, because different spectral components superimpose onto each other and yield again white light in all directions. This is why dispersion is ignored in the case of papers or white paints, and a constant real refractive index is attached to them over the visible spectrum. If the refractive index varies, the speed of light also varies and the shape of the wavefront is modified accordingly; the quickest path may follow a curved line, or a broken line as it happens at the interface between two homogeneous media.

Light and Optical Radiations

7

However, there exist many materials, qualified as diffusing, in which the refractive index varies at a very small scale, comparable to the wavelength of light. The ray concept is not adapted in this case: it is preferable to describe transfers of light power thanks to the concepts defined in radiometry.

1.3

Reflection and Refraction by a Smooth Interface

When a wave propagating in a medium of given refractive index n1 enters into a medium of different refractive index n2 , its propagation speed is modified. As a direct consequence of the Huygens–Fresnel principle, wavefronts equally spaced in medium 1 remain equally spaced in medium 2, but their distance, thereby the wavelength, is modified. Figure 1.5 illustrates this through the example of a monochromatic spherical wave emitted by a point source located in medium 1 near a flat, smooth interface with medium 2 in the case where n2 [ n1 . As the wave enters into medium 2, it is slowed down; the successive wavefronts are closer to each other and their shape is modified. The orientation of the rays is also modified, since Malus’ theorem asserts that it is perpendicular to the wavefront: they are refracted.

S

T1

T1 medium 1

n1

medium 2 n2 > n1 T1 FIG. 1.5 – Wavefronts (dashed lines) of a spherical wave emitted by a point source in medium 1 and crossing the interface with medium 2 of a higher index (wavefronts of the wave reflected are shown in pink). The rays, perpendicular to the wavefronts, are refracted into medium 2 at angles given by Snell’s law.

1.3.1

Snell–Descartes Laws

The directions of reflection and refraction of a light ray, called regular directions, satisfy Snell–Descartes laws: (1) the incident, reflected and refracted light rays belong to a same plane, called the incidence plane, which also contains the normal of the interface; (2) the angles formed by the incident ray and the reflected ray with respect to the normal of the interface are equal; (3) the angle of refraction, h2 , is related to the angle of incidence, h1 , according to the sine law (figure 1.5):

Optical Models for Material Appearance

8

n1 sin h1 ¼ n2 sin h2

ð1:3Þ

Let us consider n2 [ n1 . When light comes from medium 1, the refraction angle is always smaller than the incidence angle. At grazing incidence, i.e., h1 ¼ p=2, the refraction angle reaches a limit value hc , called the critical angle, issued from Snell’s sine law: hc ¼ arcsinðn1 =n2 Þ

ð1:4Þ

No light can be refracted into medium 2 at a higher angle. When light comes from medium 2, it is refracted into medium 1 provided the angle of incidence h2 is lower than the critical angle hc . Otherwise, Snell’s sine law (1.3) provides no real solution for angle h1 , refraction does not occur and the ray is totally reflected.

1.3.2

Fresnel’s Formulae

The fraction of light that is reflected from the interface between media 1 and 2, which depends on the orientation of incident light beam, is called angular reflectance (a definition is proposed in §2.3). It is given by Fresnel’s formulae, named after the French physicist Augustin Fresnel (1788–1827), established by writing the transition equation of electromagnetic waves at the interface [22]. It depends on the angle of incidence h1 , the polarization of the incident light, as well as the relative refractive index of the interface n ¼ n2 =n1 . The latter ratio is greater than 1 if medium 2 is the most refringent, i.e., n2 [ n1 , and lower than 1 otherwise. In many applications, the incident light is unpolarized, i.e., modelled as the sum of two linearly polarized lights without any phase relationship (see §1.1.2). Since the angular reflectance depends on the orientation of the electric field with respect to the incidence plane, one considers the cases where the electric field is either parallel or perpendicular to it. These two polarizations are often denoted by symbols p (for parallel) and s (for perpendicular, from the German senkrecht). Consider a light pencil coming from medium 1 with an angle of incidence h1 . For p-polarized light, the angular reflectance is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2



tan ðh1  h2 Þ
2

n 2 cos h1  n 2  sin2 h1



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rp12 ðh1 Þ ¼
¼
ð1:5Þ tan ðh1 þ h2 Þ

n 2 cos h1 þ n 2  sin2 h1
where h2 ¼ arcsinðsin h1 =n Þ is the angle of refraction into medium 2 defined by Snell’s law and symbol j. . .j denotes the absolute value or the modulus operator if the fraction is a complex number. For s-polarized light, the angular reflectance is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2



sin ðh1  h2 Þ
2

cos h1  n 2  sin2 h1


¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rs12 ðh1 Þ ¼

ð1:6Þ sin ðh1 þ h2 Þ

cos h1 þ n 2  sin2 h1
Remember that if medium 1 is the more refringent (i.e., n\1), light is totally reflected beyond the critical angle hc given by equation (1.4). In this case, the term n 2  sin2 h1 is negative and the square root gives a complex number. However,

Light and Optical Radiations

9

formulae (1.5) and (1.6) remain valid thanks to the modulus operator, and one obtains the expected values Rp12 ðh1 Þ ¼ Rs12 ðh1 Þ ¼ 1. In the case of unpolarized incident light, which contains same quantity of p- and s-polarized components, the angular reflectance is the average of the angular reflectances associated with the two polarizations: R12 ðh1 Þ ¼

 1 Rp12 ðh1 Þ þ Rs12 ðh1 Þ 2

ð1:7Þ

After reflection at the interface, the unpolarized light is transformed into partially polarized light, modelled as the superimposition of natural light and polarized light, or equivalently as two linearly polarized waves p and s of different amplitudes (the p and s light components thus have different powers, Fp and Fs ). A degree of polarization (DOP) is defined as follows, by assuming that Fp \Fs [127, 259]: DOP ¼

Fs  Fp Fs þ Fp

ð1:8Þ

The DOP is 0 when the two components have equal power (natural or unpolarized light), and 1 when one of the two components is zero (totally or linearly polarized light). Since Rs12 ðh1 Þ is always higher than or equal to Rp12 ðh1 Þ, the degree of polarization (DOP) of the reflected light when the incident light is unpolarized defined in equation (1.8), can be expressed as DOPðh1 Þ ¼

Rs12 ðh1 Þ  Rp12 ðh1 Þ Rs12 ðh1 Þ þ Rp12 ðh1 Þ

ð1:9Þ

At normal incidence, the angular reflectance takes the same values for p-polarized, s-polarized and unpolarized lights, given by the following simple formula and tabulated as a function of the relative refractive index in table 4.1:   n1 2 Rp12 ð0Þ ¼ Rs12 ð0Þ ¼ R12 ð0Þ ¼ ð1:10Þ nþ1 Regarding the refracted component, since no light is absorbed at the interface, the angular transmittance is T12 ðh1 Þ ¼ 1  R12 ðh1 Þ

ð1:11Þ

where the symbol , also used hereinafter, stands for s, p, or nothing in the case of unpolarized light. The angular reflectances for p-polarized and s-polarized lights are plotted in figure 1.6 as functions of the angle of incidence h1 for an interface of relative refractive index n ¼ 1:5, e.g., an air-glass interface. The angular reflectance for unpolarized light and the degree of polarization of the reflected ray are also plotted. At normal incidence, the reflectance for unpolarized light is minimal (4%) and the reflected light is unpolarized (DOP = 0). As the angle of incidence increases, the reflectance remains almost stable while the DOP increases until a remarkable angle

Optical Models for Material Appearance

10

FIG. 1.6 – Angular reflectances for p-polarized light, s-polarized light, and unpolarized light and DOP of the reflected light for an interface with relative refractive index n ¼ 1:5 as for an air-glass interface. hb ¼ arctanðn Þ at which the DOP is 1 because the p-polarized component is not reflected at all: Rs12 ðhb Þ ¼ 0. hb is called Brewster angle after the British scientist Sir David Brewster (1781–1868), who discovered the extinction of the p-polarized component at this angle [25]. It is 53.1° in the case of water (n = 1.33), and 56.3° in the case of glass (n = 1.5). Linearly polarized light can be obtained by illuminating a glass plate (“Brewster window”) at the Brewster angle (the reflectance can be increased by adding a layer of clear material with a high refractive index [22]). Beyond hb , the DOP decreases and the reflectance strongly increases: At grazing angle (90°), all light is reflected and the DOP is zero. These variations according to the angle of incidence are similar for an air–water interface (n ¼ 1:33, R12 ð0Þ ¼ 2%) and explain the color gradient that one can observe at the surface of the river in the picture displayed in figure 1.7.

1.3.3

Reciprocity Properties

When light comes from medium 2, the angular reflectance, denoted by R21 ðh2 Þ, is still given by equation (1.5) or (1.6) according to the polarization, by considering the relative refractive index n ¼ n1 =n2 . Independently of polarization, the angular reflectance is the same if light comes from medium 1 at the angle h1 or comes from medium 2 at the regular angle h2 ¼ arcsinðn sin h1 Þ: R12 ðh1 Þ ¼ R21 ðh2 Þ

ð1:12Þ

As a consequence of (1.12), one has T12 ðh1 Þ ¼ T21 ðh2 Þ

ð1:13Þ

Light and Optical Radiations

11

FIG. 1.7 – Picture of a river where the color gradient observed at the surface of water is due to the Fresnel angular reflectance of the air–water interface (n = 1.33). This equality means that for a given path of light, the angular transmittance does not depend on the direction of propagation from medium 1 to medium 2 or vice-versa. In case of total internal reflection, the angular transmittance is zero. Figure 1.8 shows the angular reflectances and transmittances as functions of the

FIG. 1.8 – Angular reflectance (solid lines) and transmittance (dashed lines) of an interface

with relative index n ¼ n2 =n1 ¼ 1:5 as functions of the angle of incidence for natural light coming from medium 1 (blue curves) or medium 2 (pink curves).

Optical Models for Material Appearance

12

angle of incidence from the normal of the surface (0°) to the grazing angle (90°) in both media 1 and 2 for an interface of relative index n ¼ 1:5. The case of absorbing media and metals, characterized by a complex refractive index, will be addressed in §4.3. Other illumination configurations than directional lighting will also be presented in chapter 4.

1.4

Radiometry

Radiometry is the metrology of radiations, which covers their emission, their detection, their propagation within materials or optical systems, etc. It thus gives rise to a profuse literature (see for example Refs. [81, 160, 163, 225, 232, 244, 246, 259]). Optical radiometry is mainly concerned with incoherent radiations and ray optics, but takes into account wave phenomena such as diffraction and interferences when these phenomena are significant. Optical radiations are evaluated in terms of radiant power, photon flow, or visual stimulus depending on the type of detection, and different units are used accordingly. The branch of radiometry considering detection by the human visual system, called photometry, focuses on visible light, i.e., electromagnetic radiations of wavelengths comprised between 380 and 780 nm. The radiant power at each wavelength is weighted by a visual sensitivity function that models human brightness sensitivity. The measured quantities are expressed in luminous units such as lumen, lux and candela, the latter being the SI unit. The interest of optical radiometry is the possibility of quantifying the amounts of light being in interaction with an object and their transfer between sources, objects, optical systems and detectors. The fundamental radiometric quantities describe the geometrical distributions of a radiation, this being the reason why we first introduce some geometrical concepts before presenting the quantities themselves.

1.4.1

Geometrical Concepts

Describing the transport of light from a source to an object, then from the object to a detector is primarily a geometry issue. Ray positions and orientations are described in a 3D coordinate system, which may be cartesian, cylindrical, or spherical according to the problem treated. A light ray is not to be defined as the trajectory of photons passing through two points P1 and P2, because the probability for a photon to meet precisely one point or to follow one direction is zero. One should rather consider a small area around each point. The small set of directions between these two diaphragms defines two solid angles under which each area is being observed from the other one. A solid angle, expressed in steradian (sr), is defined as the ratio between an area A on a sphere and the square of radius r of this sphere: X¼

A r2

ð1:14Þ

It extends the concept of angle, expressed in radian (rad), and defined as the ratio between the length of an arc on a circle and the radius of that circle. However,

Light and Optical Radiations

13

in contrast with angles which may be oriented and thus take negative values, solid angles are always positive. To specify one propagation direction, the spherical coordinate system (θ, φ) is the most appropriate, where θ is the polar angle and φ the azimuthal angle. An infinitesimally small solid angle d 2 Xðh; uÞ is attached to this direction by considering an infinitesimal angle dθ around angle θ, and an infinitesimal angle dφ around angle φ (figure 1.9). The pyramid thus defined intercepts a sphere of radius r in a small rectangle of area r 2 sin hdhdu. According to the definition of solid angles, the infinitesimal solid angle is therefore d 2 Xðh; uÞ ¼ sin hdhdu

ð1:15Þ

Solid angles may be subtended by larger portions of the sphere. They can be computed by integrating the contributions from infinitesimal solid angles. For example, the solid angle Ω associated with the whole sphere is 4π sr: ZZ Z 2p Z p 2 Xsphere ¼ d Xðh; uÞ ¼ sin hdhdu ¼ 4p sr ð1:16Þ u¼0

sphere

h¼0

and the solid angle corresponding to a cone of half-angle α (see figure 1.10) is ZZ Z 2p Z a Xa ¼ d 2 Xðh; uÞ ¼ sin hdhdu ¼ 2p½ cos ha0 ¼ 2pð1  cos aÞ ð1:17Þ cone

u¼0

h¼0

FIG. 1.9 – Infinitesimal solid angle along the direction (θ, φ).

FIG. 1.10 – Conical solid angle with half-angle α.

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14

A second geometrical concept, called geometrical extent and measured in m2.sr, enables characterizing the size of light beams. A thin light pencil can be defined as the light flowing between two elementary areas ds1 and ds2 around two points P1, respectively P2, of respective normal vectors N1 and N2 , distant from each other of a length P1P2 = h (see figure 1.11). The corresponding elementary geometrical extent is d2G ¼

1 ðds1 cos h1 Þðds2 cos h2 Þ h2

ð1:18Þ

FIG. 1.11 – Geometrical extent describing a thin pencil flowing between two elementary areas ds1 and ds2.

where h1 and h2 are the angles made by line (P1P2) with N1 , respectively N2. One can notice that ds1 cos h1 ¼ dA1 , also called apparent area, is the projection of ds1 onto the sphere of radius h centered in P2. It subtends the solid angle dX2 based in P2 dX2 ¼ dA1 =h 2 ¼ ds1 cos h1 =h 2

ð1:19Þ

Likewise, ds2 cos h2 ¼ dA2 is the projection of ds2 onto the sphere of radius h centered in P1. It subtends the solid angle dX1 based in P1 dX1 ¼ dA2 =h 2 ¼ ds2 cos h2 =h 2

ð1:20Þ

The elementary geometrical extent of the light pencil [259] can therefore be defined in three equivalent ways: d 2G ¼

1 ðds1 cos h1 Þðds2 cos h2 Þ ¼ dA1 dX2 ¼ dA2 dX1 h2

ð1:21Þ

A large pencil of light car be defined from a finite area S over which the density of light is uniform, and an elementary solid angle dX along one direction making an angle h with the normal of A (see figure 1.12). The geometrical extent in this case is dG ¼ A cos hdX

ð1:22Þ

For a finite light beam, flowing into a finite solid angle X in which the density of light is uniform, the geometrical extent is obtained by integrating the components given by equation (1.22) over the whole solid angle: ZZ G¼A cos h sin hdhdu ð1:23Þ ðh;uÞ2X

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15

FIG. 1.12 – Geometrical extent dG describing a large pencil of light based on a finite area A on a surface and a solid angle dΩ in some spatial direction.

1.4.2

The Four Basic Radiometric Quantities

Optical radiations are characterized by four fundamental quantities: flux, intensity, irradiance and radiance, and their corresponding spectral densities. Flux F, also called light power, expressed in watt (W), is the energy radiated per unit time. Radiant intensity I, expressed in W.sr−1, is the density of flux per unit solid angle that is propagating toward some specified direction ðh; uÞ towards the infinitesimal solid angle dXðh; uÞ: I ðh; uÞ ¼

d 2 F ðh; uÞ d 2 F ðh; uÞ ¼ d 2 Xðh; uÞ sin hdhdu

ð1:24Þ

Intensity applies to radiations from point sources, as well from extended sources being very far such as stars in astronomy. The angular distribution of light from a given source over space is given by an intensity diagram, defined over the sphere. A few examples are shown in figure 1.13. Sources that emit light along one main direction are directional sources, and sources with constant intensity along all directions in space are said to be isotropic sources. In this latter case, the total flux F emitted is related to the constant intensity I by F ¼ 4pI In the general case, the total flux and intensity are related by Z 2p Z p F¼ I ðh; uÞ sin hdhdu u¼0

h¼0

ð1:25Þ

ð1:26Þ

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16

FIG. 1.13 – Sectional view of intensity diagrams of different light sources. Irradiance E, expressed in W.m−2, is the density of flux that is incident on a surface around a specified point of that surface. It is a function of position on the surface. Shadows, for example, are drawn by a variation of irradiance on a surface. When at a given point P of the surface, elementary area ds ðPÞ around it receives an elementary flux dF ðPÞ, the irradiance is Exitance E ð PÞ ¼

dF ðPÞ ds ðPÞ

ð1:27Þ

Exitance M, also expressed in W.m−2, is the density of flux that emerges from a surface around a specified point P of that surface: M ð PÞ ¼

dF ðPÞ ds ðPÞ

ð1:28Þ

This quantity stands for the emission of light of extended sources, i.e., sources having a certain area.

Light and Optical Radiations

17

Radiance L, in W. m−2.sr−1, is the flux per unit geometrical extent that is incident on, passing through or emerging from an elementary area ds at a specified position P on the surface, in an elementary solid angle around a specified direction ðh; uÞ: LðP; h; uÞ ¼

d 2 F ðP; h; uÞ d 3 G ðP; h; uÞ

ð1:29Þ

The elementary geometrical extent can be written d 3 G ðP; h; uÞ ¼ ds ðPÞ cos hd 2 Xðh; uÞ ¼ ds cos h sin hdhdu

ð1:30Þ

and the defining expression of radiance can also be written LðP; h; uÞ ¼

d 3 F ðP; h; uÞ ds ðPÞ cos h sin hdhdu

ð1:31Þ

for every direction ðh; uÞ of the hemisphere based on the tangent plane of the surface containing ds(P) (figure 1.14).

FIG. 1.14 – Radiance LðP; h; uÞ emitted by a small area ds around point P on a curved surface. Angles are defined in the hemisphere based on the plane containing ds.

Radiance can also be defined between two small apertures, or two small areas ds ðP1 Þ and ds ðP2 Þ around the respective points P1 and P2 of the space as defined in figure 1.11: L ð P1 ; P2 Þ ¼

d 2 F ð P 1 ; P2 Þ 1 h 2 ds ðP1 Þ cos h1 ds ðP2 Þ cos h2

ð1:32Þ

with h ¼ P1 P2 . Radiance is the most suitable radiometric quantity for the description of thin light pencils. It is the quantity that can be attributed to each pixel in an image captured by a camera, or by each cell of the retina.

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A given extended source, or a given surface that reemits part of the light it has received (secondary source), can be characterized by an angular radiance diagram over the hemisphere. The elementary exitance around point P on the source along the direction ðh; uÞ is d 2 M ðP; h; uÞ ¼

d 3 F ðP; h; uÞ ¼ LðP; h; uÞ cos h sin hdhdu dsðPÞ

ð1:33Þ

and the total exitance around point P is obtained by integrating angularly the elements of irradiance given by equation (1.33): Z Z 2p Z p=2 M ð PÞ ¼ d 2 M ðP; h; uÞ ¼ LðP; h; uÞ cos h sin hdhdu ð1:34Þ u¼0

hemisphere

h¼0

Radiance may also be used to characterize the angular distribution of illumination of a given surface. We can still refer to figure 1.14, where the light flux incomes onto ds(P) instead of outgoing. By analogy with the exitance of a source, the irradiance of the surface around point P is Z 2p Z p=2 E ðP Þ ¼ LðP; h; uÞ cos h sin hdhdu ð1:35Þ u¼0

1.4.3

h¼0

Spectral Radiometry

The previous radiometric quantities have been defined without consideration of wavelength. The spectral distribution of a radiation is described by spectral quantities Qλ defined as the quantity per unit wavelength: Qk ¼

dQ dk

ð1:36Þ

One therefore has spectral flux (in W.nm−1), spectral intensity (in W.sr−1.nm−1), spectral irradiance and spectral exitance (in W.m−2.nm−1) and spectral radiance (in W.sr−1.m−2.nm−1). Spectral quantities are measured with instruments such as spectrophotometers that analyze the radiation in adjacent, narrow spectral bands. If spectral bandwidths Dk are small, the measured flux in each bandwidth is Fk Dk. Over a larger band ½k1 ; k2 , the measured flux is Z k2 Q ð k1 ; k2 Þ ¼ Qk dk ð1:37Þ k1

The spectral resolution of spectrophotometers varies according to the application and the method used to decompose the light spectrum. For color science applications, usual commercial instruments have a resolution ranging from 1 to 10 nm. In order to select narrow spectral bandwidths, the light is usually decomposed by means of a diffraction grating [124]. The type of photodetector determines the spectral domain of interest. Many spectrophotometers capture all the spectral bands simultaneously using an array of sensors (e.g., a CCD linear array).

Light and Optical Radiations

1.4.4

19

Photometry and Visual Units

The scientific metrology of light started before the creation of the first optical detectors, especially with the works by Bouguer and Lambert in the 18th century [13]. The light quantities thus defined were used for characterizing light perception by humans. The corresponding science was called photometry, now considered as a branch of optical radiometry where the spectral energy is weighted by the spectral sensibility of the human visual system [42]. The relative visibility (or spectral sensitivity) of monochromatic light according to its wavelength, usually denoted by V(λ), has been determined by Gibson and Tyndall and adopted by the CIE in 1924 [80], then refined several times, for example by Judd in 1951 [119] or Sharpe et al. in 2005 [208]. This function, plotted in figure 1.15, is based on psychophysical experiments made by a number of human observers, then averaged to define a “standard observer” representative of all individuals. It stands for daytime vision, or photopic vision, when the luminous level is rather high (beyond 10 cd.m−2) and sufficient to activate the photosensitive cells called cones present in the retina, responsible for the vision in color (see §3.1). A second relative visibility function, V 0 ðkÞ, has been established in 1952 for scotopic vision, i.e., lighting conditions where the luminous level is too weak (below 2 × 10−5 cd.m−2) to activate the cones but high enough to activate the other type of photosensitive cells in the retina, the rods.

FIG. 1.15 – Spectral sensitivity of the human visual system in photopic vision, V(λ), and scotopic vision, V 0 ðkÞ. Photometry relies on similar quantities as radiometry, by weighting the spectral quantities by function V(λ). The visual flux, or luminous flux, expressed in lumen (lm), is defined as: Z 780 nm Fv ¼ Km Fk V ðkÞdk 380 nm ð1:38Þ Z 780 nm Fv0 ¼ Km0

Fk V 0 ðkÞdk

380 nm

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20

where Km ¼ 683 lm/W is a scaling constant for photopic vision, and Km0 ¼ 1700 lm/W a scaling constant for scotopic vision. The equivalent visual quantities for intensity, irradiance, exitance and radiance are, respectively, luminous intensity (in candela, cd), illuminance (in lux, lx), luminous exitance (in lm.m−2), and luminance (in cd.m−2). A recap table with the names of the quantities and the units is given in table 1.2. The luminous efficacy K of a given light source, expressed in lumen per watt (lm/W), is defined as the ratio between its luminous flux Fv and its radiant flux Fe, i.e., R 780 nm Fv Km 380 nm Fk V ðkÞdk ¼ K¼ ð1:39Þ R 780 nm Fe Fk dk 380 nm

TAB. 1.2 – Radiant and visual quantities and their units. Radiant quantities Quantity Radiant flux Intensity Irradiance Exitance Radiance

1.5

Unit Watt (W) W.sr−1 W.m−2 W.m−2 W.sr−1.m−2

Visual quantities Quantity Luminous flux Luminous intensity Illuminance Luminous exitance Luminance

Unit Lumen (lm) Candela (cd) Lux (lx) lm.m−2 cd.m−2

Perfectly Diffuse Light

The notion of diffuse light is often met in lighting, solar energy or material appearance applications. It can be characterized by an angular spectral radiance distribution. However, in many optical models or simulators, it is simpler to consider “perfectly diffuse light”, i.e., isotropic radiance distribution. A light source that (nearly) verifies this property is qualified as Lambertian, after the 18th century Swiss scientist Johann-Heinrich Lambert (1728–1777) for his work on this subject, published in 1760 in the book Photometria [138]. The term Lambertian also applies to surfaces or diffusers, behaving as Lambertian secondary sources by reflection or transmission of light. Integrating spheres are spherical cavities coated on their internal side with a white material whose properties approach those of a Lambertian surface. They are convenient tools to produce a Lambertian lighting on a surface.

1.5.1

Lambertian Lighting and Lambertian Surface

The notion of perfectly diffuse light refers to an angular radiance diagram that is constant with respect to direction. When at a given point P of a surface, the elementary area around it received the same spectral radiance Lk ðPÞ from every direction of the hemisphere, the lighting can be qualified as perfectly diffuse, or Lambertian. With an angle-independent radiance, the spectral irradiance expressed by equation (1.35) becomes:

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21 Z

E k ð PÞ ¼ L k ð PÞ

2p u¼0

Z

p=2

h¼0

cos h sin hdhdu ¼ pLk ðPÞ

ð1:40Þ

A Lambertian light source is also characterized by an isotropic radiance distribution. By analogy with equation (1.40), the spectral exitance is related to the spectral radiance by Mk ðPÞ ¼ pLk ðPÞ

ð1:41Þ

This definition also stands for secondary light sources: any object able to reemit a constant radiance over the hemisphere whatever the angular geometry of illumination is qualified as Lambertian reflector, Lambertian diffuser, or Lambertian surface. The constant radiance Lk ðPÞ reemitted in every direction by the elementary area around point P is related to irradiance Ek ðPÞ of this elementary area by L k ð PÞ ¼

qðkÞEk ðPÞ p

ð1:42Þ

where qðkÞ is the fraction of incident flux that is reflected from the diffuser for each wavelength, referred to as spectral reflectance, Lambertian spectral reflectance, or spectral albedo. Equation (1.42) is known as Lambert’s law. It follows from equation (1.31) that the spectral flux emitted by a finite area Δs on a uniform Lambertian source in a finite small solid angle DX around a certain direction ðh; uÞ varies as the cosine of θ, which is known as the Lambert’s cosine law: Fk ðh; Ds Þ ¼ Lk ðDs ÞDsDX cos h

ð1:43Þ

The perfect white diffuser is a Lambertian reflector that does not absorb light in the visible spectrum, i.e., whose spectral albedo is qðkÞ ¼ 1. White tiles approaching these properties are common tools in radiometry of surfaces to calibrate measurement setups [38]. They are generally made of pressed barium sulfate or PTFE (known as Algoflon, Halon or Spectralon). An example is shown in figure 1.16.

FIG. 1.16 – A perfect white diffuser from Labsphere company, USA.

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22

1.5.2

Integrating Spheres

Integrating spheres are spherical cavities capable of producing perfectly diffuse light. They are internally coated with a powder of non-absorbing material, e.g. barium sulfate (BaSO4), behaving as an almost perfect white diffuser [184, 240]. Their spherical shape and their Lambertian coating provide them with excellent radiometric properties, which explains why they are widely used in radiometric metrology. The spherical shape ensures that the geometrical extent between any pair of small areas on the surface is constant and depends only on radius r of the sphere. In order to show this invariance property, let us consider two points A and B anywhere on the surface of the sphere, and define a small area DsA around A, and an elementary area dsB around B (see figure 1.17). By denoting by C the center of the sphere, one can observe that segments [AC] and [BC], of length r, are orthogonal to DsA and dsB , respectively. The triangle ABC is therefore isosceles, and one denotes by θ its base angles. The length of its base AB is therefore AB ¼ 2r cos h

ð1:44Þ

FIG. 1.17 – Sectional view of an integrating sphere of radius r, in the plane containing two points A and B on its surface and the center C of the sphere.

According to the defining expression (1.18) of the geometrical extent, one finds that the geometrical extent between DsA and dsB is independent of angle θ, therefore independent of the selected positions of for A and B: dG ¼

DsA cos hdsB cos h DsA dsB ¼ 4r 2 AB2

ð1:45Þ

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23

Let us assume that the small area DsA receives spectral flux Fk . Its spectral irradiance is Ek ðAÞ ¼ Fk =DsA . The fact that the coating is a Lambertian reflector ensures that spectral radiance Lk ðAÞ reemitted from DsA is the same in every direction. According to Lambert’s law, equation (1.42), this spectral radiance is Lk ð AÞ ¼

qðkÞEk ðAÞ qðkÞFk ¼ p pDsA

ð1:46Þ

where ρ(λ) is the spectral albedo of the coating. Let us now consider an elementary area dsB anywhere else on the surface of the sphere, that receives radiance Lk ðAÞ. Its irradiance is ð1Þ

Ek ð B Þ ¼

Lk ðAÞdG qðkÞFk ¼ dsB 4pr 2

ð1:47Þ

ð1Þ

We observe that Ek ðBÞ is independent of the position of point B: the whole sphere is uniformly illuminated. For B or any other point on the surface of the sphere, elementary area dsB around it reflects fraction qðkÞ of the light it receives, by producing an isotropic radiance and therefore a uniform irradiance of the whole surface of the sphere. If one considers one specific point B0 on the sphere, elementary area dsB0 around it receives from dsB the elementary spectral irradiance ð2Þ

dEk ðB0 ; BÞ ¼

ð1Þ

qðkÞEk ðBÞdsB 4pr 2

ð1:48Þ

and it receives from the whole surface of the sphere the spectral irradiance Z ð1Þ qðkÞEk ðBÞ ð2Þ ð1Þ 0 Ek ðB Þ ¼ dsB ¼ qðkÞEk ðBÞ ð1:49Þ 4pr 2 B2sphere ð2Þ

Once again, the sphere is uniformly illuminated: spectral irradiance Ek ðB0 Þ is independent of the position of B0 . This process continues indefinitely: once the k-th reflection on the sphere has occurred, the irradiance around every point P on the surface of the sphere is ðk Þ

ðk1Þ

Ek ðPÞ ¼ qðkÞEk

ð PÞ

ð1:50Þ

In a steady state, every surface element simultaneously receives all the irradiance ðk Þ components Ek ðPÞ, k ¼ 1; 2; :::. By summing them up, one obtains the total irradiance, equal at every point P at the surface of the sphere, expressed as a geometric series with common ratio ρ(λ): Ek ð P Þ ¼

1 X k¼1

ðk Þ

Ek ðPÞ ¼

1 X

ð1Þ

qk ðkÞEk ðPÞ

k¼0

qðkÞ qðkÞ Fk ð1Þ E ð PÞ ¼  ¼ 1  qð k Þ k 1  qðkÞ 4pr 2

ð1:51Þ

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24

The term qðkÞ=½1  qðkÞ is a unitless factor, higher than one, usually referred to as the sphere multiplier. It accounts for the accumulation of light over time into the sphere. For most commercial integrating spheres, the sphere multiplier value is between 10 and 30. Remark: Integrating spheres necessarily have some holes in order to place a light source, a detector, etc. These holes contribute to decrease the sphere multiplier. By denoting f the fraction of the sphere area occupied by these holes, the multiplier becomes: qðkÞ=½1  ð1  f ÞqðkÞ, and equation (1.51) becomes E k ð PÞ ¼

qðkÞ Fk  1  ð1  f ÞqðkÞ 4pr 2

ð1:52Þ

In the demonstration above, one has considered that the incident flux falls on a small area DsA . Actually, the same result would be obtained if a larger area was initially illuminated. Notice however that spectral irradiance Ek ðPÞ concerns only the points which have not been directly illuminated by the source. For those which have been directly illuminated (direct spectral irradiance Ek0 ðPÞ being a function of position), the total spectral irradiance is obviously Ek ðPÞ þ Ek0 ðPÞ. Thanks to formula (1.51), it is easy to relate the spectral irradiance of a detector placed anywhere at the surface of the sphere (provided it is not directly illuminated by the initial source). Rigorously speaking, the detector captures a flux, not an irradiance, but if its area is known, its spectral irradiance Ek corresponds to the captured spectral flux divided by the detector area. The integrating sphere is therefore a good measuring device for the spectral flux emitted by lamps. It is also a good device to produce perfectly diffused light. Both properties are also useful for the characterization of surfaces, as explained in §2.6.2.

1.6

Light Source Illuminating a Plane Surface

In many scenes studied in radiometry, a light source illuminates a flat surface. One assumes that the angular diagram of emission of the source presents a symmetry of revolution around the normal of the plane. The irradiance of the plane is maximal in the nearest point from the source and decreases radially around this nearest point according to a function which depends on whether the light source is nearly a point or has a large area. In this section, we propose to address the two possible configurations.

1.6.1

Illumination by a Point Source: Bouguer’s Law

A point source, assumed to be isotropic of spectral intensity Ik, is located in point S at distance h from the plane. Point O is the nearest point on the plane from the source. Let us define an elementary area ds around it, which subtends from S an elementary solid angle dX ¼ ds=h 2 (as ds is small, one can assume that it is equal to its projection on the sphere of radius h). The elementary spectral flux flowing into this solid angle is therefore dFk ¼ Ik dX

ð1:53Þ

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25

and the irradiance of the plane at point O is therefore E k ðO Þ ¼

dFk Ik ¼ 2 ds h

ð1:54Þ

The relation stating that the irradiance decreases with the inverse squared distance from the source is known as Bouguer’s law after the French scientist Pierre Bouguer (1698–1758), even though in his essay published in 1729, Bouguer attributes this law to Kepler [23].

FIG. 1.18 – A point source in S illuminates a plane. Point O, the closest point from the source, receives light from its normal direction and has the highest irradiance (see figure 1.18).

Now let us consider another point P on the plane, distant from O by a length u and from the source by a length hp ¼ SP ¼ h=cosh, where θ is the angle between lines (SO) and (SP). Area ds around P subtends a solid angle dXP ¼ ds cos h=hP2 ¼ ds cos3 h=h 2 (ds cos h is the projection of ds on the sphere of radius hp ). The elementary spectral flux flowing into this solid angle is Ik dXp . After some calculations, using equation (1.54), one obtains the following expression for the spectral irradiance of the surface around point P, known as the cos3 law: Ek ðPÞ ¼ Ek ðOÞ cos3 h

ð1:55Þ

If the source is not isotropic, equation (1.55) can be easily adapted by replacing the constant spectral intensity Ik with the angle-dependent spectral intensity Ik ðh; uÞ in the considered direction.

1.6.2

Illumination by an Extended Source

The point source considered in the previous section is replaced with an extended source of area A, parallel to the planar surface, and still located at distance h from the plane surface (figure 1.19). It is assumed to be uniform and Lambertian, i.e., any small area on it emits the same spectral radiance Lk in every direction.

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26

FIG. 1.19 – A square extended source illuminates a plane. The irradiance map on the plane varies according to the size of the source.

Let us first consider that the source is small, i.e., its transversal size is much smaller than h (figure 1.19a). When viewed from a point on the surface, light seems to come from a single direction. The nearest point O from the center S of the source receives radiance Lk that the source uniformly emits in every direction. This radiation flows perpendicularly to the source and to the small area ds around O, into a geometrical extent dG ðA; ds Þ ¼

Ads h2

ð1:56Þ

a2 h2

ð1:57Þ

The irradiance of ds around point O is Ek ðOÞ ¼ Lk

We can see that, as in the case of the point source, the irradiance at point O is inversely proportional to the squared distance between the plane and the small extended source. At point P distant from O by length u, therefore from S by length hp ¼ h=cosh, the geometrical extent between the small area ds around P and the source becomes dG ¼

A cos hds cos h Ads ¼ 2 cos4 h hp2 h

ð1:58Þ

and the irradiance of the surface at point P becomes Ek ðPÞ ¼ Ek ðOÞ cos4 h

ð1:59Þ

This relation is known as the cos4 law, decreasing faster from point O than the cos law applicable to point sources. 3

Light and Optical Radiations

27

Let us now consider a larger source, which is still Lambertian and uniform. Every small area on it still emits the same radiance Lk in every direction. However, one cannot assume any more that the light reaching point P on the surface comes from one direction. The geometrical extents between elementary area ds around point P of the surface and an elementary area ds 0 around point P0 of the source varies with respect to the positions of both P and P0 : d2G ¼

dsds0 cos4 hðP,P0 Þ h2

ð1:60Þ

where hðP,P0 Þ is the angle made by the normal of ds and ds 0 with (PP0 ). The spectral radiance emitted by ds0 around P0 contributes to the spectral irradiance at point P with an elementary spectral irradiance that can be written dEk ðP,P0 Þ ¼ Lk

cos4 hðP,P0 Þ 0 ds h2

The total spectral irradiance at point P is therefore: ZZ Lk Ek ð P Þ ¼ 2 cos4 hðP,P0 Þds 0 h P0 2Source

ð1:61Þ

ð1:62Þ

In the most general case where the source is non-Lambertian and non-uniform, the constant spectral radiance Lk should be replaced with a function Lk ðP0 ; h; uÞ. The irradiance of the surface at point P this becomes ZZ 1 E ðPÞ ¼ 2 Lk ðP,hðP,P0 Þ; uðP,P0 ÞÞ cos4 hðP,P0 Þds 0 ð1:63Þ h P0 2Source

Chapter 2 Spectral Radiometry of Surfaces When an optical radiation reaches a material, a large number of phenomena may occur on a small scale, according to a process the complexity of which increases rapidly beyond the representable as soon as the material microstructure of the object becomes random. The most complex situations, however, are the most common. The light from the sun or from artificial sources is strongly incoherent and often non-directional, at the opposite of a monochromatic and plane wave. Similarly, objects are rarely parallelepipedal blocks of a single solid matter with smooth surface: they most often result from a stacking of material layers themselves constituted of a more or less random arrangement of micro-objects of various compounds. However, these material microstructures are rarely known, or too complex to be described even when observed by microscopy. Thus, in everyday life, there are only rare configurations in which the incident wave and the object can be described in a simple way and Maxwell’s equations can be used to describe how, where and in what proportions the radiation propagates after its interaction with the object. Small-scale optical phenomena must be considered on average on a larger scale, as flux transfers between different positions. Certainly, the phenomena of interference and diffraction occurring locally can no longer be finely described according to this approach, but their effects in the form of well-known colored irisations with regular hardware structures tend to disappear rapidly with random structures. Thus, paper, although composed of cellulose fibrils and transparent pigments all susceptible to diffract the light, has no macroscopic iridescence, but instead a homogeneous white appearance. In the end, the result of the interaction between incoherent light and the material can be described with a rather reduced number of macroscopic radiometric quantities, accessible by optical measurement, which are the subject of this chapter. At the macroscopic scale, the light exiting from the object can be decomposed into three main components, regardless of the exact directions that it follows: the reflected spectral flux returns into the medium of incidence, the transmitted spectral flux crosses through the object, and the absorbed spectral flux remains inside the matter and is most often converted into heat (unless it is converted into other optical radiations by luminescence or fluorescence effects [109], not addressed in this book). The corresponding macroscopic radiometric quantities are, respectively, the spectral reflectance, spectral transmittance and spectral absorptance. These quantities, DOI: 10.1051/978-2-7598-2647-6.c002 © Science Press, EDP Sciences, 2022

30

Optical Models for Material Appearance

nevertheless, should be handled with caution depending on the type of lighting and materials considered, whose diversity remains important even if the description of the light-matter interaction is much simpler at a macroscopic scale than at a microscopic one. Thus, there are various types of spectral reflectance and transmittance depending on the illumination conditions (directional, hemispherical…) and the viewing angle or solid angle of detection. The ability for a surface to scatter light or not is one of the essential points to consider before undertaking its radiometric analysis, and it is with this point that we will begin this chapter, by considering first the reflected light, then the transmitted light. We will then pursue with the main radiometric functions permitting to characterize the reflecting or transmitting properties of surfaces.

2.1

Types of Reflecting Surfaces

All surfaces do not reflect light the same way even when placed under the same lighting conditions. We all know this from our daily experience, and it is even what allows us to identify many materials with the naked eye. This obviously comes from the optical phenomena underlying the reflection of light. Some surfaces qualified as glossy, or specular (from the Latin speculum, mirror), are able to reflect each light pencil by keeping its geometrical extent. Some surfaces qualified as matte are capable of diffusing light uniformly in all directions of the half-space. These two opposite types of surfaces have relatively simple radiometric properties, and the optical models developed in chapters 4–7 are dedicated to them. Between these two extreme cases, there is a continuum of intermediate reflectors, sometimes qualified as “satin” or “translucent”, whose optical characterization is more complex and will be briefly evocated in chapter 8.

2.1.1

Mirrors and Specular Reflectors

Mirrors designate surfaces which do not scatter light at all. Each light ray is reflected along one direction called the regular direction, or specular direction, determined by the incidence angle and the normal vector of the surface in the reflection point. There exist other types of specular reflectors, in particular a layer of non-scattering material presenting flat interfaces with its surrounding media, or piles of layers of this kind: a glass plate, a double-glazed window, a mirror with thin coating, etc. The fraction of incident light that is reflected from a specular reflector depends on its angle of incidence and its polarization. The photometric quantity suitable to characterize it is the spectral angular reflectance, defined in §2.3.

2.1.2

Very Matte Surfaces: Lambertian Reflectors

A perfectly matte surface is a surface whose appearance remains the same when moving around it, whatever the illumination conditions. No shine is perceptible in any direction. Since appearance is related to the spectral radiance received by the

Spectral Radiometry of Surfaces

31

observer, one concludes that the spectral radiance issued from the matte surface is the same in all directions, for every polarization state, which in radiometry is the definition for a Lambertian reflector (see §1.5.1). Lambert’s law, equation (1.42), applies. The perfect white diffuser belongs to this category of reflectors. The radiometric quantity suitable to characterize a matte surface is the spectral reflectance, or spectral albedo, presented in §2.3.

2.1.3

Intermediate Cases

Between the two extreme types of reflectors presented above, there exist many intermediate cases where light is scattered in an angle-dependent way. This is, for example, the case of silk, satin, leather, or other materials with shiny or glossy finishing: when moving around the surface, the radiance of reflected light varies. The radiometric characterization of these surfaces requires an angular function including all possible pairs of directions of incidence and reflection, the Bi-directional Reflectance Distribution Function (BRDF), presented in §2.4. Objects made of a diffusing material with well-polished surface are a special case: the reflected light may be divided into two components, a specular component reflected from the smooth air-material interface, and a diffuse component, almost Lambertian, reflected from the material itself. Photographic paper with glossy finishing or porcelain objects are in this category. They can be characterized with the reflectance factor concept, by including or excluding the specular component.

2.2

Light Transmission Through Objects

We can also classify objects into various categories based on how they transmit light if they are not opaque, i.e., if they are translucent. A first category contains the transparent objects which do not scatter light at all. A flat transparent object such as a plate allows to see through it as clearly as if it were absent. If the shape is not flat, light can locally converge or diverge and create caustics, a situation that will not be discussed in this book. Like reflection on mirrors and flat specular reflectors, transmission through a transparent plate depends upon the wavelength, the orientation and the polarization of incident light, which is described by its angular spectral transmittance. A second category contains the perfect diffusers: the transmitted radiance is constant over the hemisphere. This property is often desired in lighting and diffusing plates are designed for this purpose. Perfect diffusers are characterized with the spectral transmittance concept. In the intermediate cases where light is scattered in an angle-dependent way, it is necessary to describe the angular distribution of the transmitted flux, which is possible thanks to the concept of Bi-directional Transmittance Distribution Function (BTDF) (see below, §2.4).

Optical Models for Material Appearance

32

2.3

Angular Spectral Reflectance and Transmittance

The angular spectral reflectance and transmittance are the basic radiometric concepts for the optical characterization of a surface. Consider that around a given point P of the surface, elementary area ds ðPÞ receives from a given direction ðhi ; ui Þ ðiÞ spectral flux Fk ðP; hi ; ui Þ having a certain polarization state * (usually, the incident light is unpolarized, but one may also consider one of its polarization components s or p, see §1.1.2). The angular spectral reflectance and transmittance of the surface, R ðP; h; u; kÞ and T ðP; h; u; kÞ, are defined as the ratios between the total reflected ðrÞ ðtÞ spectral flux Fk ðPÞ, respectively, the total transmitted spectral flux Fk ðPÞ, and ðiÞ

the incident directional spectral flux Fk ðP; h; uÞ, all fluxes being defined in the same small bandwidth Dk around the considered wavelength λ: R

R ðP; h; u; kÞ ¼

ðrÞ Dk Fk ðPÞdk R ðiÞ Dk Fk ðP; h; uÞdk

R

and T ðP; h; u; kÞ ¼ R

ðtÞ Dk Fk ðPÞdk ðiÞ Dk Fk ðP; h; uÞdk

ð2:1Þ

In some special cases, the angular spectral reflectance or transmittance is constant with respect to one of the parameters. The object is thus qualified as:  achromatic when R and T are constant with respect to wavelength λ in the visible spectrum,  uniform when they are constant with respect to position P,  isotropic when they are constant with respect to azimuthal angle φ,  Lambertian when they are constant with respect to both angles θ and φ. The Fresnel formulae presented in §1.3.2 correspond to the angular reflectance of a smooth interface between two media. For most materials considered in this book, the interface is uniform and isotropic. It is also achromatic when the refractive index of the two media can be assumed constant over the visible spectrum, which is the case for many clear materials, but not for metals (see §4.3 and §8.1.5) nor strongly absorbing media such as printing inks. Angular spectral reflectance and transmittance do not describe the angular distribution of the reflected and transmitted light components. The latter is easy to determine for the two special types of objects presented at the beginning of this chapter: for the specular reflectors and the plane transparent objects, the reflected and transmitted fluxes are directional and their respective directions can be easily determined with respect to the direction of the incident beam according to Snell-Descartes laws (§1.3.1); for perfect diffusers (Lambertian objects), the reflected and transmitted radiances are isotropic over their respective hemispheres. For the other types of surfaces, however, the angle-dependent radiance reemitted by the object at both sides is described by the BRDF and BTDF concepts introduced below.

Spectral Radiometry of Surfaces

2.4

33

Bi-Directional Reflectance/Transmittance Distribution Function (BRDF/BTDF)

Although originally defined for optical surface metrology [176], BRDF and BTDF have gained popularity in the field of material appearance with the emergence of computer graphics which use them as a basic quantity for the visual rendering of objects [81]. Modeling and measuring BRDFs and BTDFs of different kinds of materials has become a very dynamic field of research, stimulated by a growing demand from the computer graphics industry (advertising, video games, animation movies…) as well as from companies with strong requirements in terms of appearance of their products (luxury, automotive industries). This impetus has also been accompanied by the development of numerous fast or more precise measuring devices. The present section is dedicated to the definition of the BRDF and BTDF and the presentation of some measurement techniques. A BRDF model will be presented later in §8.1.3.

2.4.1

Definition

The spectral BRDF is defined for a given position of the surface under test (specified by point P), a narrow spectral bandwidth Dk around a given wavelength λ, a given polarization state of the incident light (denoted by the * as with the Fresnel formulae in §1.3.2), a given solid angle dXi ðhi ; ui Þ around the direction of incidence ðhi ; ui Þ and a given direction of observation ðhr ; ur Þ. According to Nicodemus [176], the reflection process of light by a surface is embodied in the fundamental equation relating elementary radiance dLk ðP; hr ; ur Þ scattered by dsðPÞ along direction ðhr ; ur Þ: Z dLk ðP; hr ; ur ; Þdk ð2:2Þ dLðP; hr ; ur ; k; Þ ¼ Dk

and elementary irradiance dE ðP; hi ; ui ; k; Þ of dsðPÞ produced by the incident light pencil coming from direction ðhi ; ui Þ Z dE ðP; hi ; ui ; k; Þ ¼ dEk ðP; hi ; ui ; Þdk ð2:3Þ Dk

This fundamental equation is dLðP; hr ; ur ; k; Þ ¼ fR ðP; hi ; ui ; hr ; ur ; k; ÞdE ðP; hi ; ui ; k; Þ

ð2:4Þ

where function fR is the bidirectional reflectance distribution function (BRDF), in sr−1: fR ðP; hi ; ui ; hr ; ur ; k; Þ ¼

dLðP; hr ; ur ; k; Þ dE ðP; hi ; ui ; k; Þ

ð2:5Þ

Thanks to equation (1.33), it can be defined in terms of incident radiance Li ðP; hi ; ui ; k; Þ: fR ðP; hi ; ui ; hr ; ur ; k; Þ ¼

dLðP; hr ; ur ; k; Þ Li ðP; hi ; ui ; k; Þ cos hi dXi

ð2:6Þ

34

2.4.2

Optical Models for Material Appearance

BRDF of Some Typical Reflectors

Figure 2.1 shows BRDF sections in the incidence plane of various surfaces. In the case of the perfect white diffuser defined in §1.5.1, spectral exitance Mk ðPÞ is equal to incident irradiance Ek ðPÞ at every point P of the surface, for every polarization state. Since the spectral albedo is qðkÞ ¼ 1, the spectral radiance reflected along every direction of the hemisphere is Ek =p (Lambert’s law). The BRDF is therefore a constant with respect to angles: fR ðP; hi ; ui ; hr ; ur ; k; Þ ¼

1 p

ð2:7Þ

Lambertian reflectors (very matte surfaces) also have an angle-independent and polarization-independent BRDF, proportional to their spectral albedo, qðkÞ: fR ðP; hi ; ui ; hr ; ur ; k; Þ ¼

qðkÞ p

ð2:8Þ

In the case of a perfect mirror, the BRDF cannot be defined (although some models consider that is it a Dirac Delta function). A setup for mirror reflectance measurement is described in Ref. [249, p. 54].

FIG. 2.1 – Sections of BRDF in the incidence plane (φi = φr = 0 mod. π), plotted in polar coordinates as a function of θr, of (a) a perfect white diffuser, (b) a mirror-like surface, (c) a roughened aluminum surface [89] and (d) a glossy paper [21].

Spectral Radiometry of Surfaces

2.4.3

35

Lambert’s Azimuthal Equal-Area Projection

Since BRDF depends on many parameters, it is impossible to plot a full BRDF in 2D. 3D visualization by software is often preferred. However, plane mapping is a good alternative for BRDF display [58]. The Lambert azimuthal equal-area projection [220] is especially convenient as it conserves areas: a portion of the half-sphere with area A is mapped onto a portion of disk with identical area A. Every unit vector in the hemisphere specified by its spherical coordinates ðh; uÞ, hence denoted by P ¼ ðsin h cos u; sin h sin u; cos hÞ in Cartesian coordinates, is mapped onto a point P0 of polar coordinates ðr; uÞ contained within a disk of radius pffiffiffi 2 tangent to the hemisphere at the North pole N (figure 2.2). The azimuthal coordinate φ is the same in the two coordinate systems. Coordinate r ¼ 2 sinðh=2Þ corresponds the distance NP. Point P0 is also specified by the following Cartesian coordinates in the disk: pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2px = 1 þ pz pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 2 sinðh=2Þ sin u ¼ 2py = 1 þ pz

u ¼ 2 sinðh=2Þ cos u ¼

FIG. 2.2 – Mapping of the hemisphere onto a disk of radius

ð2:9Þ

pffiffiffi 2 according to Lambert

azimuthal equal-area projection applied at the North pole N.

Applying this mapping to the BRDF yields a multispectral image containing as many channels as there are wavelengths. To each pixel of the image is attached a square area in the disk corresponding to an identical solid angle. Square pixels with size d represent solid angles of d 2 steradians. An example of BRDF displayed in the representation system is shown in figure 2.3.

36

Optical Models for Material Appearance

FIG. 2.3 – Example of BRDF representation in the Lambert azimuthal equal-area projection plane, for a rough glass surface illuminated at 30° from the normal. The grey value in each pixel of the image is proportional to the radiance in each corresponding direction (the highest radiance value is displayed in white).

2.4.4

BRDF Measurement Systems

BRDFs are usually measured with a goniophotometer, or a gonio-spectrophotometer if the detection is spectral, in which the sample is illuminated by a collimated light beam. In most usual measurement setups, the elementary solid angle dXi is a finite solid angle DXi determined by the optical system. Radiance DLr ðP; hr ; ur ; k; Þ of the scattered light is finite as well. The illuminated area is generally a finite area Ds which may be large (e.g., 1 cm2 or more) and the small fluctuations of the scattering properties of the surface are averaged over this area. A classical configuration relies on rotating arms allowing the operator to change the angles of incidence and detection. Light is then transferred to a photodetector or a spectrophotometer. The goniophotometer ConDOR developed by the LNE-CNAM laboratory in Saint-Denis, France, benefits from a very high angular resolution comparable to that of the human vision (0.03°) thanks to a narrow field of view imaging system [76]. Another configuration, more compact, is proposed by the company Eldim, France: a Fourier lens technology transforms the angular distribution of the light scattered over the hemisphere into a spatial distribution onto the matrix sensor [19], allowing polarized and spectral BRDF acquisition [20]. Every system needs a photometric calibration upon a perfect white diffuser, in order to determine the instrument transfer function (see exercise 12).

Spectral Radiometry of Surfaces

37

Gonio-spectrophotometers remain expensive instruments, and their volume and fragility restrict their use for specific applications in laboratory. Fast and low-cost alternative solutions can be found if the surface under test is spherical or cylindrical: The instrument is composed of a point source and a camera so that each pixel of the captured image corresponds to a different bi-directional configuration. Hence, one camera shot provides a large portion of BRDF, and the camera or the source can be moved to enlarge the angular domain of BRDF [146, 155]. LED domes are currently being developed in order to avail of many source positions: multiple pictures of the object are taken while the LED point sources, distributed over a large hemispherical dome, are turned on sequentially in synchronization with the camera. Computational imaging methods then enable us to obtain the BRDF. These LED domes can also be used with flat objects: the camera records a partial BRDF in each pixel (the viewing position is fixed as the camera is fixed), and generates a so-called Bi-directional Texture Function (BTF) which is used as input to rendering models in computer graphics or as appearance assessment data for visual quality control of some products [49]. In order to prevent saturation of the camera sensors in the pixels receiving the highest radiances, High Dynamic Range imaging is often used. Methods enable us to remove diffraction patterns surrounding the brightest image points [147]. Image-based solutions now extend to BTDF measurement [203].

2.5

Spectral Reflectance and Transmittance Factors

Spectral reflectance and transmittance extend the basic concepts of angular spectral reflectance and transmittance to any angular distribution of illumination. This concept is frequently used for the optical and color characterization of matte surfaces such as skin, paints, paper, or prints. However, it is generally not the quantity provided by the spectrophotometers dedicated to the optical measurement of these surfaces: they rather provide a spectral reflectance/transmittance factor, or a spectral radiance factor according to the angular geometry of detection. The present section introduces these concepts before showing some usual measuring devices.

2.5.1

Reflectance, Transmittance

The terms reflectance and transmittance are specific to the characterization of diffusers. They denote any ratio of reflected/transmitted/absorbed flux to incident flux [38]. They are dimensionless quantities whose values are between 0 and 1. They may depend on the wavelength, direction and polarization of the incident light, as well as on the position on the sample. R

The spectral reflectance RðkÞ of a sample is the ratio between flux Fr ðkÞ ¼ R ðr Þ ði Þ Dk Fk dk reflected from the sample and incident flux Fi ðkÞ ¼ Dk Fk dk, both being ðr Þ

ði Þ

defined from the reflected spectral flux Fk and the incident spectral flux Fk over a narrow bandwidth Dk around the considered wavelength λ: R ð kÞ ¼

F r ð kÞ F i ð kÞ

ð2:10Þ

Optical Models for Material Appearance

38

Spectral transmittance is defined in the same way. Except in the case of a perfect diffuser, or Lambertian surface (see exercise 10), these concepts depend on the solid angles of incidence and detection. Hence, the measuring geometry should be specified each time the concepts are used. Specifications related to the standard measurement geometry will be addressed in section 2.6. In the general case, the surface may be illuminated through some solid angle Ωi that may be conical or not and inside which spectral radiance Lk ðP; hi ; ui Þ may not be isotropic. The reflected light can be detected inside another solid angle Ωr (conical or not). Exercise 8 helps to establish that, in this case, the spectral reflectance RðkÞ is related to the spectral BRDF of the surface by: RR RR ðhr ;ur Þ2Xr ðhi ;ui Þ2Xi fR ðhi ; ui ; hr ; ur ; k; Þ Lk ðhi ; ui Þ cos hi sin hi dhi dui cos hr sin hr dhr dur RR RðkÞ ¼ ð2:11Þ ðhi ;u Þ2Xi Lk ðhi ; ui Þ cos hi sin hi dhi dui i

With a uniform isotropic reflector (the spectral BRDF is independent of the azimuth angle of incidence ui and the position on the surface) and a uniform Lambertian illumination (incident spectral radiance Lk is constant with respect to direction and position), the general expression (2.11) noticeably simplifies. For example, if the incident solid angle is narrow and makes angle hi with the normal of the surface, and if the solid angle of detection is the hemisphere, one obtains the angular spectral reflectance Rðhi ; kÞ defined in §2.3, for a given polarization state of the incident light: Z 2p Z p=2 fR ðhi ; ui ; hr ; ur ; k; Þ cos hr sin hr dhr dur ð2:12Þ R ðhi ; kÞ ¼ ur ¼0

hr ¼0

The bi-hemispherical spectral reflectance r ðkÞ of the isotropic reflector is defined for a Lambertian illumination over the hemisphere and a hemispherical observation. It can be directly related to the angular spectral reflectance [98] thanks to the following equation, also derived in exercise 9: Z p=2 r ð kÞ ¼ Rðhi ; kÞ sin 2hi dhi ð2:13Þ hi ¼0

This equation will be frequently used in the next chapters to compute the amount of light reflected from specular reflectors when the incident light is perfectly diffuse. The bi-hemispherical spectral transmittance is defined in the same way from the angular spectral transmittance.

2.5.2

Spectral Reflectance/Transmittance Factor

In most reflectance or transmittance measurement devices used to characterize matte surfaces, the incident flux is measured indirectly using a perfect white diffuser able to reflect or transmit the incident light uniformly over the hemisphere without

Spectral Radiometry of Surfaces

39

absorbing it. The spectral flux captured by detector Fref ðkÞ is therefore proportional to the incident spectral flux Fi ðkÞ. ^ ðkÞ of a sample is defined in a narrow spectral The spectral reflectance factor R band Dk around wavelength λ as the ratio between flux F ðkÞ measured from the sample and flux Fref ðkÞ measured from the perfect white diffuser, both being illuminated and observed under exactly the same conditions [40]: ^ ð kÞ ¼ F ð kÞ R Fref ðkÞ

ð2:14Þ

^ ðkÞ is defined as the ratio between flux F ðkÞ Spectral transmittance factor T measured from the sample and flux Fref ðkÞ measured in the absence of the sample. ^ ðkÞ ¼ F ðkÞ T Fref ðkÞ

ð2:15Þ

In contrast with reflectance and transmittance values which are between 0 and 1, reflectance factor and transmittance factor values are between 0 and 1 [213]. Values higher than 1 are obtained when the sample reflects/transmits more light towards the detector than the perfect white diffuser sample. This is a situation that one can meet with a glossy sample under direct sunlight when looking at it in the specular direction: it looks much brighter than a perfect white diffuser or any other matte surface. An example will be analysed in §7.1.3. Like reflectance and transmittance, reflectance factor and transmittance factor may strongly depend upon the measuring geometry. In the special case where the reflector is Lambertian or perfectly matte, and in this case only, reflectance and reflectance factor are equivalent, whatever the geometry (see exercise 10).

2.5.3

Spectral Radiance Factor

A radiance factor is measured with an optical setup capturing the light scattered by the sample in a particular direction, most often along the normal of the sample. It is defined for each wavelength as the ratio between radiance LðkÞ measured from the sample and radiance Lref ðkÞ measured from a perfect white diffuser in a narrow spectral bandwidth Dk around wavelength λ: ^ ðkÞ or T ^ ð kÞ ¼ Lð kÞ R Lref ðkÞ

2.6

ð2:16Þ

Geometries and Devices for the Measurement of Reflectance Factors

For most reflectors except the Lambertian reflectors, the illumination and observation geometries have a strong influence on the measured values. It is therefore crucial to select them carefully and have them in mind when interpreting the measured values. In this section, we propose to introduce the geometries

40

Optical Models for Material Appearance

recommended by international standards for reflectance (similar geometries exist for transmittance), and focus on those which are the most frequently used. We will finally insist on the fact that the measured values also depend on the type of reflector (matte, specular…), and the effective measurement geometry can be different from the geometry of the measuring device.

2.6.1

Nicodemus’ Nomenclature for Nine Reflectance Factors

Judd [120], then Nicodemus [176], proposed a nomenclature for nine reflectance and reflectance factor geometries where each of the solid angles containing the incident flux and the captured flux is either directional, conical or hemispherical. This gives the nine typical measurement geometries featured in figure 2.4. Among these geometries, only a few of them are commonly used, in particular the bi-directional geometry, the directional-hemispherical geometry, and the hemispherical-direction geometry. As explained in the next section, the hemispherical geometry usually relies on an integrating sphere, which can be used either to illuminate the sample or to collect the reflected light. Notice that the bi-hemispherical geometry is not realizable in reflectance mode, since the integrating sphere cannot be used for illumination and collection at the same time.

FIG. 2.4 – The nine typical geometries according to Nicodemus [176].

Spectral Radiometry of Surfaces

41

In measurement devices, the spectrum of the source generally tends to reproduce the color of a standard illuminant [206], typically the D65 illuminant defined as a model for daylight (see §3.1.6), despite the difficulty to reproduce reliably standard illuminant spectra with artificial lightings (see figure 3.8, in chapter 3). Once captured, light is transferred to a spectrophotometer which measures the flux in the different spectral bands of width 1, 5 or 10 nm. Table 2.1 presents some geometries recommended by the CIE for reflectance measurement [42]. TAB. 2.1 – Some of the geometries recommended by the CIE for reflectance measurements. Appellation Diffuse/8° geometry, specular component included (di:8°) Diffuse/8° geometry, specular component excluded (de:8°) Diffuse/diffuse geometry (d:d) Alternative diffuse geometry (d:0°) 45° annular/normal geometry (45°a:0°) 45° directional/normal geometry (45°x:0°)

Illumination Capture Hemispherical Directional Hemispherical Directional Hemispherical Hemispherical Hemispherical Directional Annular Directional Directional Directional

Several companies such as BYK-Gardner, Datacolor, Konica Minolta or X-rite have developed spectrophotometers able to measure reflectances according to these geometries. They also propose lab-benches able to measure both spectral reflectance and transmittance. The latter are typically based on the di:8° and de:8° geometries in reflectance mode, and on the d°:0° geometry in transmittance mode (see table 2.1 for the meaning of the notations).

2.6.2

Geometries Using Integrating Spheres

Integrating spheres (see §1.5.2) can be used either to produce a Lambertian illumination or to collect reflected light over the hemisphere. Figure 2.5 illustrates these two possibilities.

FIG. 2.5 – Integrating spheres used in a 0°:d geometry (left) and a d:0° geometry (right).

42

Optical Models for Material Appearance

In the 0°:d geometry, the integrating sphere collects the whole flux reflected from the sample, which is illuminated by a collimated beam along the normal of the sample. The corresponding reflectance is a directional reflectance, given by equation (2.12). In the d:0° geometry, the integrating sphere plays the role of diffuser providing a Lambertian illumination. The reflected light is captured at 0° or 8° with respect to the normal of the sample. A hemispherical-directional reflectance is measured. In the case of diffusing samples having a flat surface, one may want to include or discard the specular reflection component from the measurement. This gives two modes, schematically represented in figure 2.6: the specular component included mode (denoted di:8° by the CIE), where the ray incoming in the regular direction with respect to the detector at 8° from the normal of the sample, featured in red, can be specularly reflected from the surface; and the specular component excluded mode (denoted de:8° geometry), where this light ray is removed by opening a hole on the sphere in this direction [249]. If the sample is perfectly matte, therefore perfectly diffusing, no light is specularly reflected and the same reflectance value is measured in both modes (see exercise 10). In opposition, if the sample is perfectly glossy like a mirror, light is reflected specularly, and if the hole of the sphere is open, no light is measured by the detection device. The measured value is therefore zero in the de:8° mode.

FIG. 2.6 – Reflectance measurement device based on the hemispherical-directional geometry in specular component included and excluded modes.

Spectral Radiometry of Surfaces

2.6.3

43

Bidirectional and Annular Geometries

In a device based on a bidirectional geometry, the detector captures only a fraction of the flux issued from the specimen. This fraction depends on the solid angle of detection. The so-called 45°:0° geometry, widely used in the printing industry, is a bidirectional geometry with an incident angle of 45° and a detection along the normal [240]. The sample is illuminated at a single azimuthal angle, or at all the azimuthal angles, yielding, respectively, the directional and annular variants of the 45°:0° geometry. The annular geometry, illustrated by figure 2.7 (see also exercise 2), minimizes anisotropic texture effects whereas the directional geometry tends to enhance them. This geometry naturally excludes the specularly reflected light from the measurement.

FIG. 2.7 – (a) 45° annular/normal geometry (45°a:0°) for reflectance measurements.

2.6.4

Effective Measurement Geometry

Since Lambertian diffusers reflect the same radiance in every direction, the detector can be placed anywhere and can have any size. This is also nearly true for strongly scattering surfaces like matte paper or cotton fabric. In this case, the instrument geometry and the effective geometry for the measurement coincide (figure 2.8a). In the case of a specular reflector, as already mentioned above, the reflectance factor is zero in the de:8° geometry, and non-zero in the di:8° geometry. In the latter case, actually, the light beam reflected towards the detector necessarily comes from the regular direction with respect to the detector, making an angle of 8° with the normal of the sample (see figure 2.8b). The effective measurement geometry is 8°:8°, even though the instrument used is based on the di:8° geometry. In the general case of a reflector being neither strongly diffusing nor specular, the origin of the rays reflected towards the detector is not known: it depends on the angular reflection properties of the surface, i.e., precisely what one wants to measure. The effective measurement geometry is therefore unknown, which prevents us from interpreting correctly the measured quantity. The reflectance measurement device, and even the reflectance concept, are not suitable for this type of reflector.

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FIG. 2.8 – Original directions of the rays likely to reach the detector with a d:8° reflectance measurement geometry according to the type of reflector: (a) with a Lambertian reflector, rays from the whole hemisphere are likely to be scattered towards the detector, and the effective measuring geometry is d:8°; (b) with a specular reflector, only the pencil coming along the regular direction with respect to the detector can reach it, and the effective geometry is 8°:8°; (c) in the general case, rays likely to reach the detector are not precisely identified, the effective geometry is not known.

FIG. 2.9 – Transmittance measurement of a non-scattering filter with a d:0° geometry. Only the radiance incident at 0° is captured and the light coming from other directions is ignored by the detector. The effective measurement geometry is the 0°:0° geometry.

Similar situations in transmittance exist. For example, if an instrument based on the d:0° geometry is used to measure the transmittance of a non-scattering sample, such as a non-scattering filter, only the radiance normal to the sample is captured by the detector (figure 2.9). The effective measurement geometry is therefore the 0°:0°

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geometry. The incident radiance at 0° is obtained by measurement without the sample. Thus, the ratio of measurements with the sample and without it provides the directional transmittance factor at 0°.

2.7

Surface Distribution Functions

In the case of translucent materials, light penetrates deeply inside the material and can travel a significant lateral distance before exiting. The BRDF does not suffice to completely characterize the light being reflected from the surface. The functions presented in this section describe how the exiting light is distributed over the surface.

2.7.1

Bidirectional Scattering-Surface Reflectance Distribution Function (BSSRDF)

Bidirectional Scattering-Surface Reflectance Distribution Function (BSSRDF) Sr, expressed in m−2.sr−1, is a more complete function than BRDF because it incorporates the distance travelled laterally by light inside the material before emerging into the air. The BSSRDF concept has been widely used in computer graphics since the work by Jensen et al. [115]. It is defined for two elementary areas ds ðPÞ and dsðP0 Þ, respectively located around two points P and P0 of the surface, corresponding to the illuminated and observed areas (figure 2.10). It is also defined for a narrow spectral bandwidth Dk around a given wavelength λ, a given polarization state of the incident light (denoted by the  as for the Fresnel formulae in §1.3.2), an elementary solid angle d 2 Xi ðhi ; ui Þ around a given direction of incidence ðhi ; ui Þ and an elementary solid angle d 2 Xr ðhr ; ur Þ around a given direction of observation ðhr ; ur Þ. It relates elementary radiance dLðP0 ; hr ; ur ; k; Þ of ds ðP0 Þ along direction ðhr ; ur Þ, defined in the same way as equation (2.2), and incident elementary flux dFi ðP; hr ; ur ; k; Þ received by ds ðPÞ from direction ðhi ; ui Þ:

FIG. 2.10 – Angular and spatial parameters used in the definition of BSSRDF.

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Sr ðP; P0 ; hi ; ui ; hr ; ur ; k; Þ ¼

dLðP0 ; hr ; ur ; k; Þ dFi ðP; hi ; ui ; k; Þ

ð2:17Þ

The BRDF of a surface around point P is the integral of its BSSRDF over area B from which light outgoes: Z fr ðP; hi ; ui ; hr ; ur ; k; Þ ¼ Sr ðP; P0 ; hi ; ui ; hr ; ur ; k; Þds 0 ð2:18Þ P0 2B

2.7.2

Point Spread Function (PSF)

The Point Spread Function (PSF), expressed in m−2, describes the pattern of light exiting from the surface of a scattering material when illuminated by a thin collimated beam. As BSSRDF, it is defined for two elementary areas ds ðPÞ and ds ðP0 Þ representing with illuminated and observed areas and located around points P and P0 , for a narrow spectral bandwidth Dk around given wavelength λ, a given polarization state of the incident light, and a given direction of incidence ðhi ; ui Þ. It is the ratio between exitance M ðP0 ; k; Þ of the surface at point P0 and incident flux Fi ðP; hi ; ui ; k; Þ received by dsðPÞ from an elementary solid angle d 2 Xi ðhi ; ui Þ around direction ðhi ; ui Þ: PSFðP; P0 ; hi ; ui ; k; Þ ¼

M ðP0 ; k; Þ Fi ðP; hi ; ui ; k; Þ

ð2:19Þ

The PSF is the integral of the BSSRDF over the half-space: Z p=2 Z 2p PSFðP; P0 ; hi ; ui ; k; Þ ¼ Sr ðP; P0 ; hi ; ui ; hr ; ur ; k; Þ cos hr sin hr dhr dur hr ¼0

ur ¼0

ð2:20Þ Most frequently, what is measured is the light pattern observed from one direction by means of a camera. Hence, by language misnomers, many patterns called PSFs are in fact radiance maps or BSSRDFs in a bi-directional configuration. Figure 2.11 shows two examples of patterns frontally observed with a camera ðhr ¼ ur ¼ 0Þ on two samples illuminated by a laser beam from direction ðhi ¼ 15 ; ui ¼ 0Þ. In this case, the value of each pixel is proportional to its radiance, or to Sr ðP; P0 ; 15 ; 0; 0; 0; kÞ, where P is the central point of the image. The modulation transfer function (MTF), expressed in terms of spatial frequencies, is sometimes preferred to the PSF. It is the modulus of the Fourier transform of the PSF.

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FIG. 2.11 – Examples of PSFs from Ref. [214] of a glass plate on top of a white diffusing background, captured by a camera, (a) when the plate and the background are in optical contact, (b) when they are separated by a thin layer of air. At the center of each figure, the bright disk corresponds to the impact of the incident light pencil (laser beam).

Chapter 3 Visual Characterization of Surfaces We all know how important appearance is in the perceived quality or attractiveness of manufactured products. Long before the Industrial Age, artists and craftsmen devoted all their talent and know-how to the work of materials in order to obtain the best appearance allowed by the techniques available. Far from diminishing, this trend is on the contrary being reinforced to meet ever higher expectations for quality. This obviously prevails in the luxury industry, but is also true in almost all other manufacturing domains. Appearance and its various attributes, among which color, gloss, transparency, and texture to mention only the main ones [187], are not physical quantities. They are sensations produced by the human brain in response to a stimulation of the retina by a given light signal issued from the scene viewed. Their sensory nature therefore makes them not measurable a priori. The mechanism transforming the light signal, which is measurable, into a visual sensation is complex and far from completely understood. Several reasons explain this difficulty. The first reason is that the different attributes are interdependent. For example, two objects made of the same polymer can be perceived as having different colors simply because their surface state is smooth for one, therefore glossy, and rough for the other, therefore matte. Likewise, a textured surface tinted by very small colored dots can have a completely different color when viewed far away or at short distance – pointillist artists have extensively used this principle, on which also halftone printing is based. A second reason is the variability of perceptual mechanisms from one individual to another one. Beyond the physiological differences due to genetic or epigenetic specificities, which can affect vision as any other organ or function of the human body, the visual perception mechanisms are not innate, but result from a long learning process during the first years of life. A third reason is the difficulty of translating a sensation into accurate values understandable by everyone. Most of the time, visual attributes are described by elements of language, which are too imprecise: when two individuals express a different sensation, it is difficult to know whether the difference comes from the sensation itself, and therefore from the psycho-physical mechanisms which generated it, or from the language used to express it, or both. A fourth reason is the fact that appearance is often relative and dependent on the surroundings. Nevertheless, it seems that one can rather easily DOI: 10.1051/978-2-7598-2647-6.c003 © Science Press, EDP Sciences, 2022

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agree within a group of persons on the fact that a shirt is red or a table is matte, which tends to show that there is an objectifiable part in the appearance of objects. The importance of appearance in human activities is such that this objectifiable part is actively studied, despite the immense obstacles mentioned previously. The question is, how these visual attributes can be quantitatively evaluated? For a long time, visual inspection was the only way to characterize surfaces, and perceptual analysis methods have improved over time. Sensory metrology enables us to classify objects according to pre-defined criteria on a custom or standardized scale. For example, the Munsell scale for color (see §3.1.2), the NCS scale for gloss (see §3.3), or the Fischer–Saller scale for hair color shades [72] are widely used worldwide today. Several companies strongly concerned by the appearance of their products employ panels of trained observers performing these classifications, and the average value given by them on the chosen scale is used as a measure for the attribute under consideration. These methods are still the most reliable today for complex structures combining several visual attributes, such as hair or leathers. However, they are expensive and sometimes considered too subjective. New optical methods and devices, which have been developing rapidly since the early 2000s, have therefore aroused great enthusiasm – sometimes with a risk of a misunderstanding regarding their capacity to reliably account for appearance as one perceives it in complex situations. Appearance assessment based on optical measurement relies on the assumption that there exists a relationship between the physical and geometrical characteristics of the light signal issued from an object and the visual attributes perceived when watching it. Schematically, even if reality is often more complex and there are many counterexamples, one can say that the main visual attributes are linked to the following characteristics:
color is related to the spectral distribution of light,
texture is linked to its spatial distribution,
gloss and translucency are linked to its geometric variations. For each attribute considered separately, a perceptual model is established on the one hand based on several samples measured optically thanks to an instrument, and measured perceptually thanks to psychophysical experiments, on the other. The curves of correspondence between the two types of measurements issued from this process are averaged over several individual viewers, and the resulting average curve, called “standard observer”, is assumed to be representative of all humans with “normal” vision. With this curve, the optical measurement can be mathematically converted into a perceptual attribute value. This method was operated one century ago for color [38, 42], later for gloss [9, 10, 111], and many affordable instruments have been recently designed for specific applications. We understand their rapid success by the fact that they give quantified results, deemed objective. Since the models are standardized around the world, communication from one point to another of the globe is facilitated, while solving linguistic and cultural issues. Their limitation is mainly due to the fact that the psychophysical experiments underlying the perceptual models have been carried out in simple scenarios where only one attribute is exhibited, therefore much simpler than real objects in scenes of everyday

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life; for example, color patches are homogenous and placed in a homogeneous environment; gloss analysis is performed on achromatic surfaces, etc. Their use therefore requires some expertise in optics in order to interpret correctly the attribute values deduced from the measured light quantities. This is what we will see in this chapter, by focusing on two main visual attributes, gloss and gloss.

3.1

Color and Colorimetry

As the name suggests, colorimetry is the science of measuring colors. It is often associated with photometry, and the connection between these two disciplines is founded: without light, the notion of color vanishes. It is by measuring the properties of light, in particular its spectrum in the visible domain, that the physical evaluation of color is performed. However, as any visual attribute, color is not a physical quantity but a sensation produced by the human brain. The term “colorimetry” therefore seems to be based on an oxymoron: the sensation of color is not a universal quantity and not a very reproducible phenomenon which is measured like a photodetector measures a luminous flux; it is therefore not immediately evident that this is a measurable quantity. However, color perception is also not a purely subjective individual experience: most humans perceive the sky as blue, the setting sun as red, oak foliage in summer as green, etc. There is therefore a certain degree of reproducibility, sufficient in any case for scientists to have undertaken to make it a measurable quantity. The scientific approach to color really began in the 19th century and led throughout the 20th century, under the aegis of the International Commission on Illumination (CIE, for Commission Internationale de l’Eclairage), to the establishment of worldwide recognized standardized representation spaces and recommendations for physical measurements.

3.1.1

A Relative Perception

One of the first difficulties that a scientist faces when addressing color issues is the fact that there is no absolute perception of it. The color of a light source or of an object depends on its surroundings, and more generally on one’s overall understanding of the observed scene. An obvious example of this relativity is the simultaneous contrast, first studied by the chemist Michel-Eugène Chevreul at the beginning of the 19th century [37]: identical colored objects are not perceived in the same way depending on the surrounding color (see figure 3.1a). For a long time, classical painters were using this visual effect extensively to reinforce the yellow tints by almost systematically juxtaposing a blue tint to them, blue being its complementary in the color wheel. Figure 3.1b shows one example among many others. After Chevreul’s work, the principle of simultaneous contrast and juxtaposition of opposite colors has been widely used by painters like Van Gogh (one can refer to his famous Sunflowers on a blue background painted in 1889, or the blue irises on an ochre background in Still Life: Vase with Irises Against a Yellow Background painted in 1890), Seurat and Signac who founded the artistic movement called Pointillism

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based on the juxtaposition of small paint touches with opponent colors, or Delaunay who founded a movement called Orphism, even though he himself seemed to prefer the term “Simultanism” in reference to Chevreul’s work. The origin of this phenomenon relies on the physiological functioning of the visual system [112]. The first cellular layer in the retina is made of photosensitive cells with different sensitivities and spectral responses: cones and rods. Cones, operated by daytime vision, are of three kinds for most people, denoted by S, M and L for their maximum of sensitivity in the short, middle and large visible wavelengths, respectively. They are the source of vision in color. They are mainly located at the center of the retina, i.e., at the center of the visual field. In the second retinal layer, cells connected to the cones perform logical operations to compare the responses of neighboring cones; this comparison process continues in the third and last retinal layer formed by ganglion cells, whose axons constitute the optical nerve [112]. The information that is sent to the brain via the optical nerve includes only local variations in the light signal, for example: a certain “green” cone has received light but its “red” neighbor has not. This process of comparisons between neighboring areas then continues in the visual cortex by encompassing larger and larger areas of the visual field, until a colorized image of the whole visual field is achieved. The relationship between the light signal falling on the retina, formed by the spectrum of the different light rays, and the color image reconstructed by the brain is rarely linear and often difficult to establish, as shown by the phenomenon of simultaneous contrast and many other optical illusions.

FIG. 3.1 – (a) Simultaneous contrast: the nine gray discs are the same color, but seem to be of different colors (fix your gaze on the central point). (b) Detail of a painting by Lorenzo Lotto (Holy Family with the Family of St John the Baptist, 1536, Musée du Louvre), illustrating the use of painters to reinforce yellow hues by juxtaposing a solid blue color. In this process of “colorization” by the visual system, white holds a special place: the visual system defines at every time a color that will be perceived as white, thanks to a gain control mechanism applied to the three kinds of cones, S, M and L [67]. This white color, often but not always attributed to the most intense light signal in

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the visual field, conditions the perception of all the other colors. In a sense, it is comparable to the white balance operation that some photographers perform using digital processing software to adjust colors of the image or correct the effect of overly chromatic lighting. In vision, this process is called chromatic adaptation (see §3.1.8). In addition, a global analysis of the observed scene allows the observer, to some extent, to perceive the intrinsic color of objects he/she is looking at regardless of the spectrum of the lighting: a yellow balloon looks the same yellow under a cloudy sky, under a setting sun, or under some LED lamp. This principle is called color consistency [55]. Although one cannot perceive colors on an absolute scale, unlike musicians with “perfect pitch” who can perceive the fundamental frequency of sounds by giving the exact pitch, one has however a good capacity to compare colors, especially those of juxtaposed flat areas. Color matching, the operation of adjusting the color of an object or that of a light panel until it becomes indistinguishable from that of the neighboring object or panel, is the ground concept for most colorimetric models.

3.1.2

Color Mixing and Early Color Representation Systems

Even though the number of perceptible colors is infinite, it has been progressively established in the 18th Century that mixing a limited number of colors generates many other ones. For a long time, it was known that the mixture of various colored materials like pigments or dyes generates a new color (subtractive color mixing, see figure 3.2a). Isaac Newton, then James Clerk Maxwell, studied colors perceived from superimpositions of colored rays (additive color mixing, figure 3.2b).

FIG. 3.2 – Mixing colors produces other colors: (a) Additive color mixing, resulting from the superimposition of 3 colored light beams, red, green and blue; (b) Subtractive color mixing resulting from the dying of a white support with 3 inks, cyan, magenta and yellow.

The observation that many colors can be obtained by mixing three colors – usually blue, red and yellow, called primary colors – subtended a long time ago the

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idea of ordering colors according to a tridimensional representation system. Smithson et al. found in a short text in Latin written in 1225 by the English theologian Robert Grosseteste, entitled De Colore, what they consider to be the first intent of a color classification system composed of three axes [219], based on the oppositions of sensorial attributes: obscura–clara (dark–bright), pauca–multa (little–much), impurum–purum (impure–pure). This idea has been formalized more concretely during the 18th Century. The German cartograph and astronomer Tobias Mayer, at the origin of the triangle reproduced in figure 3.3a [157], also proposed a tridimensional representation system for colors under the form of a hexahedron, that Philippe Otto Runge then transformed in 1805 into a sphere (figure 3.3b) [194].

FIG. 3.3 – Color plates showing early color representation systems, designed by (a) Lichtenberg in 1775, replication of Mayer’s triangle [157], (b) Runge in 1805 [194], (c) Goethe in 1809 [82], and (d) Chevreul in 1861 (© Linda Hall Library, 2016).

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FIG. 3.4 – (a) Plates of the Grammar of Color, published in 1921 by the Strathmore Paper Company, explaining the Munsell Color System [226]. The plate on the top shows the three dimensions of the Munsell Color System, the plate on the bottom shows the ten hues regularly arranged on the color wheel and the degrees of chroma, (b) Picture of Munsell Color Tree (© Hannes Grobe, 2017 [85]), (c) Various editions of the Munsell Atlas of Color and the Munsell Books of Color, with at the foreground some removable plates (© Mark Fairchild, 2005 [66]).

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Runge’s sphere already incorporates, on its equator, the emerging idea of color wheel, which represents all pure colors by ordering them in the same way as they appear is the rainbow, and by connecting red and purple. In his famous Theory of Color published in 1810 [82, 209], Johann Wolfgang von Goethe observed the opponent process of colors, i.e., the fact that some colors are in opposition with one another: for example, red cannot be greenish, and blue cannot be yellowish. This observation is at the origin of the opponent color theory formulated by Ewald Hering in 1892 [107], and color wheels were then more systematically designed in order to place complementary colors at diametrically opposite locations, even though many color wheels of the late 18th Century have already been designed in this way. Michel–Eugene Chevreul produced, as of 1839, a set of remarkable color plates, one of which is reproduced in figure 3.3d. Notice that with the development of printing techniques at the beginning of the 20th century, the blue, red and yellow primary colors have been supplanted by cyan, magenta and yellow, which give a wider set of reproducible colors, i.e., a wider color gamut. In 1898, Munsell proposed a first version of his famous color representation system, predecessor of The Munsell Atlas of Color published in 1915 and The Munsell Books of Color which are still widely used today in industry (see figure 3.4c), and which contain the concepts presented above: a tridimensional space, within which colors are ordered according to their lightness (or value in the Munsell system) along a vertical axis, to their hue disposed along a circle comparable to the color wheel, and their chroma which ranges from achromatic grey near the value axis to the most intense shade at the most external position [136] (see figure 3.4a and b).

3.1.3

Trichromacy

In parallel to the development of tridimensional color representation systems, the tridimensionality of color perception has been given a more scientific framework. The English savant Thomas Young sensed the presence of three different photoreceptors in the eye [257], what Helmholtz confirmed later [105, 166]. Humans with normal vision are thus qualified as trichromats. However, there exist different forms of color blindness, or dyschromatopsia, that Young also studied. Individuals with deuteranopia, for example, are dichromats, and it is also probable that some individuals are tetrachromats [114]. Young also stated that the mixture of three primary colors is enough to produce an infinite number of other colors, called color gamut, but not all perceptible colors: with a given set of three primaries, the color gamut is always restricted, as we will see later. Even more remarkable were the works of the German scientist Hermann Günther Grassmann (1809–1877), who established in his Laws of additive color mixtures, published in 1853 [84], important rules at the origin of colorimetry. The principle of trichromacy is reaffirmed and becomes more concrete: three criteria are necessary and sufficient to classify colors, and they can be represented in a space of dimension 3. Three primary colors P1, P2 and P3 are therefore enough to produce an infinite number of other colors. Conversely, each color C, provided it is in the gamut of the

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set of selected primaries, can be obtained by mixing the primaries with a unique set of proportions, a1 , a2 , a3 . Using symbol  (mixing operator) to mean a mixing of colors and symbol  (matching operator) to mean that two colors are perceived as similar, one can write: P  a1 P1  a2 P2  a3 P3 ;

ð3:1Þ

and one can associate with color C the three coordinates ða1 ; a2 ; a3 Þ, thus describe the color under the form of a vector: 0 1 a1 C ¼ @ a 2 A: ð3:2Þ a3 Grassman’s additivity law allows saying that if another color C 0 was obtained by mixing the same three primaries P1, P2 and P3 in proportions a10 , a20 , a30 – it is thus   represented by the vector C0 ¼ a10 ; a20 ; a30 – then color C 00 obtained by mixing colors C and C 0 in the respective proportions u and v can also be obtained by mixing directly the primary colors with the respective proportions ua1 þ va10 , ua2 þ va20 and ua3 þ va30 , which is equivalent to operating an addition between vectors C and C01 : 0 1 0 01 a1 a1 C00 ¼ uC þ vC0 ¼ u @ a2 A þ v @ a20 A: ð3:3Þ a30 a3 This is how colors could be described in an elegant mathematical framework, taking the form of a vector space whose three primaries are the orthogonal basis vectors. It may be not a coincidence if Grassmann, the person at the origin of the trichromacy principle, also left produced some major contributions in mathematics (and also in other disciplines such as linguistics), in particular pioneering works on the notion of vector space. Of course, the coordinates of colors only make sense whether the three primary colors are specified.

3.1.4

CIE 1931 RGB and XYZ Color Spaces

The best-established color space in the world is based on psychophysical experiments of color matching based on three precise monochromatic primaries: 700 nm (red), 546.1 nm (green), and 435.8 nm (blue). The visual field of view occupies a solid angle of apex angle 2° along the optical axis of the eye, thus falling on the fovea, a region of the retina rich in cones. The experiment, described in detail in the world reference book on colorimetry by Wyszecki and Stiles [249], consisted in reproducing – by mixing these three primaries – all the other monochromatic colors Ck of the visible spectrum, wavelength by wavelength, and thus in determining for each one its   triplet of values, called tristimulus values, and denoted r; g; b in this case. The spectral curves r ðkÞ, g ðkÞ, and bðkÞ thus obtained, called color matching functions (CMFs), are plotted in figure 3.5. It should be noted, however, that all of these monochromatic colors are in fact outside the gamut of the three primaries. Matching

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is only possible by mixing two primaries on the one hand, and the monochromatic color Ck and the third primary on the other. For example, at λ = 500 nm, the equalization is done as follows: Ck  r ðkÞR  g ðkÞG  bðkÞB ) Ck  g ðkÞG  bðkÞB  r ðkÞR which leads to a negative value for r ðkÞ. Thanks to the law of additivity, the color of the visual stimulus from of spectrum S ðkÞ is obtained by summing vectors Ck associated monochromatic colors proportionally to spectral function S ðkÞ. This well-known defining equations associated with the CIE 1931 RGB color Z 700 nm R¼k S ðkÞr ðkÞdk Z G¼k B¼k

380 nm 700 nm

380 nm Z 700 nm

S ðkÞg ðkÞdk

ð3:4Þ

some light with the yields the space:

ð3:5Þ

S ðkÞbðkÞdk

380 nm

where R, G, B are the tristimulus values attached to the light stimulus of spectrum S ðkÞ, and k is a normalization constant. In order to avoid dealing with negative tristimulus values, the CIE proposed to operate a linear transformation on tristimulus values ðR; G; B Þ associated with each   color (therefore also with the values r; g; b associated with the monochromatic colors), to obtain values ðX; Y ; Z Þ of the CIE 1931 XYZ color space: 0 1 0 10 1 X 0:4887180 0:3106803 0:2006017 R @ Y A ¼ @ 0:1762044 0:8129847 0:0108109 A@ G A ð3:6Þ Z 0 0:0102048 0:9897952 B

FIG. 3.5 – Color matching functions associated with (a) the CIE 1931 RGB color space and (b) the CIE 1931 XYZ color space [42].

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This operation gives the color matching functions plotted in the right of figure 3.5. The tristimulus values of a light stimulus with a given spectrum S(λ) can be directly calculated by the following equations, much more commonly used than equation (3.5): Z 750 nm X ¼k S ðkÞx ðkÞdk 380 nm 750 nm

Z Y ¼k Z ¼k

380 nm Z 750 nm

S ðkÞy ðkÞdk

ð3:7Þ

S ðkÞz ðkÞdk

380 nm

where the normalization factor k is Km ¼ 683 lm/W. The X, Y, and Z tristimulus values are all positive. The color matching function y ðkÞ coincides with the sensitivity function of the human visual system used in photometry, V(λ), plotted in figure 1.15. The CIE recommends for S(λ) a spectral resolution of 5 nm or less [42]. The tristimulus values can also be computed for a non-luminous object, from the spectrum of light reflected or transmitted by the object with spectral reflectance R(λ) or T(λ), under a certain lighting with spectral distribution power S(λ). In this case, they are given by Z 750 nm X ¼k RðkÞS ðkÞx ðkÞdk 380 nm 750 nm

Z Y ¼k Z ¼k

RðkÞS ðkÞy ðkÞdk

380 nm Z 750 nm

ð3:8Þ

RðkÞS ðkÞz ðkÞdk

380 nm

with k ¼ R 750 nm 380 nm

100 S ðkÞy ðkÞdk

ð3:9Þ

For a perfect white diffuser (R = 1 for all visible wavelengths) or a perfect transparent object (T = 1), the Y value is 100.

3.1.5

CIE 1931 Chromaticity Diagram

In order to simplify the representation of colors in documents, which by nature are two-dimensional, another transformation is performed, corresponding to a projection, yielding the CIE 1931 xy chromaticity diagram shown in figure 3.6. The chromaticity values (x, y) are defined from the X, Y, Z tristimulus values as:

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X X þY þZ Y y¼ X þY þZ Z ¼1x y z¼ X þY þZ

x¼

ð3:10Þ

Since the chromaticity diagram is calculated from spectra using linear operators, it is not surprising that the chromaticity values associated with colors obtained by mixing two primary colors form a segment the ends of which are the chromaticity values of the two primaries. Likewise, for mixtures of three primaries, the chromaticity values are contained in a triangle whose vertices are the chromaticity values of the three primaries. This triangle corresponds to the color gamut associated with these three primary colors.

FIG. 3.6 – CIE 1931 xy chromaticity diagram. The monochromatic colors draw the boundary of the diagram, called “spectrum locus”. The wavelengths corresponding to some of them are written in red. By mixing three colors with respective chromaticity points A, B, and C in the diagram, one obtains colors whose chromaticities are in the triangle ABC (color gamut). The chromaticity diagram does not contain any lightness information (black, white and all grays are projected onto the same point), but is very useful for the color characterization of light sources to which the notion of lightness is not

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applicable: light cannot be gray, for example. However, the CIE 1931 XYZ space and xy chromaticity diagram fails to evaluate differences between colors. As shown by MacAdam in 1942, a segment of the same length in the green or blue areas of the chromaticity diagram do not correspond at all to the same perceived color differences, which are much larger in blues than in greens. Moreover, the phenomenon of chromatic adaptation, requiring the definition of a reference white, is not considered in these spaces. Color appearance models have been developed for these purposes.

3.1.6

Color of Light Sources and Illuminants

Standard light sources have been defined by the CIE in order to facilitate the communication worldwide on color appearance of materials under lightings of different spectral power distributions (SPDs). Standard lightings with specific SPDs are called illuminants [42]. The equal energy illuminant E is a theoretical illuminant with uniform relative SPD, not feasible in practice with artificial light sources. Other illuminants are inspired of the SPDs of incandescent light (illuminant A), daylight (D illuminant series), fluorescent lighting (illuminants F), and LEDs. Some of them are plotted in figure 3.7. The relative SPD of illuminant A is issued from the spectral radiance of a black body at the temperature TA ¼ 2856 K given by Planck’s law, normalized to the value 100 at the wavelength k0 ¼ 560 nm:    5 exp hc  1 kk0 TA k0   Sk ¼ 100 ð3:11Þ k exp hc  1 kkTA

where c ’ 2:998  108 m.s−1 is the speed of light in vacuum, h ’ 6.626  1034 J.s is the Planck constant and k ’ 1.380  1023 J.K−1 is the Boltzmann constant.

FIG. 3.7 – Relative spectral power distributions of CIE standard illuminants A, D65, D50 and F11.

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FIG. 3.8 – Spectral power distribution of the D65 illuminant and two artificial light sources having same chromaticity values ðx; y Þ ¼ ð0:312; 0:329Þ as D65.

The D series of illuminants was constructed by Judd, MacAdam, and Wyszecki to represent natural daylight [121]. The D50 and D65 illuminants are especially used in the graphical industry and paper industry. Their spectra, plotted in figure 3.7, are known to correspond to horizon daylight and noon daylight spectra. They are easy to characterize mathematically since they may be derived from the linear combination of three spectra. However, they are difficult to produce artificially. Figure 3.8 shows two examples of light sources considered as D65 illuminants with noticeably different spectra, but same CIE 1931 xy chromaticity values: the light source in the SpectroEye spectrophotometer from X-rite and the ‘Color Control Classic Line’ light table from Just Normlicht. The ability of real light sources to reproduce the D65 illuminant can be assessed with the CIE metamerism index [43]. The F series of illuminants represent various types of fluorescent lighting. The F11 illuminant, plotted in figure 3.7, is a narrow triband illuminant consisting of three narrowband emissions in the red, green and blue regions of the visible spectrum, obtained by a composition of rare-earth phosphors.

3.1.7

CIE 1976 L*a*b* Color Space and Color Appearance Models

The CIE 1976 L*a*b* model, sometimes called “CIELAB”, is a three-dimensional color representation system where the L* coordinate corresponds to the lightness, between 0 (black) and 100 (white), and the other two coordinates, a* and b*, are used to describe the hue and chroma. They are calculated from the CIE 1931 tristimulus values X; Y ; Z of the considered light and Xw ; Yw ; Zw of the light signal defined as reference white for the chromatic adaptation:

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63

L ¼ 116f ðY =Yw Þ  16 a  ¼ 500½f ðX=Xw Þ  f ðY =Yw Þ

ð3:12Þ



b ¼ 200½f ðY =Yw Þ  f ðZ =Zw Þ with

 f ðx Þ ¼

x 1=3 7:787x þ 16=116

if x [ 0:008856 if x 0:008856

ð3:13Þ

Two views of the 3D space are displayed in figure 3.9. The following quantities   and hue hab : serve as correlates of chroma Cab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Cab ¼ a 2 þ b2 ; ð3:14Þ  hab ¼ arctanðb =a  Þ

FIG. 3.9 – Representations of the CIE 1976 L*a*b* color space as viewed (a) from one side, (b) from the top [64].

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The further away from the L* axis, the higher the chroma (brighter colors); on the contrary, the L* axis contains the achromatic grays. This space is one of the first color appearance models, and by far the most widely used. It is considered perceptually uniform because the Munsell colors where used to create it (under a D65 illuminant) [131]. Beyond considering the chromatic adaptation (although imperfectly),  the CIE 1976 L*a*b* model enables estimating the difference between two colors L1 ; a1 ; b1   and L2 ; a2 ; b2 with metrics, denoted ΔE. The original 1976 metric, usually denoted  by DEab , is simply the Euclidean distance between the points representing colors to be compared: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð3:15Þ DEab ¼ ðDL Þ2 þ ðDa  Þ2 þ ðDb Þ2 with DL ¼ L1  L2 , Da  ¼ a1  a2 , Db ¼ b1  b2 . The just noticeable difference (JND) between colors, i.e., the value below which two different colors look identical for most people, has been established around 2.2 units for a non-trained observer. A more reliable metric was then proposed in 1994: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2  2  2   DCab DHab DL  ; ð3:16Þ þ þ DE94 ¼ kL SL kC S C kH SH with  DCab ¼ C1  C2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ci0 ¼ ai02 þ b02 i ¼ 1; 2 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2 ¼ ðDa  Þ2 þ ðDb Þ2  DCab DHab

ð3:17Þ

and where SL ¼ 1, SC ¼ 1 þ K2 C1 , SC ¼ 1 þ K1 C1 , kL , K1 and K2 are constants depending on the application domain. For example, in the graphical arts industry, one considers: kL ¼ 1, K1 ¼ 0:045 and K2 ¼ 0:015, whereas in the textile industry, one rather considers: kL ¼ 2, K1 ¼ 0:048, K2 ¼ 0:014. The just noticeable difference with this metric has been established around 1 unit. In 2000, a new metric which tends to prevail today, the DE2000, or DE00 , has been proposed to correct perceptual non-uniformity of the previous metrics. It relies on more complex formulae [207]: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi      DL0 2 DC 0 2 DH 0 2 DC 0 DH 0 ð3:18Þ þ þ þ RT DE00 ¼ kL SL kC SC kH SH kC SC kH SH

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65

with L0 ¼ L DL0 ¼ L01  L02   L ¼ L01 þ L02 =2   C ¼ C1 þ C2 =2 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 u u C7  a B C 0  a ¼ a þ @1  t 7 A 2 7 C þ 25 b0 ¼ b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 0 ¼ a 02 þ b02 DC 0 ¼ C10  C20   C 0 ¼ C10 þ C20 =2 h 0 ¼ arctanðb0 =a 0 Þ modð360 Þ Dh 0 ¼ h10  h20   h 0 ¼ h10 þ h20 =2 qffiffiffiffiffiffiffiffiffiffiffi Dh 0  DH 0 ¼ 2 C10 C20 sin 2  2 0:015 L  50 SL ¼ 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 20 þ L  50 SC ¼ 1 þ 0:045C 0 SH ¼ 1 þ 0:015C 0 T     T ¼ 1  0:17 cos h 0  30 þ 0:24 cos 2h 0     þ 0:32 cos 3h 0 þ 6  0:20 cos 4h 0  63 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  0 7 " "  u  ## C 0  275 2 u h  sin 60 exp  RT ¼ 2u t 0 7 25 C þ 257 kL ¼ kC ¼ kH ¼ 1 A space like CIE 1976 L*a*b* based on a colored appearance model is suitable to characterize the color of objects. However, one must be aware of its limits, if one wants the model to give a realistic representation of colors perceived in a scene: it is preferable that the object be opaque and matte, since transparency and shine may influence significantly the perceived color. It is also preferable to control the environment of the object: neutral and uniform background to avoid simultaneous contrast phenomena, absence parasitic light in the visual field of the observer (a screen or a warning light, even far away, may change the white reference with respect to which other colors are perceived). Light booths with controlled lighting have been specially designed to avoid these perceptual artefacts.

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More advanced color appearance models CIECAM02 [44], iCAM [68], iCAM06 [132], or CAM16 [143] have been recently proposed in order to better evaluate colors of objects in complex surroundings or images, by considering many parameters such as shape, size, contrast, spatial frequencies, radiance level, etc.

3.1.8

Von Kries Chromatic Adaptation

As said earlier, chromatic adaptation is the capacity of the visual system to preserve the color of objects when the lighting color changes. The CIE 1976 L*a*b* color space takes chromatic adaptation into account by dividing the tristimulus values ðX; Y ; Z Þ of the light signal issued from an object under a certain illuminant with the ones ðXw ; Yw ; Zw Þ of the signal issued from a white reflector under the same illuminant (the reference white on which the chromatic adaptation is based). However, this basic method is not accurate and can yield false colors when the lighting is noticeably chromatic. A more accurate model relies on the von Kries coefficient law (1904) which asserts that the visual system individually tunes the gains of the three cone cell responses, S, M and L, according to their responses from the white point. After adaptation, the cones’ responses associated with the white point are identical under any illuminant [67]. Let L, M, and S be initial cone responses to a light signal issued from an object under a certain illuminant (e.g., D65), and Lw , Mw and Sw one responses to a white reflector under the same illuminant. According to the modern interpretation of the von Kries model, these responses are scaled by the respective gain controls kL ¼ 1=Lw , kM ¼ 1=Mw , and kS ¼ 1=Sw , and the post-adaptation cone responses to the signals issued from the object are ðLa ; Ma ; Sa Þ ¼ ðkL L; kM M ; kS S Þ, or (1, 1, 1) for the white reflector. Under a second illuminant, the initial cone responses are L0 , M 0 , and S 0 for the object, and L0w , Mw0 , and Sw0 for the white reflector. The gain controls 0 become kL0 ¼ 1=L0w , kM ¼ 1=Mw0 , and kS0 ¼ 1=Sw0 , and the post-adaptation cone responses remains (1, 1, 1) for the white reflector, whereas they can be expressed as functions of the ones under the first illuminant thanks to a diagonal matrix: 0 0 1 0 1 0 0 10 1 Lw =Lw 0 0 L L L @ M 0 A ¼ D@ M A ¼ @ 0 Mw0 =Mw 0 A@ M A ð3:19Þ 0 0 Sw0 =Sw S S S0 The white reference reflector is therefore perceived as white under every illuminant, and the other object colors are accordingly perceived. The L, M, and S cone responses can be deduced from the CIE 1931 XYZ tristimulus values thanks to a chromatic adaptation transform (CAT), which is generally a linear transform. The so-called von Kries transform is given by matrix MCAT defined as: 0 1 0 1 0 10 1 L X 0:4002 0:7076 0:0808 X @ M A ¼ MCAT @ Y A ¼ @ 0:2263 1:1653 0:0457 A@ Y A ð3:20Þ S Z 0 0 0:9182 Z

Visual Characterization of Surfaces

CAT02 is another transform defined by the CIE [131]: 0 1 0:7328 0:4296 0:1624 MCAT02 ¼ @ 0:7036 1:6975 0:0061 A 0:0030 0:0136 0:9834

67

ð3:21Þ

By combining equations (3.19) and (3.20) [or (3.21) if one prefers using CAT02], one obtains the tristimulus values X 0 , Y 0 , Z 0 of the object color perceived under the second illuminant after chromatic adaptation, as functions of the ones X, Y, Z of the object color perceived under the first illuminant: 0 01 0 1 X X @ Y 0 A ¼ M1 D MCAT @ Y A ð3:22Þ CAT Z Z0   The new tristimulus values ðX 0 ; Y 0 ; Z 0 Þ, as well as Xw0 ; Yw0 ; Zw0 corresponding to the white reflector, can then be used to compute the CIE 1976 L*a*b* coordinates of the object.

3.2

Color Measurement and Color Imaging

For a long time in art workshops and industry, the control of colors and their faithful reproduction on a production line was the field of craftsmen with their vision as the only measuring tool. During the past two decades, optics-based measurement tools, in particular spectrophotometers and spectrocolorimeters, have become more widespread, assisted by software for a more reliable and reproducible evaluation of colors. Companies like X-rite, Byk-Gardner, Konika-Minolta, and many other ones, offer models at affordable price, with varying precision depending on the model but often acceptable, and ergonomic versions adapted to certain specific materials. However, the use of these instruments requires some knowledge of surface photometry to correctly analyze the spectra or colorimetric values they deliver. In particular, the measurement geometry should be well chosen (see §2.6.1) depending on the type of surface, especially if it is shiny or has colored reflections in certain directions [156]. These spectrophotometers remain the tools of choice for obtaining reliable colorimetric values, but they can only deliver an average value over the entire observed area, which is poorly suited to non-uniformly colored surfaces. Imaging techniques, capable of acquiring a color value in each pixel of the image, therefore on many small contiguous areas, are therefore arousing great interest in the analysis of some materials. It is nevertheless necessary to calibrate them carefully to obtain colorimetric values conforming to CIE standards.

3.2.1

Color Characterization of a Surface Using a Spectrophotometer

The best option for the color characterization of a surface is to measure its spectral reflectance. The measuring geometry must be carefully selected, except if the surface is perfectly matte since the geometry has no importance in this case (see §2.5.2). It is also important to consider the context in which the object to characterize is placed,

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especially the SPDs of the lighting, S ðkÞ, and of the white point (the light signal that is perceived as white and fixes the chromatic adaptation process), Sw ðkÞ. In general, the white point is the light reflected from a white surface under the same lighting as the object to characterize. For example, in printing, the white reflector is the unprinted paper. In scenes where no white surface exits, it is recommended to insert a white diffuser. Reference spectrum Sw ðkÞ is the product of S ðkÞ and spectral reflectance qw ðkÞ of the white diffuser. First, one measures the spectral reflectances of the surface to be tested, qðkÞ, and that of the white diffuser, qw ðkÞ. Then, one computes the tristimulus values ðX; Y ; Z Þ of the surface and those ðXw ; Yw ; Zw Þ of the white point in the CIE 1931 XYZ space [equations (3.8) and (3.9), with qðkÞ, respectively qw ðkÞ, in place of RðkÞ]. The CIE L*a*b* color coordinates are finally computed according to equation (3.12).

FIG. 3.10 – Spectral reflectances of some common colors issued from the Munsell Book of

Color Matte Collection, and their representation in the CIE 1931 xy chromaticity diagram and CIE 1976 L*a*b* color space.

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69

Figure 3.10 shows characteristic spectral reflectances of surfaces with common colors (red, green, blue, cyan, magenta, yellow, gray, black and white), issued from the Munsell Book of Color Matte Collection considered as a standard for color classification. The corresponding CIE 1931 xy chromaticity values and CIE 1976 L*a*b* color coordinates are also represented. In some domains, especially in the branches of physics where new materials are developed, colors are often represented in the CIE 1931 xy chromaticity diagram. Even though this space is not the best option for representing surface colors (a space like CIE 1976 L*a*b* based on a color appearance model is more suitable), it can be justified in the case of diffractive components or nanostructures, which often display highly saturated colors, rarely black or white (see for example [137]).

3.2.2

Metamerism

Two light signals of different spectral power distributions may be perceived as being of the same color. They are said to be metameric. For example, yellow color can be produced by the addition of red and green monochromatic lights, as shown in 1881 by Lord Rayleigh in his color matching experiment [238]. On a mathematical point of view, equation (3.7) can be viewed as an operation of projection: the spectral functions are projected into a 3-dimensional space, and several spectra may be projected onto the same vector (X, Y, Z). This can of course occur with materials also: under a lighting with a certain spectral power distribution (SPD), they have the same color, but under a lighting with different SPD, their colors look different. The color deviation between the two surfaces can be assessed by computing the CIE L*a*b* color coordinates of the two materials for the considered lighting’s SPD, according to the method presented in   the previous section, then using a color distance metric such as the DEab , DE94 or  DE00 . If the distance is low, or below the value of the just noticeable difference associated with the metric, the two surfaces are metameric. In figure 3.11, the spectral reflectance of two surfaces are plotted, one corresponding to a sample of the Munsell Book of Colors, the other being a reproduction of the first color by color printing on paper. Under the standard illuminant D65, the two surfaces are metameric: the deviation between their respective L*a*b* color coordinates computed  with the DE94 metric is 0.33 unit, therefore below the just noticeable difference of around 1 unit. Under standard illuminant A, the surfaces have slightly different colors: the deviation is 2.3 units. Metamerism is often an issue in manufacturing, when an object made of various materials is expected to have a uniform color. In the automotive industry, the interior design may include objects in leather, plastic, metal or fabric. In the textile industry, the color of the threads for the seam stiches should match the one of the fabrics. In the packaging industry, products like drinks and sodas can be presented into the form of metallic cans, plastic bottles with paper labels and plastic cap, glass bottle with metallic cap, and gathered into a printed board box. All these materials are expected to match precisely the color defined by the graphic charter of the trademark. When color matching between the materials has been achieved under a

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certain lighting, it is possible that shade differences appear under another lighting. The only way to prevent this issue is to ensure that all the materials have the same spectral reflectance (ideally the same spectral BRDF, if the object is not matte), but this is generally a challenge since each material has its proper coloration technique and coloring substances.

FIG. 3.11 – Spectral reflectances of two surfaces being metameric under the D65 illuminant, but not under the A illuminant.

3.2.3

Color and Spectral Measurement by Imaging Techniques

Although spectrophotometers are well-suited for measurement on uniformly colored areas, the fact that they collect the light reflected from a certain area, typically a disk of diameter ranging from 5 to 20 mm, is a drawback when the surface is heterogenous and a color characterization of small details expected. This, for example, is the case of skin, where it is interesting for the dermatologist to distinguish nevus or melanomas [200]. Color imaging is a good alternative to spectrophotometers to get color information at each point of the considered object, or each pixel of the image. Color imaging systems present many advantages: they are widely available, low-cost, and contactless, the latter property being much appreciated in cultural heritage [168] and medical applications [33]. The spectral responses in the three channels of an RGB camera, including the spectral response of the photoreceptor itself and spectral transmittances of the color filters, are usually different from the ones of the human visual system. Camera manufacturers target spectral responses as close as possible to a linear combination of the color matching functions CIE 1931 x ðkÞ, y ðkÞ, z ðkÞ. When this target is reached, the camera is said to be colorimetric

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71

(Luther condition, also referred to as Maxwell-Ives criterion, defining the gold standard for color sensors [71]). However, this condition is hardly met in practice due to technical limitations in the spectral transmittances achievable in color filter arrays [46]. The spectral power distribution of the lighting also strongly influences colors in the image. Under a given lighting, a color calibration method is necessary to obtain colors related to the CIE standards. In comparison to a spectral reflectance, RGB or XYZ color coordinates provide poorer information on the analyzed surface. In many applications, color is not sufficient to characterize finely the components of the material, whereas spectral reflectances are able to retrieve chromophores, colorants or pigments by their characteristic spectral patterns. The combination of spectral measurement and imaging has led to multispectral imaging systems in which the three classical red, green, blue channels of the color image are replaced with more channels with custom spectral bandwidths [87, 211] as shown in figure 3.12b. It has also led to hyperspectral imaging systems containing a high number of channels with narrow and contiguous spectral bandwidths (figure 3.12c) [83]. The image with its different channels is often called a “hypercube” in reference to its three dimensions: two spatial dimensions, and one spectral dimension. The acquisition of multispectral images can be done in one shot (snapshot camera) if micro-filters corresponding to the different bandwidths are placed directly on the pixels of the imaging sensor [139]; since the filters share the number of pixels on the sensor, the spatial resolution of the image after ‘demosaicing’ decreases as the number of channels increases [242]. Alternatively, a rotating wheel with interferential filters can be placed in front of the lens of a grey-level camera, which offers the advantage of simplicity in fixed scenes while maintaining the full spatial resolution of the camera. However, a geometrical calibration is often needed [24, 260]. The spectral bandwidths can also be defined by the lighting, as it is now allowed by a large panel of LEDs of different spectral power distributions [144].

FIG. 3.12 – Spectral characteristics of (a) a color camera, (b) a multispectral camera, and (c) a set of interferential filters for hyperspectral imaging.

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For hyperspectral imaging, the number of spectral bands is too high to use microfilters on the sensor. One dimension is therefore acquired over time. Most systems are based on the “wavelength scanning approach”: the 2D sensor captures the whole spatial image at successive times, one time per spectral bandwidth; fast and automated technologies have arisen with tunable spectral filters driven electronically [31, 50, 78, 122]). Other systems are based on a “line-scanning approach”: the scene is captured line by line thanks to a scanning system; at a given time, the light contained in the line meets a diffraction grating or a dispersive prism, which projects onto the 2D sensor a spectrum per point in the line. This technique can offer excellent spatial and spectral resolution [188], but the scanned object must be fixed and stable along the whole scanning process. Hyperspectral imaging has provided fantastic results in cultural heritage [148]. Elias and Cotte, using this technology, analyzed the famous painting Mona Lisa by Leonardo da Vinci, and determined the pigment mixture in every point of the painting [56]. From the pigment mixtures, thanks to light scattering model, they could simulate the painting before aging of the binder and reveal amazing colors (without certainty that these colors where the original colors, since the effective tint of the binder is not precisely known). Hyperspectral imaging has also shown its interest in skin analysis, as illustrated figure 3.13 with a human face. The spectral image of the face was taken on the human face with the SpectraFace hyperspectral camera from the Newtone Technologies company (France), based on a tunable filter [79]. An image of a perfect white diffuser was also taken with the same setup. Then, the ratio of spectral radiance measured on the face to the one measured on the white diffuser was computed in order to obtain a spectral radiance factor (see §2.5.2) in each pixel. The lighting is polarized in cross polarization configuration in order to remove the light specularly reflected at the skin surface (gloss). At 420 nm, skin is strongly scattering and absorbing, therefore rather opaque; one can see many thin details such as melanin stains. At 590 nm, in the spectral domain where hemoglobin has a peak of absorptance, well contrasted blood vessels appear. At 700 nm and in the infrared, skin is very translucent and light travels deeper inside: the PSF is large. Consequently, all details are blurred and the aspect is very smooth. Photographic retouching methods have been proposed based on this principle to erase skin imperfections in advertising portraits [227]. From the spectral image, a color image can be computed for any spectral power distribution of light source; the example displayed in figure 3.13c relies on the D65 standard illuminant. An analysis of the spectral image based on a light scattering model provides maps of skin composition parameters: the melanin concentration map highlights some spots, the blood volume fraction highlights the small vessels on the eyelids, whereas the oxygen saturation level of hemoglobin reveals deep veins almost invisible in the color image. Whatever the imaging technique, color measurement of non-flat objects remains a challenge because the spectral irradiance inevitably varies over the object’s surface [78]. Unless the spectral irradiance is evaluated at each point of the object and considered in the model, the darker reflectance values in shadowed areas are interpreted as an increase in absorptance by the material.

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FIG. 3.13 – Example of usage of a hyperspectral image for face skin analysis [79]. (a) Three channels of the hypercube corresponding to three narrow wavebands around 420 nm, 590 nm and 700 nm. (b) Maps of skin composition parameters deduced from the spectral reflectances acquires in each pixel: concentration in melanin, volume fraction of blood, oxygen saturation level. (c) Color image reconstruction under a D65 illuminant. © Courtesy of Lou Gevaux, Marie Cherel and Newtone Technologies.

3.2.4

Measuring the Spectral Response of an RGB Camera

In some applications, it is necessary to measure the spectral response of the red, green and blue channels of a color camera. Camera manufacturers sometimes provide the spectral response or the spectral quantum yield in the documentation of the sensor, but most of the time, the spectral transmittance of the optics and other influencing parameters are not considered. Measuring the spectral response of the camera can be done in a laboratory, using a monochromator, a power meter (or spectroradiometer), and a powerful light source. The monochromator, illuminated on the one side by the light source, has two slits whose widths determine the spectral bandwidth of the exiting monochromatic light. The wavelength can be selected thanks to a scroll wheel or by software. For each selected wavelength, a picture of the exit slit is taken (see figure 3.14), and the average Rk ; Gk ; Bk values in an area at the

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74

center of the image of the slit is recorded. Because the radiant flux of the light source and the diffraction efficiency of the grating in the monochromator are not known, flux Fk exiting the monochromator is measured using a powermeter, for each selected wavelength. The spectral responses of the sensor in the three channels are given by: SR ð k Þ ¼

Rk Gk Bk ; SG ðkÞ ¼ ; SB ðkÞ ¼ ; Fk Fk Fk

ð3:23Þ

FIG. 3.14 – Spectral response of the color camera in a Kinect from Microsoft company. On the left a selection of images of the monochromator exit slit for various wavelengths is shown. Usually, the spectral responses obtained are relative and dimensionless. Getting absolute curves requires prior knowledge of the gain of the sensor, the exposition time and/or the aperture, and considering the geometrical factors associated with the camera and power meter measurements. It is recommended to verify that these parameters are suited to have the best signal-to-noise ratio over the whole spectrum and prevent saturation of the sensor at certain wavelengths. If possible, every color correction option such as white balancing should be deactivated; if not possible, it is recommended to introduce a white light pattern in the field of the camera, slightly more intense than the signal issued from the monochromator, which will guarantee that the gain and the white balance is the same for all wavelengths.

3.2.5

Color Calibration of an RGB Imaging System

The digital color camera is establishing itself as an instrument for measuring surface colors, thanks to its affordable price, its ergonomics, its transportability, and its high spatial resolution. However, the RGB values contained in the image, even in raw format, have no precise relation to those as defined by the CIE from a spectrum of light and the color matching functions: these colors depend in particular on the spectral power distribution of the lighting, which is often unknown in a scene of everyday life. To obtain colors such as those which would be deduced from spectral

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75

reflectance measurements for a given illuminant, a color calibration is carried out using a colored target whose colorimetric values are rigorously defined and controlled. Color targets are provided by several manufacturers. Two of them are shown in figure 3.15a and b, the most popular being the Xrite Color Checker. The aim of the color calibration is to reduce these deviations using a mathematical transformation, the same for all colors. According to the type of sensor and the type of light source, colors may be very far from their target values. Preprocessing steps, described in [35] or [174], allow bringing them closer to the target and thus facilitate the work of reducing the deviations required from the mathematical transformation. This transformation, generally a polynomial transform of order 2 or 3, is searched by optimization on the basis of colors of the target. Once set, it is applied to all pixels in the image. This kind of transform needs less computation and less color patches in the target than methods based on machine learning [36, 125] or look-up tables.

FIG. 3.15 – Color targets for camera color calibration: (a) the ColorChecker target from X-rite, (b) the Kodak Q-60 target.

The color calibration method can be applied with CIE 1931 XYZ tristimulus values (the RGB values of the camera being considered as uncalibrated XYZ values), or CIE 1976 L*a*b* coordinates. In the following, we present the method with color coordinates in this latter space. Let us consider that the color chart contains N color patches. For each patch of m m the target labelled j, one denotes as Lm j aj , bj the ground truth color coordinates issued from the spectral reflectance measurement, and as Lj , aj , bj the color values captured by the camera after possible pre-processing and conversion into L*a*b* coordinates. For patch j, vector wj of length k is defined in terms of Lj , aj and bj values issued from the camera, depending in the polynomial order of the transformation. For the polynomial order 1, wj is   wj ¼ 1; Lj ; aj ; bj ð3:24Þ

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76

For the polynomial order 2, it is   wj ¼ 1; Lj ; aj ; bj ; L2j ; Lj aj ; Lj bj ; aj2 ; aj bj ; b2j and for a polynomial order 3, it is  wj ¼ 1; Lj ; aj ; bj ; L2j ; Lj aj ; Lj bj ; aj2 ; aj bj ; b2j ; L3j ; L2j aj ; L2j bj ; Lj aj2 ; Lj aj bj ; Lj b2j ; aj3 ; aj2 bj ; aj b2j ; b3j

ð3:25Þ



ð3:26Þ

The color transformation is represented by a k × 3 matrix U aiming at converting the values Lj , aj , bj incorporated into the vectors wj into values as close as m m possible to the target values Lm j aj , bj . The optimal matrix U can be easily obtained with a calculation software, for example function mldivide or operator\in MathWorks® MATLAB, which returns a least-square solution for the equation: 0 1 0 c 1 L1 a1c bc1 w1 B .. C B . .. C .. ð3:27Þ @ . A U ¼ @ .. . A . wN

LcN

aNc

bcN

Selecting the best polynomial order depends on the object being analyzed: orders 1, 2 and 3 are to be tested. The higher the order of the polynomial, the better the correction of colors of the patches, but the higher the risk of divergence for other colors than the ones present in the target [36]. This is the reason why polynomial orders higher than 3 often give low satisfaction results. Figure 3.16 shows an example of image calibrated with polynomial transforms of orders 2 and 3. Order 3 performs better than order 2 for colors of the target, but it gives false colors on the skin. Order 2 looks therefore preferable in this case. NB: In optimization, colors in the target all have the same weight. As bright and dark colors are poorly represented, the transformation cannot favor reducing the deviations between the initial and target values for these colors, which are, however, extremely important for the quality of the calibrated image. We therefore

FIG. 3.16 – Image of a hand with small color target: (a) original image, (b) image after calibration with a polynomial transformation of order 2, (c) image after calibration with a polynomial transformation of order 3. (Courtesy of Chloé Sliwak, Coraline Hillairet from Institut d’Optique Graduate School, Saint-Etienne, France).

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recommend replicating color values attached to the white patch and the black patch 10 times and consider them as independent patches of the target, which will force the optimization process to find a fairly precise transformation for these extreme points of the gamut.

3.3

Gloss

Gloss is a visual attribute produced by the human visual system when analyzing the geometrical distribution of the light reflected on the surface of the object together with the respective positions of the light source and the observer. The visual system is helped in this perceptive process by various clues, in particular the fact that the variation of perceived radiance is consistent with the shape of the object and the environment, and evolves in correlation with any change of position of the object (translation, inclination…), of the light source or of the observer. Two images associated with two positions can be sufficient to discriminate radiance variations due to the object’s color or to its glossiness, and therefore perceive the two attributes separately. In the case of a curved surface, e.g. a spherical or cylindrical object, gloss can be perceived immediately by correlating the radiance variation with the position of the light source and the shape of the surface.

3.3.1

Definition for Gloss

Scientific definition for gloss as visual attribute is more recent than for color. The first one proposed by the CIE, in the third edition of International Lighting Vocabulary dating from 1970, insisted on the physical property of the material of reflecting light more or less specularly. In the fourth edition published in 1987, the following definition was proposed, including the visual dimension: gloss is “the mode of appearance by which reflected highlights of objects are perceived as superimposed on the surface due to the directionally selective properties of that surface”. According to Wills et al. [248], the modern notion of gloss was formalized by the American Society for Testing and Materials (ASTM) as the angular selectivity of reflectance, responsible for the degree to which reflected highlights or images of objects may be seen as superimposed on a surface [10]. As for any visual attribute, gloss assessment requires a model taking into account the response of the human visual system to the observed optical signal. This latter can be correlated to the optical properties of the surface. The optical phenomenon underlying the glossy appearance is mainly the light reflection, with more or less scattering, onto the air-matter interface (the phenomenon is also referred to as “insurface reflection” [70]). However, the latter is not sufficient to wholly explain gloss evaluation by the human visual system, which is inherently multidimensional [142]. As shown by Pfund [186], the contrast between the maximal radiance in the shiniest area on the object and the surrounding areas matters: since the contrast is higher if the surrounding is black than if it is white, a black object looks glossier than a white object.

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3.3.2

Gloss Measurement

It is now well accepted that the most appropriate radiometric function to account for the sensation of gloss is the BRDF [9, 178]. However, studies carried out in the 1930s based on fixed, much simpler angular configurations, led to an instrument prevailing now on the market place: the glossmeter. It can be found in the form of small, portable and affordable devices. The fact that the angular configuration is fixed is necessarily a limiting factor on the ability of the glossmeter to assess the gloss sensation, but it nevertheless constitutes a basic assessment. These first attempts of gloss assessment are mainly due to Hunter [110, 111] who also subdivided gloss into several sub-attributes, each one being assessed by measurement with a specific θi:θr bi-directional geometry [21] (see figure 3.17): specular gloss, measured by the glossmeter, is the perceived brightness associated with the specular reflection from a surface (measurement geometries can, for example, be 20°: 20°, 45°:45° or 60°:60°), sheen is the perceived shininess from matte surfaces at grazing angles (85°:85°), Distinctness of image (DOI) is the perceived sharpness of images reflected from a surface (30°:30.3°). Bloom, also called 2° Haze, is the perceived cloudiness in reflections near the specular direction (30°:32°), Haze is the ‘shininess’ measured at 5° to the specular direction (30°:35°), diffuseness is the perceived brightness for diffusely reflecting areas (30°:45°) and contrast gloss is the perceived relative brightness of specularly and diffusely reflecting areas (45°:45° and 45°:0° geometries).

FIG. 3.17 – Angular configurations attached to the different sub-attributes related to gloss. Regarding the specular gloss, which is the most commonly measured, the glossmeter gives a gloss index value, in gloss units (GU), ranging on a one-dimensional scale between 0 for perfectly matte surfaces, to a few thousands for perfect mirrors; the value 100 corresponds to a specular reference surface, most often a smooth glass surface having a specified refractive index. Values below 10 are generally considered to describe matte surfaces, and values above 70 describe highly glossy surfaces. The principle of the measurement is comparable to the radiance factor defined in §2.5.2, except that the reference sample is not a perfect white diffuser, but a specular

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reflector (the aforementioned glass surface). The gloss index, in GU, is defined as follows: gloss ¼ 100 

F Fref

ð3:28Þ

where F and Fref are the fluxes measured by the detector in the glossmeter on the sample, respectively, on the specular reflector of reference. The spectral power distribution of the light source and the spectral response of the detection system should reproduce the relative spectral sensitivity of the human visual system V(λ), presented in §1.4.4. In a glossmeter, the incident beam is directional and forms angle θ with the normal of the sample; it can be collimated, or slightly convergent [201]. The detector is placed along the specular direction, thus also forming an opposite angle −θ with the normal of the sample. The angular detection aperture is precisely fixed by a diaphragm, which collects a certain fraction of light scattered around the specular direction. The detected signal determines the gloss index in GU: it is high with specular samples, and low with matte samples. If the surface is rather smooth, thanks to the collimated incident light beam, the insurface reflected light (light component carrying the gloss information) largely prevails over the subsurface scattered light (light component carrying the color information), as shown in §7.1.3. This is not true anymore if the surface is rough and scattering: for a same surface state, a glossmeter may provide different gloss index values for a white object and a black object having similar surface states [183]. Between the amount of light detected with the device and the gloss perceived by human observers, there is a non-linear transfer function, that psycho-physical experiments can attempt to determine. Figure 3.18 reproduces transfer functions after Hunter’s work [111] for devices based on three different angles of incidence: 20°, 60° and 85° [41]. It can be seen that these transfer functions depend very strongly upon the angle of incidence, and each one presents its advantage: the 75° curve

FIG. 3.18 – Correspondence between optical and visual quantities related to gloss, after Hunter’s work [41].

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allows better evaluation of matte surfaces, while the 20° curve is better suited to glossy surfaces. The NF EN ISO 2813 standard [113] recommends making first a measurement at 60°, then using a measurement at 20° if the gloss index at 60° is beyond 70, or a measurement at 85° if it is below 10. Note that these curves are the averages of psychophysical evaluations carried out by several observers. Secondary samples, calibrated in gloss units, are manufactured to allow everyone to calibrate their own gloss evaluation system, as for example the NCS color chart shown in figure 3.19 made of coated black paper, whose gloss index is measured at 60° according to the NF EN ISO 2813 standard. The main issue with glossmeters is the existence of multiple standards based on different angles of incidence. ASTM [11] and ISO [113] standards rely on measurement at 20°, 60° or 85°. In the papermaking industry, TAPPI standards rely on measurements at 75° for low- to high-gloss [230], and at 20° for very high-gloss [231]. Each standard also specifies different optical geometries for the incident beam and the detected beam, as well as a different reference reflector, in particular its refractive index. From one system to another, the gloss index of a given surface differs significantly, and it is difficult to interpret it in an absolute manner. For reliable gloss assessment, the surface should preferably be black in order to eliminate the light scattered by the material below the material-air interface, and plane in order to prevent caustic effects. The glossmeter is therefore an indicator of gloss, or a tool for comparing surfaces of a similar nature.

FIG. 3.19 – Picture of the 7 black samples in the NCS gloss scale, from the glossiest (on the left) to the most matte (on the right): Full gloss (95 GU), Glossy (75 GU), Semi-glossy (50 GU), Satin (30 GU), semi-matt (12 GU), matte (6 GU), Full matte (2 GU).

Chapter 4 Plane Optical Interfaces and Transparent Layers Almost all materials used in manufacturing have refractive index larger than air. The change in refractive index at the interface between the material and air induces partial reflection and transmission of light, which play an important role in the global reflectance or transmittance of the material, and consequently on its appearance. For example, an object made of a colored material with a smooth surface appears glossy because of the reflection of light rays from the air-material interface, and the shine of metallic objects is also due to the reflection of light from the air-metal interface. A clear glass pane is visible because of the specular reflection of light from its two interfaces, and we will see that some light can be reflected multiple times between the two interfaces before exiting the plate definitely. Interfaces also play an important role in the scattering of light: the matte appearance of a frosted glass plate comes from the roughness of the air-glass interfaces, and the opacity of white paint layer comes from the multiple reflections of light between the interfaces between the binder (e.g., acrylic polymer, whose refractive index is around 1.5) and the pigments (e.g., titanium dioxide, whose refractive index is around 2.7). Given the major role played by interfaces in the appearance of materials, it is to some of them, namely the plane interfaces between homogeneous isotropic media, that we will first focus our attention in this series of chapters dedicated to light-matter interaction models that begins here. The Snell-Descartes laws and Fresnel formulae introduced at the beginning of the book in §1.3 will be combined with the radiometric quantities used for the characterization of light beams and illumination geometries, especially in order to derive the reflectance and transmittance of the interface when the incident light is diffuse. We will study in more depth the case of a transparent plate, made of a layer of homogeneous non-scattering medium bordered by two parallel plane interfaces. The plate looks transparent since light is transmitted through it without being scattered,

DOI: 10.1051/978-2-7598-2647-6.c004 © Science Press, EDP Sciences, 2022

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which does not exclude coloration due to spectrally selective absorption. As mentioned above, the two air-medium interfaces play an important role in the optical properties of the plate: a multiple reflection process occurs between them and determines the reflectance and transmittance of the plate. We exclude the case of thin plates in which wave interferences due to the multiple reflection process produces iridescence; this happens when the plate thickness is smaller than the temporal coherence length of the incident white light, i.e., ordinarily, a few micrometers [73]. Predicting the reflectance and transmittance of the plate in this case relies on the thin film theory [90]. The plates considered in this chapter are much thicker, which prevents interferences and enables describing the multiple light reflections in the framework of ray optics (incoherent light, see §1.1). The aim is to derive analytical formulae for the reflectance and transmittance of the plate under directional and Lambertian illuminations, as functions of the optical properties of the plate and its thickness.

4.1

Radiance Reflection and Transmission at an Interface

In radiometry, light pencils are described by the radiance concept (see §1.4.2). When a pencil falls on the interface between media of different refractive indices, the geometrical extent of the refracted beam is modified, and the radiance is modified accordingly (figure 4.1). The relationship between the incident, reflected and refracted radiances is derived from geometrical arguments issued from Snell’s laws presented in §1.3.1. ð1Þ Incident spectral radiance Lk is defined as elementary spectral flux ð1Þ

d 3 Fk ðh1 ; u1 Þ flowing within elementary solid angle d 2 X1 ðh1 ; u1 Þ ¼ sin h1 dh1 du1 in a given direction ðh1 ; u1 Þ, and illuminating an elementary area ds of the interface ð1Þ

ð1Þ

Lk ¼

d 3 F k ð h1 ; u 1 Þ ds cos h1 sin h1 dh1 du1

ð4:1Þ

The denominator in equation (4.1) denotes the geometrical extent of the incident light pencil. Since the reflected and incident pencils form the same angle with the ðRÞ normal, they have the same geometrical extent. Reflected spectral radiance Lk is ð1Þ

therefore incident spectral radiance Lk multiplied by angular reflectance R12 ðh1 Þ of the interface given by the Fresnel formulae, equation (1.7): ðRÞ

Lk

ð1Þ

¼ R12 ðh1 ÞLk

ð4:2Þ

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83

FIG. 4.1 – Incident, reflected and refracted radiances at the interface between two media of indices n1 and n2 [ n1 .

The angles of incidence and refraction satisfy Snell’s sine law, equation (1.3). By differentiating this equation (1.3), one obtains n1 cos h1 dh1 ¼ n2 cos h2 dh2

ð4:3Þ

The azimuthal angles of the incident and refracted pencils form fixed angle π. A small variation in the one implies the same variation in the other one, i.e., du1 ¼ du2 . One can write: n12 ds cos h1 sin h1 dh1 du1 ¼ n22 ds cos h2 sin h2 dh2 du2

ð4:4Þ

n12 dG1 ¼ n22 dG2

ð4:5Þ

i.e.,

where dG1 and dG2 denote the geometrical extents of the pencil in media 1 and 2, respectively. Equation (4.5) shows that the geometrical extent of the refracted pencil is the geometrical extent of the incident pencil multiplied by factor ðn2 =n1 Þ2 . The refracted spectral radiance is therefore
2 n2 ð2Þ ð1Þ Lk ¼ T12 ðh1 ÞLk ð4:6Þ n1 where T12 ðh1 Þ ¼ 1  R12 ðh1 Þ is the Fresnel transmittance of the interface (see §1.3.2). When n2 [ n1 , the light beam is condensed into a narrower solid angle and the spectral radiance is increased.

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84

 2 More generally, it is multiplied by factor nj =ni each time it goes from medium i to medium j, while quantity ni2 dGi remains invariant.

4.2

Bi-Hemispherical Reflectance and Transmittance of an Interface

Let us now consider a flat interface illuminated by a Lambertian lighting, providing equal spectral radiance from every direction over the hemisphere. The relation between angular and bi-hemispherical reflectances has been established in §2.5.1, equation (2.13). In the case of an interface with relative refractive index n ¼ n2 =n1 [ 1 illuminated from medium 1, the bi-hemispherical reflectance r12 , is given by Z p=2 R12 ðh1 Þ sin 2h1 dh1 ð4:7Þ r12 ¼ h1 ¼0

r12 depends only on relative refractive index n. It may be computed by discrete summation with a small sampling step, e.g. Dh1 ¼ 0:001 rad. Alternatively, it is given by the following analytical formula, introduced by Duntley in 1942 [52], valid when n > 1 (i.e., when the interface is lit from the less refringent medium): 1 ðn  1Þð3n þ 1Þ 2n 3 ðn 2 þ 2n  1Þ þ  4 2 ð n  1Þ ð n 2 þ 1Þ 6ð n þ 1Þ 2  
 2 8n 4 n 4 þ 1 logðn Þ n 2 ðn 2  1Þ n1 þ þ log nþ1 ð n 2 þ 1Þ 3 ðn 4  1Þ2 ðn 2 þ 1Þ

r12 ¼

ð4:8Þ

The reflected flux fulfils the whole hemisphere, but is no longer Lambertian as the Fresnel reflectance, thereby the reflected spectral radiance, varies angularly. The transmitted flux is concentrated into the cone of apex given by critical angle hc ¼ arcsinð1=n Þ. The conservation of energy at the interface implies that the bi-hemispherical transmittance is t12 ¼ 1  r12

ð4:9Þ

When the Lambertian light comes from medium 2 (the most refringent medium), bi-hemispherical reflectance r21 is given by Z p=2 R21 ðh2 Þ sin 2h2 dh2 ð4:10Þ r21 ¼ h2 ¼0

Even though R12 ðh1 Þ and R21 ðh2 Þ are equal owing to the reciprocity principle [see equation (1.12)], reflectances r12 and r21 are different due to total reflection which occurs in medium 2 and not in medium 1. The relation between them, developed in exercise 14, is:

Plane Optical Interfaces and Transparent Layers

r21

85


 r12 1 ¼ 2 þ 1 2 n n

ð4:11Þ

where the terms into brackets correspond to the fraction of light subject to total reflections. It comes from equation (4.11) that t21 ¼

1 t12 n2

ð4:12Þ

The following formula for r21 , valid for n ¼ n2 =n1 [ 1, comes from equations (4.8) and (4.11): 1 ðn  1Þð3n þ 1Þ 2n ðn 2 þ 2n  1Þ þ  4 2n 2 ðn  1Þðn 2 þ 1Þ 6n 2 ðn þ 1Þ2  
 2 8n 2 n 4 þ 1 logðn Þ ðn 2  1Þ n1 þ þ log nþ1 ð n 2 þ 1Þ 3 ðn 4  1Þ2 ðn 2 þ 1Þ

r21 ¼ 1 

ð4:13Þ

For an air-glass interface of typical relative index n = 1.5, one has r12 ’ 0:1, t12 ’ 0:9, r21 ’ 0:6 and t21 ’ 0:4. Their values for other indices are listed in table 4.1. TAB. 4.1 – Normal and bi-hemispherical reflectance and transmittance (in %) of an interface for different relative refractive indices n. n 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.33 1.34 1.36 1.38 1.4 1.42

R12(0) 0 0.01 0.04 0.08 0.15 0.23 0.32 0.43 0.55 0.68 0.83 0.98 1.15 1.32 1.51 1.70 1.90 2.01 2.11 2.33 2.55 2.78 3.01

T12(0) 100 99.99 99.96 99.92 99.85 99.77 99.68 99.57 99.45 99.32 99.17 99.02 98.85 98.68 98.49 98.30 98.10 97.99 97.89 97.67 97.45 97.22 96.99

r12 0 0.61 1.14 1.63 2.08 2.52 2.93 3.32 3.70 4.07 4.43 4.78 5.12 5.46 5.79 6.11 6.43 6.59 6.75 7.06 7.37 7.68 7.99

t12 100 99.39 98.86 98.37 97.92 97.48 97.07 96.68 96.30 95.93 95.57 95.22 94.88 94.54 94.21 93.89 93.57 93.41 93.25 92.94 92.63 92.32 92.01

r21 0 4.46 8.60 12.45 16.05 19.43 22.61 25.61 28.43 31.10 33.63 36.03 38.29 40.45 42.50 44.45 46.30 47.20 48.07 49.75 51.36 52.9 54.36

t21 100 95.54 91.40 87.55 83.95 80.57 77.39 74.39 71.57 68.90 66.37 63.97 61.71 59.55 57.50 55.55 53.70 52.80 51.93 50.25 48.64 47.1 45.64

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TAB. 4.1 – (continued). n 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2

4.3

R12(0) 3.25 3.5 3.75 4 4.26 4.52 4.79 5.05 5.33 5.60 5.88 6.16 6.44 6.72 7.01 7.29 7.58 7.87 8.16 8.46 8.75 9.04 9.34 9.63 9.93 10.22 10.52 10.81 11.11

T12(0) 96.75 96.5 96.25 96 95.74 95.48 95.21 94.95 94.67 94.40 94.12 93.84 93.56 93.28 92.99 92.71 92.42 92.13 91.84 91.54 91.25 90.96 90.66 90.37 90.07 89.78 89.48 89.19 88.89

r12 8.29 8.59 8.88 9.18 9.47 9.76 10.05 10.34 10.62 10.91 11.19 11.47 11.75 12.03 12.31 12.59 12.86 13.13 13.41 13.68 13.95 14.21 14.48 14.75 15.01 15.28 15.54 15.80 16.06

t12 91.71 91.41 91.12 90.82 90.53 90.24 89.95 89.66 89.38 89.09 88.81 88.53 88.25 87.97 87.69 87.41 87.14 86.87 86.59 86.32 86.05 85.79 85.52 85.25 84.99 84.72 84.46 84.20 83.94

r21 55.77 57.11 58.4 59.64 60.82 61.95 63.04 64.08 65.09 66.05 66.98 67.87 68.73 69.56 70.36 71.12 71.87 72.58 73.27 73.94 74.58 75.20 75.81 76.38 76.95 77.49 78.01 78.52 79.01

t21 44.23 42.89 41.6 40.36 39.18 38.05 36.96 35.92 34.91 33.95 33.02 32.13 31.27 30.44 29.64 28.88 28.13 27.42 26.73 26.06 25.42 24.80 24.19 23.62 23.05 22.51 21.99 21.48 20.99

Metals and Strongly Absorbing Media

The color of a homogenous medium comes from its capacity to absorb radiations of specific wavelengths in the visible domain, which corresponds to a non-zero spectral extinction index κ(λ). Spectral absorptance of the material is also described by a spectral absorption coefficient α(λ), in m−1, which is related to the extinction index by the formula [22]: að kÞ ¼

4p jðkÞ k

ð4:14Þ

This relation is valid for any absorbing medium, e.g. colored glass or metal. The particularity of metals is their high extinction index, which makes them very opaque

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87

and reflecting. The complex refractive indices of many metals are tabulated with respect to wavelength in Ref. [30], pp. 235–265. For each wavelength, the angular reflectance of air-metal interfaces is given by the same Fresnel formulae (1.5) and (1.6) as for air-dielectric interfaces, but the ^ ¼ n þ ij including the extinction coefficient refractive index is a complex number n [22]: refraction angle h2 is also a complex number. Nevertheless, the angular reflectance is real and it is still given by equation (1.5) or (1.6) according to the polarization, or equivalently by pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 a þ z  2 cos h1 þ a  z Rs12 ðh1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 a þ z þ 2 cos h1 þ a  z ð4:15Þ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 a þ z  2 sin h1 tan h1 þ a  z Rp12 ðh1 Þ ¼ Rs12 ðh1 Þ  pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 a þ z þ 2 sin h1 tan h1 þ a  z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with z ¼ n 2  j2  sin2 h1 and a ¼ z 2 þ 4n 2 j2 . For unpolarized incident light, the angular reflectance is the average of formulae (4.15). At normal incidence, for any polarization, the reflectance is given by: R12 ð0Þ ¼

ð n  1 Þ 2 þ j2 ðn þ 1Þ2 þ j2

:

ð4:16Þ

Figure 4.2 illustrates how the reflectance at normal incidence varies as the ^ ¼ 1:5 þ ij. From κ = 0 to extinction index increases, by considering relative index n 0.2, the reflectance remains close to 0.04, the value corresponding to real index 1.5. This justifies the fact that for weakly absorbing dielectrics the extinction index is not included in the Fresnel formulae. When κ > 0.2, the angular reflectance increases rapidly.

FIG. 4.2 – Variation in the angular reflectance at normal incidence of an interface with ^ ¼ 1:5 þ ij as a function of j. relative refractive index n

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The variation in the reflectance as a function of the angle of incidence is noticeably different between absorbing and non-absorbing media. Figure 4.3 shows three examples based on the refractive indices of a strongly absorbing glass (^ n ¼ 1:5 þ i), gold at 550 nm (^ n ¼ 0:42 þ 2:47i [116]) and silver at 550 nm (^ n ¼ 0:06 þ 3:60i [116]). In the three cases, the angular reflectance for s-polarized light is a strictly increasing function of the incident angle, while the one for p-polarized light decreases to a minimum at an angle sometimes called pseudo-Brewster angle [62], without reaching zero, in contrast with what happens with isotropic transparent materials. The reflected light is therefore partially polarized, but there is no angle at which its polarization is total.

FIG. 4.3 – Angular reflectance as a function of the incident angle, for p-polarized, s-polarized

and unpolarized lights, of (a) strongly absorbing glass (^ n ¼ 1:5 þ i), (b) gold at 600 nm (^ n ¼ 0:42 þ 2:47i), and (c) silver at 550 nm (^ n ¼ 0:06 þ 3:60i).

4.4

Angular Reflectance and Transmittance of a Clear Plate

Consider a clear plate made of a non-scattering and non-absorbing material with refractive index n2 . Its interfaces with air on both sides are assumed to be plane and parallel. Every light pencil falling on the plate undergoes a multiple reflection process between the two interfaces. One wants to derive the analytical expressions for the reflectance and transmittance of the plate for directional light at any angle of incidence, then for a perfectly diffuse light.

4.4.1

Angular Reflectance and Transmittance

Let us follow the path of a unpolarized light pencil coming from air and falling on the first interface at a certain angle h1 from the normal of the plane (see figure 4.4).

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89

It splits into a pencil reflected back into air and a pencil transmitted into the medium at angle h2 ¼ arcsinðsin h1 =n2 Þ. The fraction of flux that is being reflected corresponds to the Fresnel reflectance R12 ðh1 Þ of the interface, given by equation (1.5) for p-polarization, and by equation (1.6) for s-polarization (symbol * meaning either p or s). Even though the incident pencil is unpolarized, the reflected pencil is partially polarized due to the difference between Rs12 ðh1 Þ and Rp12 ðh1 Þ, except if h1 ¼ 0. The two polarized components of the light pencil follow exactly the same paths along the multiple reflection process, but the reflectance and transmittance values are different for each of them. The fraction of flux that is being refracted, 1  R12 ðh1 Þ, then falls on the second interface where it splits again into a pencil internally reflected into the medium, and a pencil refracted into air at the opposite side of the plate. The fraction of flux reflected, given by the Fresnel reflectance R21 ðh2 Þ, is equal to R12 ðh1 Þ according to the reciprocity principle stated by equation (1.12), while the fraction of flux exiting into air is 1  R21 ðh2 Þ ¼ 1  R12 ðh1 Þ. The reflected pencil returns to the first interface where fraction R12 ðh1 Þ is again internally reflected and fraction 1  R12 ðh1 Þ exits into air at the input side, and this process continues indefinitely.

FIG. 4.4 – Multiple reflections of light in a non-absorbing plate illuminated by directional flux

F at angle θ1. The blue arrows draw the paths corresponding to one of the incident rays. The Fresnel reflectance of the interfaces, denoted R here, is the same at the air side and the medium side due to the reciprocity principle.

The reflectance of the plate, denoted by R121 ðh1 Þ (we simplify here the notation for R12 ðh1 Þ as R ), is obtained by summing the flux components exiting the plate at the input side, and dividing the sum by the incident flux: R121 ðh1 Þ ¼ R þ ð1  R Þ2 R þ ð1  R Þ2 R3 þ ð1  R Þ2 R5 þ      ¼ R þ ð1  R Þ2 R 1 þ R2 þ R4 þ    1 X ¼ R þ ð1  R Þ2 R R2k  : k¼0

ð4:17Þ

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90

 1 The geometric series of common ratio R2 converges towards 1  R2 . The angular reflectance R121 ðh1 Þ is therefore written R121 ðh1 Þ ¼ R þ

ð1  R Þ2 R 1  R2

ð4:18Þ

The angular transmittance of the plate is obtained in a similar way: T121 ðh1 Þ ¼ ð1  R Þ2

1 X k¼0

R2k  ¼

ð1  R Þ2 1  R2

ð4:19Þ

After some simplifications, equations (4.18) and (4.19) can also be written R121 ðh1 Þ ¼

2R12 ðh1 Þ 1  R12 ðh1 Þ and T121 ðh1 Þ ¼ 1 þ R12 ðh1 Þ 1 þ R12 ðh1 Þ

ð4:20Þ

Since the incident light is unpolarized and contains the two polarized components in equal proportions, the angular reflectance of the plate is the average of Rs121 ðh1 Þ and Rp121 ðh1 Þ:  1 ð4:21Þ R121 ðh1 Þ ¼ Rs121 ðh1 Þ þ Rp121 ðh1 Þ 2 and the angular transmittance is the average of Ts121 ðh1 Þ and Tp121 ðh1 Þ. Notice that these angular reflectances and transmittances are independent of the thickness of the plate, which is just assumed to be larger than the temporal coherence length of light in order to prevent interferences. One can easily verify that for each polarization, thereby for unpolarized light, the sum of R121 ðh1 Þ and T121 ðh1 Þ is 1 – which is expected as the material is non-absorbing and energy is conserved at the interfaces – all the incident flux exits the plate, and, on both sides, it is oriented with angle h1 with respect to the normal of the plate. Notice that using equation (4.18) directly with the Fresnel reflectance R12 ðh1 Þ defined in equation (1.7) for unpolarized light is incorrect because after crossing the first interface, light is not unpolarized anymore: it is partially polarized. The error compared to the correct formula (4.21) can reach 10% at certain angles for a plate of refractive index 1.5 surrounded by air. Figure 4.5 shows the curves of the angular reflectances and transmittances of a plate with refractive index 1.5 for s-polarized light, p-polarized light, and unpolarized light, as well as that of the degree of polarization, as functions of the angle of incidence. For comparison, the corresponding curves for the first interface alone, plotted in figure 1.6, are reproduced here in dashed lines. We notice that the reflectance of the plate is always higher than that of the first interface. At normal incidence, it is 7.7%, whereas it is 4% for the first interface alone. The difference is the contribution of the second interface to the plate’s reflectance. At the Brewster angle hb , p-polarized light cannot be reflected from the first interface, nor by the second interface: it is therefore totally transmitted. The degree of polarization is 1 at this angle. Notice that exactly same curves would be obtained for a plate of refractive index n2 ¼ 2 in water (n1 ¼ 1:33) since the relative refractive index of the interfaces is also 1.5 in this case.

Plane Optical Interfaces and Transparent Layers

91

FIG. 4.5 – Angular reflectances for p-polarized light, s-polarized light, and unpolarized light as well as DOP of the reflected light for a plate with refractive index n2 ¼ 1:5 surrounded by air (solid lines) and for the first air-material interface (dashed lines). We have considered so far that the refractive index of the plate is higher than the one of the surrounding medium (air), the configuration where the relative refractive index of the interfaces n ¼ n2 =n1 > 1. We can also consider the opposite configuration: a layer of clear medium with refractive index n1 (e.g., air) surrounded by a medium with higher index n2 , which is the same on both sides of the layer. We thus have n ¼ n2 =n1 \1. Total reflections occur in this case, at angles of incidence larger than critical angle hc . However, equation (1.12) remain valid: the total reflections are automatically incorporated into the formulae. The curve of angular reflectance, which can be denoted by R212 ðh2 Þ, is similar to that of R21 ðh2 Þ plotted in figure 1.6, i.e., its value is equal to 1 beyond hc .

4.4.2

Bi-Hemispherical Reflectance and Transmittance

If the incident light is perfectly diffuse, i.e., Lambertian, the bi-hemispherical reflectance r121 of the plate is obtained by integrating angular reflectance R121 ðh1 Þ given by equation (4.21) over the hemisphere, in a similar manner as shown in §2.5.1, equation (2.13): Z p=2 r121 ¼ R121 ðh1 Þ sin 2h1 dh1 ð4:22Þ h1 ¼0

The reflectance depends only on the refractive index of the material. It can be given an analytical expression, comparable to Duntley’s formula for one interface given by equation (4.8), valid when n ¼ n2 =n1 [ 1:

Optical Models for Material Appearance

92

n n3  4 2 ðn þ 1Þ "  
#
4 2 n2 n4  1 n2  1 p n 1 n1  þ arctanðn Þ þ log 2 2 n4 þ 1 nþ1 ð n 4 þ 1Þ 2

r121 ¼ 1 

ð4:23Þ

The bi-hemispherical transmittance t121 is given by similar integral as in equation (4.22), with T121 ðh1 Þ in place of R121 ðh1 Þ. Since no light is absorbed, one also knows that: t121 ¼ 1  r121

ð4:24Þ

When the layer has a smaller refractive index than that of the surrounding medium, the bi-hemispherical reflectance r212 is related to r121 in a similar manner as r21 and r12 in the case of a single interface [see equation (4.11)]:
 r121 1 r212 ¼ 2 þ 1  2 ð4:25Þ n n Table 4.2 gives the bi-hemispherical reflectance and transmittance values for r121 , t121 , r121 and t121 for various relative refractive indices n of the interfaces.

TAB. 4.2 – Bi-hemispherical reflectance and transmittance values (in %) for a clear layer of different relative refractive indices n of its two interfaces. n 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.33 1.34

r121 0 1.03 1.92 2.74 3.50 4.21 4.89 5.54 6.16 6.77 7.35 7.92 8.48 9.02 9.56 10.08 10.59 10.85 11.10

t121 100 98.97 98.08 97.26 96.50 95.79 95.11 94.46 93.84 93.23 92.65 92.08 91.52 90.98 90.44 89.92 89.41 89.15 88.90

r212 0 4.87 9.32 13.44 17.26 20.84 24.18 27.32 30.26 33.04 35.66 38.14 40.48 42.70 44.80 46.79 48.69 49.60 50.49

t212 100 95.13 90.68 86.56 82.74 79.16 75.82 72.68 69.74 66.96 64.34 61.86 59.52 57.30 55.20 53.21 51.31 50.40 49.51

n 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86

r121 14.91 15.36 15.80 16.25 16.69 17.12 17.55 17.98 18.40 18.82 19.23 19.64 20.05 20.45 20.85 21.25 21.65 22.04 22.42

t121 85.09 84.64 84.20 83.75 83.31 82.88 82.45 82.02 81.60 81.18 80.77 80.36 79.95 79.55 79.15 78.75 78.35 77.96 77.58

r212 62.18 63.37 64.50 65.59 66.62 67.62 68.58 69.50 70.39 71.24 72.05 72.84 73.59 74.32 75.02 75.70 76.35 76.97 77.58

t212 37.82 36.63 35.50 34.41 33.38 32.38 31.42 30.50 29.61 28.76 27.95 27.16 26.41 25.68 24.98 24.30 23.65 23.03 22.42

Plane Optical Interfaces and Transparent Layers

93

TAB. 4.2 – (continued). 1.36 1.38 1.4 1.42 1.44 1.46 1.48

4.5

11.60 12.09 12.57 13.05 13.52 13.99 14.45

88.40 87.91 87.43 86.95 86.48 86.01 85.55

52.20 53.84 55.39 56.88 58.30 59.65 60.94

47.80 46.16 44.61 43.12 41.70 40.35 39.06

1.88 1.9 1.92 1.94 1.96 1.98 2

22.81 23.19 23.57 23.94 24.32 24.69 25.05

77.19 76.81 76.43 76.06 75.68 75.31 74.95

78.16 78.72 79.27 79.79 80.30 80.79 81.26

21.84 21.28 20.73 20.21 19.70 19.21 18.74

Spectral Transmittance of Absorbing Layers

Let us now consider an absorbing medium. Absorption leads to the attenuation of light due to the conversion of the electromagnetic energy into another form of energy, typically because its frequencies are resonant with transition frequencies of electrons in the medium [73]. Absorption depends upon wavelength (and sometimes on polarization). It is responsible for the colored aspect of most objects, such as stained glasses, dyes, pigments, inks, etc. The light absorption property of a (homogeneous) medium is assessed by its spectral linear absorption coefficient α(λ), in m−1, which depends upon wavelength and is related to the extinction coefficient of the medium according to equation (4.14). Before showing how absorptance is considered in the formulae for the angular reflectance and transmittance of a plate, we first consider the plate without its bordering interfaces, i.e., the transmission of light through the absorbing medium itself.

4.5.1

Bouguer’s Law and Beer’s Law

According to Bouguer’s law (not to be confounded with the law presented in §1.6.1, also named after the same French scientist Pierre Bouguer) [23], when propagating into a homogeneous absorbing medium, light is exponentially attenuated as a function of the travelled distance, and a spectral absorption coefficient α(λ). This law derives from the fact that the flux loss by absorption at a very small scale is linear: when crossing a very thin sublayer of medium at a given depth z, incident flux F ðk; z Þ, here assumed to flow perpendicularly to the layer, is attenuated into flux F ðk; z þ dz Þ. The portion of flux that has been lost is proportional to incident flux F ðk; z Þ and to sublayer thickness dz, the proportionality constant being absorption coefficient aðkÞ at the considered wavelength (see figure 4.6a): F ðk; z þ dz Þ  F ðk; z Þ ¼ aðkÞF ðk; z Þdz

ð4:26Þ

This equation yields a differential equation of the first order, dF ðkÞ ¼ aðkÞdz F ðkÞ

ð4:27Þ

F ðk; z Þ ¼ eaðkÞz F0 ðkÞ

ð4:28Þ

whose solution is

94

Optical Models for Material Appearance

where F0 ðkÞ is the incident spectral flux at depth z = 0. Spectral flux F ðk; h Þ exiting a layer of thickness h, divided by incident spectral flux F0 ðkÞ is by definition a spectral transmittance, called intrinsic spectral transmittance of the layer and denoted by t ðkÞ, or tk hereinafter (see figure 4.6b). Its expression, coming directly from equation (4.28), is known as Bouguer’s law [249]: t ðkÞ ¼ eaðkÞh

ð4:29Þ

FIG. 4.6 – (a) Derivation of the differential equation describing the lost flux in a thin layer of medium; (b) intrinsic spectral transmittance t(λ) and (c) angular spectral transmittance T(λ, θ) of the absorbing layer. In equation (4.29), the intrinsic spectral transmittance is dimensionless, as is the term aðkÞh since a ðkÞ is in m−1 and h in m. This quantity aðkÞh is called spectral optical thickness. In 1852, the German scientist August Beer (1825–1863) stated with colored liquids [15] that by modifying the concentration in colorant, for example by factor c, the absorption coefficient is modified in the same proportion c. Intrinsic spectral transmittance tc ðkÞ of a layer with thickness h of this modified medium becomes tc ðkÞ ¼ ecaðkÞh ¼ t c ðkÞ

ð4:30Þ

which is referred to as Beer’s law, Lambert’s law, or Beer-Lambert law. Notice that same intrinsic spectral transmittance would be obtained with the original medium by modifying the layer thickness by factor c. Thus, a change of colorant concentration has similar effect as a change in layer thickness. If the beam crosses the layer at angle θ to the normal, the distance travelled in the layer becomes h=cos h (see figura 4.6c). Angular spectral transmittance T ðk; hÞ of the layer can be expressed in terms of layer thickness h, propagation angle θ and spectral absorption coefficient aðkÞ, or in terms of intrinsic spectral transmittance t ðkÞ and propagation angle θ: T ðk; hÞ ¼ eaðkÞh=cos h ¼ t 1=cos h ðkÞ

ð4:31Þ

Plane Optical Interfaces and Transparent Layers

4.5.2

95

Piles of Absorbing Layers and Mixing of Absorbing Media

When N layers of different absorbing materials with same the real refractive index but possibly different thicknesses hi and spectral absorption coefficients ai ðkÞ are placed on top of each other, they form again a non-scattering, absorbing layer whose intrinsic spectral transmittance is the product of the layers’ individual intrinsic spectral transmittances, and whose optical thickness is the sum of the individual optical thicknesses: t ð kÞ ¼

N Y

t i ð kÞ ¼ e



N P

ai ðkÞhi

i¼1

ð4:32Þ

i¼1

If one now considers the homogeneous mixture of different colorants with respective spectral absorption coefficients ai ðkÞ with respective proportions ci , one obtains similar formula as equation (4.32), which extends the Beer-Lambert law presented above. Absorption coefficient aðkÞ of the resulting material is the sum of the individual spectral absorption coefficients weighted by their concentrations (Beer-Lambert-Bouguer law): að kÞ ¼

N X

c i ai ð kÞ

ð4:33Þ

i¼1

The intrinsic spectral transmittance is therefore t ð kÞ ¼ e



N P i¼1

ci ai ðkÞh

¼

N Y i¼1

c

t i i ð kÞ

ð4:34Þ

It may also happen that the spectral absorption coefficient aðk; z Þ varies continuously as a function of depth z (while being constant at each depth). This is the case, for example, with the atmosphere and the sea, or an ink bottle in which the dyes begin to settle. The normal transmittance in this case is given by Rh  aðk;z Þdz t ð kÞ ¼ e 0 ð4:35Þ

4.6

Spectral Reflectance and Transmittance of an Absorbing Plate

A non-scattering absorbing plate is a layer of material in which Beer’s law applies and a multiple reflection process occurs, as presented in §4.4 if the plate is surrounded by a medium of different index, e.g., air. Colored filters and stained glasses are examples of absorbing plates of this kind. The aim of this section was to derive analytical formulae for the plate’s angular reflectance and transmittance.

96

4.6.1

Optical Models for Material Appearance

Angular Reflectance and Transmittance

The material composing the plate has spectral absorption coefficient aðkÞ and refractive index n2 (which may be also a function of wavelength). The plate has thickness h, therefore intrinsic spectral transmittance t ðkÞ given by equation (4.29), that we will write tk for shorter notations.

FIG. 4.7 – Multiple reflections of light in an absorbing plate of intrinsic spectral transmittance tλ illuminated by a directional spectral flux F ðkÞ at angle θ1. The blue arrows draw the paths corresponding to one of the incident rays. As in §4.4.1, one considers a light pencil coming from air with an angle of incidence h1 . A multiple reflection process occurs, similar to the one taking place in the non-absorbing plate, but in which one now considers the attenuation of light along its path within the medium (figure 4.7): each time it crosses the plate, at angle 1=cos h2 h2 ¼ arcsinðsin h1 =n2 Þ, it undergoes spectral attenuation tk corresponding to the angular spectral transmittance given by equation (4.31). This attenuation can also be expressed as a function of h1 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1sin2 h1 =n22 1=cos h2 ¼ tk ð4:36Þ tk By summing, the flux components exiting the plate at the side of incidence (see figure 4.7), and dividing the sum by the incident flux, one obtains the angular spectral reflectance of the plate, still denoted by R121 ðh1 ; kÞ (we use again the shortened notation R for R12 ðh1 Þ): 2=cos h2

R121 ðh1 ; kÞ ¼ R þ ð1  R Þ2 R tk

1 X k¼0

2k=cos h2

R2k  tk

ð4:37Þ

Plane Optical Interfaces and Transparent Layers

97 2=cos h

2 geometric series of common ratio R2 tk converges towards

1 2 2=cos h2 1  R  tk . Using the relation (4.36), angular spectral reflectance

The

R121 ðh1 ; kÞ for p- and s-polarized lights is therefore

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2= 1sin2 h1 =n22 ½1  R12 ðh1 Þ2 R12 ðh1 Þtk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R121 ðh1 ; kÞ ¼ R12 ðh1 Þ þ 2= 1sin2 h1 =n22 1  R212 ðh1 Þtk

ð4:38Þ

and the one for unpolarized light is R121 ðh1 ; kÞ ¼

 1 Rs121 ðh1 ; kÞ þ Rp121 ðh1 ; kÞ 2

ð4:39Þ

In a similar way, for flux components exiting the plate at the opposite side, one obtains angular spectral transmittance T121 ðh1 ; t Þ for p- and s-polarized lights, expressed as 1=cos h2

T121 ðh1 ; kÞ ¼ ð1  R Þ2 tk ¼

ð1 

1 X k¼0

1= R12 ðh1 ÞÞ2 tk 2=

1  R212 ðh1 Þtk

2k=cos h2

R2k  tk

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 1ðsin h1 =n2 Þ

ð4:40Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 1ðsin h1 =n2 Þ

and the angular spectral transmittance for unpolarized light is the average of Ts121 ðh1 ; kÞ and Tp121 ðh1 ; kÞ. In order to see how the plate’s angular reflectance varies according to the angle of incidence and the intrinsic normal incidence, one can refer to figure 4.5 where refractive index 1.5 is considered. The curves in solid lines correspond to the case of a non-absorbing plate, i.e., t ðkÞ ¼ 1 for all wavelengths. The curves dashed lines correspond to the case of a very thick, totally absorbing plate. In this latter case, since only the first interface can reflect light, one has: R121 ðh1 Þ ¼ R12 ðh1 Þ. For intermediate t values, the reflectance curves are between these two extreme curves.

4.6.2

Bi-Hemispherical Reflectance and Transmittance

When the incident light is Lambertian, the spectral bi-hemispherical reflectance r121 ðkÞ is obtained by integrating the angular spectral reflectance R121 ðh1 ; kÞ given by equation (4.39) in a manner similar to that shown in §2.5.1, equation (2.13): Z p=2 r121 ðkÞ ¼ R121 ðh; kÞ sin 2hdh ð4:41Þ h¼0

Similarly, the spectral bi-hemispherical transmittance is given by Z p=2 t121 ðkÞ ¼ T121 ðh; kÞ sin 2hdh h¼0

ð4:42Þ

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Optical Models for Material Appearance

FIG. 4.8 – Bi-hemispherical reflectance and transmittance of a plate with refractive index 1.5 as functions of intrinsic spectral transmittance t. These spectral bi-hemispherical reflectance and transmittance depend on both refractive index n2 and intrinsic spectral transmittance t ðkÞ. Their variation as functions of the t value is plotted in figure 4.8 for a material of refractive index 1.5. When t = 1, the plate is non-absorbing and one retrieves the reflectance and transmittance values given in table 4.2. When t = 0, the plate is opaque: its transmittance is 0 and its reflectance is the one of the first interface: one retrieves the value given in table 4.1. The transmittance varies almost linearly between these two extreme cases, whereas the reflectance varies weakly, in a parabolic way. Same curves would be obtained by plotting these reflectance and transmittance as functions of the optical thickness from infinity to 0 with a log scale in abscissa.

4.6.3

Obtaining the Intrinsic Parameters of an Absorbing Plate

The reflectance and transmittance of the plate depend on two intrinsic parameters: refractive index n2 , and the intrinsic normal transmittance t ðkÞ. These can be deduced from reflectance and transmittance measurements, performed at normal incidence, or under a very weak angle of incidence which gives numerically similar results since the plate’s reflectance and transmittance do not vary at low incidence angles (see figure 4.5). A number of instruments are based on an illumination or an observation at 8° from the normal, as recommended by the CIE, see table 2.1. However, it is preferable to avoid instruments based on a bi-directional geometry – 0°:0° or 8°:8° geometries, for example – because it may happen that the plate has small defects and scatters some light out of the regular direction where the detector

Plane Optical Interfaces and Transparent Layers

99

is situated, and this missing light would be interpreted as absorbed light. Instruments providing a diffuse illumination thanks to an integrating sphere, and observing the sample close to the normal (e.g., d:8° geometry) are to be preferred. As explained in §2.6.4, with transparent objects, their effective measurement geometry is bi-directional as expected, and the small amount of light possibly scattered is incorporated into the measurement. Let us therefore consider that spectral reflectance Rm ðkÞ and transmittance Tm ðkÞ of the plate are measured at normal incidence (h1 ¼ 0). By combining equations (1.10), (4.38) and (4.40), one can express them as  2 2   n2  1 ðn2  1Þ2 n22  6n2 þ 1 t 2 ðkÞ Rm ðkÞ ¼ R121 ð0; kÞ ¼ ð4:43Þ ðn2 þ 1Þ4 ðn2  1Þ4 t 2 ðkÞ Tm ðkÞ ¼ T121 ð0; kÞ ¼

16n22 t ðkÞ 4

ðn2 þ 1Þ ðn2  1Þ4 t 2 ðkÞ

ð4:44Þ

If the material’s refractive index is known, one can use only measured transmittance T121 ð0Þ and inverse formula (4.44) (see Ref. [249], p. 30): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4 64n24 þ 1  n22 Tm2 ðkÞ  8n22 ð4:45Þ t ð kÞ ¼ ð1  n2 Þ4 Tm ðkÞ If the refractive index is not known, one can deduce it as well as normal transmittance t ðkÞ, by resolving analytically the system of equations (4.43) and (4.44). One obtains  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V ð kÞ 1  R m ð kÞ  þ U ðkÞ  ½1  Rm ðkÞV ðkÞ þ 2Tm2 ðkÞ n2 ðkÞ ¼ ð4:46Þ U 2 and V ð kÞ  U ð kÞ 2Tm ðkÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with U ðkÞ ¼ ½1  Rm ðkÞ2 Tm2 ðkÞ and V ðkÞ ¼ U 2 ðkÞ þ 4Tm2 ðkÞ. t ð kÞ ¼

4.7

ð4:47Þ

Extensions and Applications

The models developed in this chapter prove useful to characterize the optical properties of transparent polymer sheets, stained-glasses, color filters, solar glasses in the domain of photovoltaics, glass panes in the domain of architecture for the natural lighting design, and many other domains. The model is also applicable when different plates with same refractive index are stacked in optical contact, i.e., without any discontinuity of refractive index between them. Optical contact is important to prevent Fresnel reflections at the internal interfaces, which would

100

Optical Models for Material Appearance

modify noticeably the reflection and transmission properties of the layered plate. Hence, the case of double-paned windows, for example, cannot be addressed with these models but with the ones presented in the next chapter. If the stacked plates are all made of the same material and are in optical contact with each other, one obtains a new uniform plate whose thickness is the sum of the individual thicknesses. If the intrinsic spectral transmittance has been defined for a plate with thickness h and the sum of the thicknesses is H, then the reflectance and transmittance of the new plate is predicted with the intrinsic spectral transmittance t H =h ðkÞ (Lambert’s law). If the plates are made of different materials of different optical thicknesses, the resulting intrinsic spectral transmittance is related to the individual ones by equation (4.32). When the layer is bordered by something else than flat interfaces with air, in particular diffusing elements (rough interface, diffusing layer, etc.), the optical paths within the layer may be harder to describe. Analytical reflectance and transmittance formulae are not always obtained, except in the case addressed in chapter 7 where the layer is bordered by a strongly diffusing medium. In the other cases, ray tracing models are often the best option [245]. In many cases, refractive index n2 and the absorption coefficient (thereby the intrinsic spectral transmittance) are functions of wavelength. This, of course, is the case of colored materials the color of which is related to the variation in the spectral absorption coefficient in the visible spectrum. This is also the case of silicate glass which is strongly absorbing in the infrared, or titanium dioxide (rutile form) which is highly transparent in the visible spectrum but more absorbing in the ultraviolet. The models apply for any wavelength, and the reflectance and transmittance values can be computed for each of them.

Chapter 5 Transparent Multilayers: Two-Flux Models for Directional Light After studying in the previous chapter how the interaction of light with a (thick) non-scattering plate can be described in the framework of ray optics, and deriving from this model the analytical expressions for the angular reflectance and transmittance of the plate depending on the refractive index, thickness and absorption coefficient, we propose to pursue this approach by considering piles of non-scattering layers, plates or films. Piles of non-scattering glass plates are certainly not the most common items in everyday life. They therefore have modest applicative interest – except the special case of double-glazed windows used in architecture, since a good estimation of indoor lighting during daytime requires determining the transmittance of the windows according to the different outdoor lighting configurations. However, they have a high didactic interest in the theoretical progression that we wish to pursue in this book, because they allow to address important notions related to the propagation of light in the depth of materials, which will be useful in the next two chapters dedicated to diffusing layered materials. This didactic value has also marked the entire modern history of optics, through the example of piles of glass plates (naturally separated by a layer of air): during the first half of the 18th century, Pierre Bouguer [23] and Jean-Henri Lambert [138] used them to modify the quantities of light in experiments that would later become photometry (with the naked eye, the only optical detector available at that time). A century later, in 1860, George Gabriel Stokes gave the most rigorous physical model to predict the reflectance and transmittance of piles of glass plates based on properties of each plate, their number, and properties of the light beam: orientation and polarization; other famous people in optics such as Augustin Fresnel [74] and Lord Rayleigh [229] also studied this kind of object. Here, we generalize this historical case to many other configurations where the plates are possibly different, absorbing, non-symmetrical (their reflectances on their two faces are different), in optical contact or separated by media with various indices. To take into account the multiple reflections and refractions that occur at all air-matter interfaces in the pile, which can be indefinitely numerous, the approach introduced in the previous chapter for the simple window, i.e., the count of light DOI: 10.1051/978-2-7598-2647-6.c005 © Science Press, EDP Sciences, 2022

102

Optical Models for Material Appearance

trajectories leading to a geometrical series, is too tedious to be followed. Instead, a mathematical formalism based on flux transfer matrices that appeared in the literature only recently eases the calculations considerably. This formalism and the set of stratified structures that it allows to address constitute what we call the discrete two-flux model (meaning that the number of layers is an integer) or the directional two-flux model (meaning that it applies to collimated light beams, which keep their directionality since light is not scattered). At the end of the chapter, a specular background, e.g., a mirror, is added behind the multilayer. This is a configuration that one can find for instance in the photovoltaic domain when the silicon cells are protected with encapsulant layers [65].

5.1

Piles of Transparent Plates Separated by Air

Consider different plates placed on top of each other, separated by layers of air. Basically, their optical properties are similar to those studied in the previous chapter, characterized by a refractive index, a thickness, and a spectral absorption coefficient. Their spectral reflectance and transmittance are the same on both sides: they are symmetric. Given that what happens with one plate, an incident light pencil illuminating the pile is decomposed into many pencils by reflections and refractions at the interfaces. The multiple reflection process taking place within each plate has already been described in the previous chapter, where analytical expressions have been derived for the angular spectral reflectance and transmittance of the plate [equations (4.38) and (4.40), valid for either s or p polarization]. Next, the multiple reflections of light between the different plates will be described. When the pile is illuminated with an angle of incidence h1 , all the pencils exiting from a plate reach the next plate with the same angle of incidence h1 . The individual spectral reflectances and transmittances of the plates are therefore all expressed at this angle, and all angular spectral functions f ðh1 ; kÞ will be simply written f in order to simplify the notations. Notice that the light exiting definitively the pile from both sides is also oriented with this angle h1 : the pile is therefore a specular reflector and a transparent object. We can intuitively understand that the spectral reflectance of a pile of plates is increased and its spectral transmittance decreased when a new plate is added: the two additional air-plate interfaces back-reflect some light, to an extent which depends on the plate’s refractive index (a higher index increases the Fresnel reflectance of each interface) and absorptance (a higher absorption coefficient attenuates the amount of light back-reflected). These back-reflected flux components contribute to the total reflected flux, while they represent a loss for the transmitted flux. The picture in figure 5.1 illustrates this with clear films, and various other examples will be shown at the end of this chapter. We will first describe the multiple reflections process between two plates in order to express the angular spectral reflectance and transmittance of this multilayer, then we will extend the line of reasoning to N plates using an iterative method. We will also introduce the method based on flux transfer matrices and finally show that

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103

FIG. 5.1 – Four clear polymer films partially overlap each other, showing the increase in reflectance as the number of superimposed films increases. these methods can extend to non-symmetric plates, i.e., plates having different reflectances on both sides.

5.1.1

Angular Reflectance and Transmittance of Two Plates

Consider two non-scattering plates, labelled 1 and 2. Each plate has an angular spectral reflectance denoted by Ri and an angular transmittance denoted by Ti , i ¼ 1; 2, whose expanded expressions given by equations (4.38) and (4.40), respectively, are not needed here. One just recalls that they depend on the refractive index of the material, its intrinsic spectral transmittance, as well as the angle of incidence, the wavelength, and the polarization of the light (symbol  standing for s- or p-polarization). Plate 1 is placed on top of plate 2. An incident light pencil falls on plate 1 at an angle of incidence h1 . It undergoes a multiple reflections process within the first plate, similar to the one represented in the previous chapter in figure 4.4, which results in an infinity of pencils reflected (angular spectral reflectance R1 ) and transmitted (angular spectral transmittance T1 ), all oriented according to h1 . The pencils reflected contribute to the global reflected flux. The pencils transmitted then fall on the second plate, and each one undergoes again a multiple reflection process within the second plate similar to the one that occurs in the first plate, after what many pencils are reflected back [angular spectral reflectance R2 ] into the air layer and will fall again on the first plate, and many other pencils will be transmitted [spectra angular transmittance T2 ] into air at the opposite side, contributing to the global transmitted flux. one can pursue this description indefinitely. Figure 5.2 shows these light transfers between the two plates, as well as the flux components exiting at each side. By summing up the flux components exiting at the front side and dividing the result by the incident flux, one obtains the frontside angular spectral reflectance for the considered polarization, denoted by R1;2 : 2 R1;2 ¼ R1 þ T1 R2

1 X k¼0

½R1 R2 k

ð5:1Þ

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104

Since the geometric series converges towards ð1  R1 R2 Þ1 , one obtains: R1;2 ¼ R1 þ

2 T1 R2 1  R1 R2

ð5:2Þ

Likewise, by summing up the flux components exiting at the back side and dividing them by the incident flux, one obtains the front-side angular spectral transmittance T1;2 for the considered polarization: T1;2 ¼ T1 T2

1 X

½R1 R2 k ¼

k¼0

T1 T2 1  R1 R2

ð5:3Þ

FIG. 5.2 – Multiple reflections between two non-scattering plates illuminated by directional flux F (polarized s or p) at a given angle θ1. The blue arrows represent light transfers, not optical paths; all angular reflectances and transmittances are expressed at angle θ1 and are spectral quantities. In the most general case, the reflectances and transmittances for light coming from the back side, denoted by R0i and Ti0 , can be different from the ones coming from the front side, R0i and Ti , i ¼ 1; 2 (see §5.1.2).

One can easily show that when the incident light pencil comes from the back side at the same angle h1 , the angular spectral reflectance of the pile becomes: R2;1 ¼ R2 þ

2 T2 R1 1  R1 R2

ð5:4Þ

and the angular spectral transmittance remains the same as when illuminated from the front side.

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105

As explained in the previous chapters, for unpolarized incident light, the angular spectral reflectance or transmittance is the average of those attached to the two polarizations.

5.1.2

Generalization to Non-Symmetric Plates

In order to be more general, one can extend the method to non-symmetric components, i.e., plates with different reflectances on both sides. Some kinds of optical filters, glass plates used in modern buildings that are coated with a thin film on one side for thermal protection, or films printed with inks belong to this category. We will therefore use the following terminology: one side is the front side, the other one is the back side; each plate labelled i is characterized by its front-side reflectance Ri , forward transmittance Ti , back-side reflectance, R0i , and backward transmittance Ti0 . These four terms are generically designated as transfer factors; they are angular and spectral functions, even though their dependence on angle of incidence and wavelength is not written in order to shorten the equations. If one computes again the sums of flux components on each side, this time by distinguishing the angular spectral reflectances and transmittances on each side of the plates as shown in figure 5.2, one obtains the following expressions which generalize equations (5.2)–(5.4): 0 T1 T1 R2 ; 0 1  R1 R2 0 T2 T2 R01 ¼ R02 þ ; 1  R01 R2

T1 T2 1  R01 R2 0 0 T1 T2 ¼ 1  R01 R2

R12 ¼ R1 þ

T12 ¼

R012

0 T12

ð5:5Þ

NB: These formulae (5.5) are comparable to Kubelka’s compositional formulae introduced by Kubelka in 1954 for stacked layers of strongly scattering media (see §6.8.1).

5.1.3

Generalization to N Plates: Iterative Model

If one adds a third plate labelled 3 at the back side of the two plates, one can describe the multiple reflection process as previously, between the first two plates and the third one. The resulting expressions for the angular reflectances and transmittances would be similar to the ones given by equation (5.5) by replacing the transfer factors labelled 1 with the ones labelled “1, 2” and the transfer factors labelled 2 with the ones labelled 3. This procedure can be repeated iteratively by adding a fourth plate, a fifth plate, etc.

5.1.4

Generalization to N Plates: Flux Transfer Matrix Model

We introduce here a more convenient method to obtain the transfer factors of a pile of plates, based on a matrix formalism. This formalism relies on flux transfer

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106

matrices which relate the incoming and outgoing spectral fluxes, often denoted as I(λ) for the flux propagating forwards, and J(λ) for the fluxes propagating backwards, as illustrated by figure 5.3. Using the notations for the fluxes defined in the figure (all being spectral functions and forming the same angle h1 with the normal of the sample, as previously), one can write the following equations: ( 0 J0 ¼ R1 I0 þ T1 J1 ð5:6Þ 0 I1 ¼ T1 I0 þ R1 J1 which can also be written this way: ( 0 R1 I0  J0 ¼ T1 J1 T1 I0 ¼ I1  R01 J1 or as a vector equation:  R1 T1

1 0



I0 J0



 ¼

0 1

0 T1 0 R1



ð5:7Þ

I1 J1

 ð5:8Þ

FIG. 5.3 – Fluxes incoming to or outgoing from two non-symmetric plates, and flux transfers (reflectances, transmittances) by each plate. The arrows, here, just indicate forward or backward directions, not the direction of propagation of rays, which all make an angle h1 with the normal of the plates, as featured in figure 5.2). All quantities are spectral functions, defined for one polarization component s or p. If transmittance T1 is non-zero (which is always the case with plates surrounded by a medium with lower refractive index, e.g., air, but may be not true in the general case), one can invert the matrix on the left and left-multiply it on both members of equation (5.8):

Transparent Multilayers: Two-Flux Models for Directional Light 

I0 J0



1 ¼ T1



0 T1

1 R1



0 1

0 T1 0 R1



I1 J1



107 

¼ M1

I1 J1

 ð5:9Þ

Matrix M1 is the flux transfer matrix attached to the first plate, for the considered wavelength, angle of incidence θ1 and polarization component s or p:   1 1 R01 M1 ¼ ð5:10Þ 0  R1 R01 T1 R1 T1 T1 It comes naturally that the inward and outward fluxes around the second layer are related by the vector equation     I I1 ¼ M2 2 ð5:11Þ J1 J2 where matrix M2 attributed to second plate is defined as M1 with the corresponding transfer factors labelled 2. The combination of equations (5.9) and (5.11) yields     I I0 ¼ M1 M2 2 ð5:12Þ J0 J2 The product of the individual flux transfer matrices is once again a flux transfer matrix since it relates forward and backward fluxes. It is the flux transfer matrix attached to the two plates considered jointly, denoted by M1;2. Likewise, the flux transfer matrix representing N plates is given by the product of the individual transfer matrices, in which the left-to-right position of the matrices reproduces the front-to-back position of the corresponding plates: M123:::N ¼ M1 M2 M3 :::MN ð5:13Þ   From the entries of the obtained matrix M ¼ mij , ði; j ¼ 1; 2Þ, one deduces the four spectral angular transfer factors as follows: R ¼ m21 =m11 ; R0 ¼ m12 =m11 ;

T ¼ 1=m11 T0 ¼ detðMÞ=m11

ð5:14Þ

where “det” designates the matrix determinant operator. In exercise 17, we propose to verify that these equation (5.14), applied to matrix M1;2 , product of M1 and M2 , give as expected the reflectance and transmittance expressions (5.5) obtained by describing the multiple reflection process. NB: the flux transfer matrix of a symmetric plate is of the form   1 1 R1 ð5:15Þ 2  R21 T1 R1 T1 0 ¼ T1 . However, in a pile of different plates, even if the since R01 ¼ R1 and T1 plates are symmetric, the resulting pile is not symmetric anymore, as illustrated with superimposed colored films in figure 5.4. The non-commutativity of the flux transfer matrices renders this non-symmetry.

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FIG. 5.4 – Stackings of cyan and green films, where the cyan film is (a) in front or (b) behind the green film.

Similar matrices, often called “transfer-matrices” in reference to the scattering matrices and scattering transfer matrices in electronics which also have similar properties, are used in wave optics to model the propagation of electric fields [90] or incoherent light beams [123, 167] in thin multilayers.

5.2

Piles of Identical Plates

A special case of the model introduced above, viz., the case where all the plates stacked are identical, is worth considering, not just for the attention that has been paid to it in the history of optics over the last three centuries as evoked in the introduction of this chapter. On a theoretical point of view, the flux transfer matrix model demonstrates its capacity to provide easily closed-form reflectance and transmittance formulae. On a practical point of view, one can easily observe the influence of the different parameters on the reflection and transmission properties of the pile: (1) the number of plates, overall “thickness” of the pile, (2) their refractive index, which determines the amount of light reflected at the interface, thereby their ability to back-reflect light (specularly) at the input side, and (3) their intrinsic spectral transmittance tk , which determines the amount of light being absorbed. We will see in chapter 6 that three parameters have comparable influence on the reflection and transmission properties of a homogeneous layer of strongly scattering medium: thickness, spectral (back-)scattering coefficient, and spectral absorption coefficient.

5.2.1

Angular Reflectance and Transmittance

One considers here symmetric plates (non-symmetric plates will be addressed later, in §5.2.4). They are characterized by their angular spectral reflectance R121 ðh1 ; tk Þ and transmittance T121 ðh1 ; tk Þ given by equations (4.38) and (4.40), which depend on the intrinsic spectral transmittance tk of the material layer, therefore on

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109

wavelength, and depend also on the angle of incidence h1 in air and the polarization component s or p (see §4.5.1). They will be simply written R and T for shorter notations. According to the definition given in equation (5.10), the transfer matrix representing one plate is   1 1 R M ¼ ð5:16Þ T R T2  R2 It can be transformed by diagonalization as   l 0  E1 M ¼ E  0 m with

 E¼

1 aþb

1 ab

ð5:17Þ

 ð5:18Þ

and 1 1 ½1  ða þ bÞR  m ¼ ½1  ða  bÞR  T T pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ R2  T2 a¼ b ¼ a2  1 2R

l¼

ð5:19Þ

which are to be evaluated for each wavelength, angle of incidence and polarization component. The transfer matrix representing a pile of N films, obtained by multiplying these N identical transfer matrices, is therefore MN  . The matrix diagonalization (5.17) yields:  N  l 0 N ð5:20Þ  E1 M ¼ E  0 mN After computation, and using equation (5.14), the bottom-left entry of this matrix divided by its top-left entry provides the following closed-form reflectance expression: RN ðh1 ; kÞ ¼

m N  lN ða þ bÞmN  ða  bÞlN

ð5:21Þ

and the inverse of its top-left entry provides the following transmittance expression: TN ðh1 ; kÞ ¼

2b ða þ bÞmN  ða  bÞlN

ð5:22Þ

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110

These equation (5.21) and (5.22) have been given different equivalent forms since the works by Stokes in 1862 [224], then Tuckerman in 1947 [234]. Mazauric recently proposed the following equations [158]: RN ðh1 ; kÞ ¼

sinhðN log mÞ a sinhðN log mÞ þ b coshðN log mÞ

ð5:23Þ

TN ðh1 ; kÞ ¼

b a sinhðN log mÞ þ b coshðN log mÞ

ð5:24Þ

where the hyperbolic sine and cosine functions are defined as 1 1 sinhðx Þ ¼ ðex  ex Þ and coshðx Þ ¼ ðex þ ex Þ 2 2

ð5:25Þ

Equations (5.23) and (5.24) are characteristic of the generalized two-flux model in its discrete form, whose interest relies on their similarity with the so-called Kubelka–Munk reflectance and transmittance formulae (6.6) and (6.7) for a homogenous diffusing layer. The Kubelka–Munk model (1931) is historically known as the classical 2-flux model (in its continuous form: the integer number N is replaced with a real number related to depth into the layer, see section 6.1). As N increases, the pile contains more layers and interfaces and thus becomes more absorbing and more reflecting, as noticed at the beginning of this chapter through the picture in figure 5.1 and shown by the top and bottom graphs in figure 5.5. The spectral reflectance asymptotically increases until a limit value corresponding to the spectral reflectance of an infinite pile, denoted by R1 ðh1 ; kÞ. Since the term raised at the power N in equation (5.21) is smaller than 1, it tends to zero. The angular spectral reflectance of the infinite pile is therefore R1 ðh1 ; kÞ ¼ a ðh1 ; kÞ  bðh1 ; kÞ

ð5:26Þ

In opposition to reflectance, transmittance decreases as the number of plates increases, because light has less chance to cross the pile without being absorbed or back-reflected from interfaces. It approaches zero more rapidly as the layers are more absorbing, as shown by the middle graph in figure 5.5. The absorptance of the pile, for each wavelength, angle of incidence and polarization, is expressed as AN ðh1 ; kÞ ¼ 1  RN ðh1 ; kÞ  TN ðh1 ; kÞ ða  1Þ sinhðN log mÞ þ b½coshðN log mÞ  1 ¼ a sinhðN log mÞ þ b coshðN log mÞ

ð5:27Þ

Absorptance increases as the number of plates increases, and as the t value decreases (i.e., the absorptance of one plate increases). However, it is capped at the value 0.96, which corresponds to 1  R12 ð0 Þ: whatever the number of plates and

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111

FIG. 5.5 – Reflectance, transmittance and absorptance at normal incidence of 1 plate of refractive index 1.5, and piles of 2, 3, 4, 5, 10, 100 identical plates as functions of the intrinsic spectral transmittance of one plate.

their absorptance, at least 4% of the incident light (for n = 1.5) is reflected from the first air-plate interface and cannot be absorbed. The case of non-absorbing plates, i.e., t = 1, is a limit case of the model. Since R þ T ¼ 1 for every angle of incidence and polarization component, then a ¼ l ¼ m ¼ 1, b ¼ 0, and equations (5.23) and (5.24) are undetermined. Nevertheless, the expressions for these parameters tend to simple formulae that we propose to derive in exercise 16. They were first demonstrated by Lambert in 1760 (without consideration of the polarization) [138]:

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112

NR ðh1 Þ 1 þ ðN  1ÞR ðh1 Þ 1  R ðh1 Þ TN ðh1 Þ ¼ 1 þ ðN  1ÞR ðh1 Þ

RN ðh1 Þ ¼

ð5:28Þ

These formulae can also be expressed in terms of the Fresnel angular reflectance of one interface, by replacing R with its expression given by equation (4.20): 2NR12 ðh1 Þ 1 þ ð2N  1ÞR12 ðh1 Þ 1  R12 ðh1 Þ TN ðh1 Þ ¼ 1 þ ð2N  1ÞR12 ðh1 Þ

RN ðh1 Þ ¼

ð5:29Þ

Angular reflectance RN ðh1 Þ of a pile of non-absorbing plates is plotted in figure 5.6a as a function of angle h1 for different N values, for s-polarized, p-polarized, and unpolarized incident light. When the plates are non-absorbing, an infinite number of plate in the pile yields R1 ð1Þ ¼ 1: all the incident light is specularly reflected and the pile behaves as a perfect mirror (on a photometric point of view, not on an imaging point of view because the exiting rays undergo a transversal shift: an infinity of shifted images are superimposed or juxtaposed). However, at the Brewster angle hb ¼ arctanðn Þ, the reflectance for the p polarized component is always zero, and the reflectance for natural incident light cannot exceed 0.5 (see figure 5.6). In practice, clear plates are rarely perfectly non-absorbing. If their absorptance can sometimes be neglected for one plate, absorption is more evident with a pile of several plates. One can see in figure 5.6b that with a pile of weakly absorbing plates of intrinsic spectral transmittance t = 0.99, the reflectance of an infinite pile of plates converges to 63% only at normal incidence. If absorption is strong (low t values), the plate may be opaque and the reflectance, for any number of plates, does not exceed the reflectance of the first air-plate interface (e.g., R12 ð0 Þ = 0.04 at normal incidence for plates of refractive index 1.5 surrounded by air). Figure 5.7 shows similar graphs for the transmitted light. Transmittance always decreases as the number of plates increases. The decrease can be very fast when the plates are absorbing.

5.2.2

Degree of Polarization

Piles of transparent plates can be used as polarizers for the transmitted light [22], with the advantage of having almost spectrally invariant properties. The unpolarized incident light that falls on the pile at non-normal incidence is partially polarized on each refraction. The degree of polarization of the light transmitted through the pile therefore depends on number N of plates, angle of incidence h1 , and intrinsic spectral transmittance tλ of each plate. It is defined from s-polarized and p-polarized spectral transmittances expressed in equation (5.24), or by equation (5.29) if the plates are non-absorbing:

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113

FIG. 5.6 – Angular reflectances of piles of (a) non-absorbing plates, (b) weakly absorbing plates and (c) absorbing plates of refractive index 1.5, for p-polarized, s-polarized and unpolarized incident light as functions of the angle of incidence h1 . In each graph, the numbers N of plates in the pile are 1, 2, 3, 4, 5, 10, 100, and 1000.

114

Optical Models for Material Appearance

FIG. 5.7 – Angular transmittance of piles of (a) non-absorbing plates, (b) weakly absorbing

plates and (c) absorbing plates of refractive index 1.5, for p-polarized, s-polarized and unpolarized incident light as functions of the angle of incidence h1 . In each graph, the numbers N of plates in the pile are 1, 2, 3, 4, 5, 10, 100, and 1000.

Transparent Multilayers: Two-Flux Models for Directional Light

DOPðN ; h1 ; kÞ ¼

115

Ts;N ðh1 ; kÞ  Tp;N ðh1 ; kÞ Ts;N ðh1 ; kÞ þ Tp;N ðh1 ; kÞ

ð5:30Þ

At the Brewster angle, hb ¼ arctanðn Þ where n is the refractive index of the plates, p-polarized reflectance of each air-plate interface is zero, and its p-polarized transmittance is therefore 1. If the plates are non-absorbing, using Rs12 ðhb Þ ¼ 2 ½ðn 2  1Þ=ðn 2 þ 1Þ and Rp12 ðhb Þ ¼ 0 in equation (5.29), one obtains: 2

DOPðN ; hb Þ ¼

1 þ ðN  1ÞRs12 ðhb Þ N ð n 2  1Þ ¼ 1 þ ð2N  1ÞRs12 ðhb Þ 4n 2 þ N ðn 2  1Þ2

1  Rs12 ðhb Þ 1 n2 ¼ þ T N ð hb Þ ¼ 1 þ ð2N  1ÞRs12 ðhb Þ 2 2n 2 þ N ðn 2  1Þ2

ð5:31Þ

As number N of plates increases, the DOP of the transmitted light increases but transmittance TN ðhb Þ of the pile decreases. The graph of figure 5.8 helps us find an optimal number of plates so as to maximize both DOP and transmittance of the pile: for each number of plates, the transmittance of the pile is plotted versus the DOP

FIG. 5.8 – Angular transmittance vs. DOP of piles of 1–1000 non-absorbing plates of refractive index 1.5.

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Optical Models for Material Appearance

while the angle of incidence is varied from 0 to π/2. The highest DOP is obtained with an illumination at the incident angle indicated by a purple ring dot, but an illumination at the Brewster angle (indicated by green ring dots) gives a good trade-off for maximizing both the pile transmittance and the DOP, which is appreciable for a polarizer. For example, with 30 non-absorbing plates of refractive index n = 1.5, one obtains a DOP of 0.84 and a transmittance of 0.54. Plane-parallel plates polarizers can also be designed by spacing the plates with a large distance in order to prevent multiple reflections between the plates [8]. At the Brewster angle, the DOP of the transmitted light and the transmittance of the pile are then given by  N N N N Tp1 ðh1 Þ  Ts1 ð h1 Þ 1 þ n 4 ð2n 2 Þ DOPðN ; hb Þ ¼ N ¼ N ðh Þ Tp1 ðh1 Þ þ Ts1 ð1 þ n 4 ÞN þ ð2n 2 ÞN 1 ð5:32Þ i 1 1  2n 2 N 1h N N TN ðhb Þ ¼ Tp1 ðh1 Þ þ Ts1 ðh1 Þ ¼ þ 2 2 2 1 þ n4 This configuration gives the same performances in terms of transmittance vs. DOP as when the plates are close to each other, but the number of plates needed to achieve a certain DOP is much lower. For example, a DOP of 0.84 and a transmittance of 0.54 are obtained with only 8 clear plates of refractive index 1.5 if they are spaced from each other, whereas 30 plates are necessary to obtain the same values if they are not spaced.

5.2.3

Bi-Hemispherical Reflectance and Transmittance

When the lighting is Lambertian, the spectral bi-hemispherical reflectance of the pile of N identical plates is given by Z p=2

1 RpN ðh1 ; kÞ þ RsN ðh1 ; kÞ sin 2h1 dh1 rN ðkÞ ¼ ð5:33Þ 2 h¼0 The spectral bi-hemispherical transmittance tN ðkÞ is given by a similar formula. Both rN ðkÞ and tN ðkÞ depend on number N of plates, their refractive index and intrinsic spectral transmittance t ðkÞ. Their variation according to N is plotted in figure 5.9, for refractive index n2 = 1.5 and various t values. For comparison, the reflectance and transmittance at normal incidence for these N, n2 and t values are also plotted. The number of plates beyond which the reflectance converges towards the limit value R1 ð0Þ or r1 depends on the t value, therefore on the absorptance: as the absorptance increases, the amount of light reaching the deepest plates in the pile is much lower. The pile of non-absorbing plates (t = 1) has not reached its limit value 1 with 1000 plates, whereas the limit value is reached with 100 plates when t = 0.99, and with 10 plates when t = 0.9. These curves help to estimate the spectral transmittance of insulated glass. For example, N = 2 corresponds to a double panned window. The bi-hemispherical

Transparent Multilayers: Two-Flux Models for Directional Light

117

FIG. 5.9 – Reflectance (left) and transmittance (right) of piles of identical plates as functions of the number of plates, for directional illumination at normal incidence (dashed lines) and Lambertian illumination (solid lines), various intrinsic spectral transmittance values, and a refractive index 1.5.

transmittance corresponds to the case where the light is diffuse (e.g., overcast weather), and the normal transmittance the case where direct sunlight falls on the window at normal incidence.

5.2.4

Generalization to Non-Symmetric Plates

The equations presented in the previous section can be generalized to non-symmetric plates, for which R0 ðh1 ; kÞ 6¼ R ðh1 ; kÞ and T0 ðh1 ; kÞ 6¼ T ðh1 ; kÞ. The line of reasoning developed in §5.2.1 can be repeated using the flux transfer matrix   1 1 R0 ð5:34Þ M ¼ T R T T0  R R0 One obtains the following expressions for parameters μ, ν, a, and b: pffiffiffiffiffiffiffiffiffiffiffi

1 1  R R0 ða þ bÞ ; 0 T 1 þ R R0  T T0 pffiffiffiffiffiffiffiffiffiffiffi a¼ 2 R R0

l¼

pffiffiffiffiffiffiffiffiffiffiffi

1 1  R R0 ða  bÞ T pffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ a2  1 m¼

ð5:35Þ

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118

The reflectance and transmittance expressions become (see Ref. [158], p. 68): sffiffiffiffiffiffi R sinhðN logðcmÞÞ RN ðh1 ; kÞ ¼ 0 R a sinhðN logðcmÞÞ þ b coshðN logðcmÞÞ bcN a sinhðN logðcmÞÞ þ b coshðN logðcmÞÞ R0 ðh1 ; kÞ R0N ðh1 ; kÞ ¼ RN ðh1 ; kÞ  R ðh1 ; kÞ TN ðh1 ; kÞ ¼

ð5:36Þ

0 TN ðh1 ; kÞ ¼ TN ðh1 ; kÞ=c2N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with c ¼ T ðh1 ; kÞ=T0 ðh1 ; kÞ.

5.2.5

Invariance of Parameter a

Parameter a defined in equations (5.19) or (5.35) is expressed in terms of the transfer factors of one plate. This parameter, always larger than 1, depicts the absorptance of the plate: it is nearly proportional to the optical thickness (product of the absorption coefficient and real thickness, see §4.5.1) when this latter is below 1. We can show that in the defining expression for a, the transfer factors attached to one plate can be replaced with the ones attached to N identical plates (denoted by aN in this case): a¼

0 1 þ R R0  T T0 1 þ RN R0N  TN TN pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ aN 2 R R0 2 RN R0N

ð5:37Þ

This invariance property of parameter a with respect to the pile “thickness”, valid for each wavelength, angle of incidence and polarization component, is characteristic of the discrete 2-flux model. It is verified each time the plates satisfy correctly the assumptions of the model, as in the illustrative examples proposed in the next section. In contrast, noticeable variations in aN according to N can indicate that slight scattering, or fluorescence occur. Scattering induces a change in the direction of rays, thereby of the Fresnel reflectances at the interfaces and the attenuation across the plates. Fluorescence is the conversion of the incident radiation at some wavelengths (excitation band) into radiation at longer wavelengths (emission band). In photometric measurements, the emitted light is confounded with the lights reflected and transmitted, unless a spectrofluorimeter is used to relate quantitatively the excitation and absorption spectra. In general, parameter aN varies in the spectral band of emission.

5.3

Layers of Different Refractive Indices in Optical Contact

The flux transfer matrix model previously introduced is not restricted to piles of transparent plates separated by layers of air; it can also be used with any

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transparent multilayer containing several layers of different refractive indices, different absorption coefficients, and different thicknesses (provided these thicknesses are larger than the temporal coherence of light). The model thus uses flux transfer matrices representing each layer and each interface, by taking into account the specific orientation of light in each layer, determined by its refractive index. The angular reflectance and transmittance of the multilayer can be easily obtained with this method.

5.3.1

Flux Transfer Matrices for Layers and Interfaces

The components of a transparent multilayer, i.e., layers and interfaces, are represented by flux transfer matrices, defined as in equation (5.10) for directional light polarized s or p (again denoted by the generic symbol *). They are expressed in terms of angular transfer factors. The flux transfer matrix representing an interface is defined in terms of the Fresnel reflectance and transmittance, or equivalently in terms of the Fresnel reflectance at the front side only thanks to the properties (1.11), (1.12) and (1.13):   1 1 Rij ðhi Þ Fij ðhi Þ ¼ ð5:38Þ 1  Rij ðhi Þ Rij ðhi Þ 1  2Rij ðhi Þ Thanks to the reciprocity property of the Fresnel formulae, one also may write   ð5:39Þ Fji hj ¼ Fij ðhi Þ The flux transfer matrix representing a transparent layer is expressed in terms of the layer’s angular transmittance (its reflectance is zero), which is itself expressed in terms of the intrinsic spectral transmittance as detailed in equation (4.31). It depends on wavelength, but generally not on polarization:   1 1 0 Lðh; tk Þ ¼ 1=cos h ð5:40Þ 2=cos h 0 tk tk These flux transfer matrices can be multiplied from left to right by respecting the front-to-back order of the different components as well as the orientation of the incident light for each of them. According to Snell’s law, the orientation of light in each layer of refractive index ni is related to the angle of incidence on the front side h1 in air as hi ¼ arcsinðn1 sin h1 =ni Þ

ð5:41Þ

The four transfer factors are deduced from the transfer matrix of the multilayer thanks to equation (5.14), for the considered angle of incidence and polarization. For unpolarized light, the two transfer factors attached to the two polarizations s and p are averaged, and for a Lambertian illumination, the bi-hemispherical transfer factors are obtained by integrating the angular transfer factors in a similar way as in equation (2.13).

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The transparent multilayer is a specular reflector. The reflected spectral radiance flows in a geometrical extent similar to the one of the incident spectral

ðRÞ Lk ð h1 Þ

ð1Þ

radiance Lk ðh1 Þ. By denoting as Rðh1 ; kÞ the spectral angular reflectance of the multilayer for unpolarized light, one thus has ðRÞ

ðRÞ

Lk ðh1 Þ ¼ Rðh1 ; kÞLk ðh1 Þ

ð5:42Þ

ðT Þ

Regarding the transmitted spectral radiance Lk ðhN Þ, the refractions may induce a changing of geometrical extent, as explained for the case of one interface in §4.1. According to whether the transmitted flux is spread to a larger solid angle or concentrated into a smaller solid angle, the transmitted spectral radiance is lower, respectively higher than the incident spectral radiance multiplied by the angular spectral transmittance T ðh1 ; kÞ of the multilayer. The change in extent is taken into account by a geometrical factor which depends only on refractive indices n1 and nN of the surrounding media at the front and back sides:  2 nN ðT Þ ð1Þ Lk ð hN Þ ¼ T ðh1 ; kÞLk ðh1 Þ ð5:43Þ n1

5.3.2

Examples of Multilayers

We have already derived in §4.6.1 the angular reflectance and transmittance expressions for a transparent plate surrounded by air by adding the light components exiting the plate on each side and reducing geometric series. We can equivalently derive these formulae using flux transfer matrices, i.e., by multiplying the matrices representing the front interface, the layer, and the back interface: M121 ðh1 ; kÞ ¼ F12 ðh1 ÞLðh2 ; tk ÞF21 ðh2 Þ

ð5:44Þ

where h2 is given by equation (5.41). When two layers of same refractive index n2 , with respective intrinsic spectral transmittances t2 ðkÞ and t3 ðkÞ, are in optical contact, i.e., the air between them is removed or replaced with a medium of refractive index n2 , then equation (5.44) applies by simply replacing Lðh2 ; tk Þ with the product of the two transfer matrices representing the two layers, L2 ðh2 ; t2 ðkÞÞL3 ðh2 ; t3 ðkÞÞ, or equivalently by transfer matrix Lðh2 ; t2 ðkÞt3 ðkÞÞ: M1231 ðh1 ; kÞ ¼ F12 ðh1 ÞLðh2 ; t2 ðkÞt3 ðkÞÞF21 ðh2 Þ

ð5:45Þ

This configuration is illustrated in figure 5.10 by a stacking of two films, the cyan and green films already displayed in figure 5.4, where this time the two films are in optical contact thanks to a layer of gel instead of an air layer. We can notice the presence of some brighter zones where some air remains. Since the relative refractive index of the gel-film interfaces is close to unity, their reflectance approaches zero and the total reflectance of the pile is strongly decreased in comparison to what was

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FIG. 5.10 – Stacking of green and cyan films, the same as those in figure 5.4, where a layer of gel replaces the layer of air between the films. (a) the cyan film is behind the green film, (b) it is in front of the green film. Same color is displayed in areas with gel, as shown by the inserts. observed with air between the films. We can also see that the pile becomes symmetric: it has the same reflectance on both sides, and therefore displays same color, as shown by the two pieces of image placed close from each other at the center of the figure. This symmetry is found in the fact that the transfer matrices representing transparent layers are commutative since they are diagonal matrices: L2 ðh; t2 ðkÞÞL3 ðh; t3 ðkÞÞ ¼ L3 ðh; t3 ðkÞÞL2 ðh; t2 ðkÞÞ ¼ Lðh; t2 ðkÞt3 ðkÞÞ

ð5:46Þ

If the two layers, still in optical contact, have different refractive indices, n2 and n3 , the flux transfer matrix becomes M1231 ðh1 ; kÞ ¼ F12 ðh1 ÞLðh2 ; t2 ðkÞÞF23 ðh2 ÞLðh3 ; t3 ðkÞÞF31 ðh3 Þ

5.3.3

ð5:47Þ

Optical Characterization of Liquids

The optical characterization of a colored, non-scattering liquid can be achieved by filling a clear parallelepipedal cuvette with the liquid and placing it into a spectrophotometer in order to measure its spectral transmittance factor. The walls of the cuvette and the liquid volume form a transparent multilayer to which the model presented in the previous sections applies. An analytical expression relates the spectral absorption coefficient of the liquid with the measured transmittance factor. Usually, the setup ensures that the detected light beam is perpendicular to the walls of the cuvette. The reflectance and transmittances of the components (interfaces of the walls of the cuvette, and liquid) are therefore evaluated at normal incidence, and are independent of polarization. A similar configuration, i.e., a bottle of perfume, is studied in exercise 19. With the refractive index of cuvette material, n2 , and the refractive index of the liquid, n3 , one can compute the Fresnel reflectance of the interfaces bounding the walls of the cuvette, at normal incidence: for the external

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interfaces, it is R12 ¼ ðn2  1Þ2 =ðn2 þ 1Þ2 , and for the internal interfaces, R23 ¼ ðn2  n3 Þ2 =ðn2 þ n3 Þ2 . Regarding the liquid, its intrinsic spectral transmittance is related to its absorption coefficient according to Beer’s law: t ðkÞ ¼ eaðkÞd , with d the internal width of the cuvette. As shown in exercise 19, the spectral ^ ðkÞ of the filled cuvette is written: transmittance factor T ^ ð kÞ ¼ T

ð1  R12 Þ2 ð1  R23 Þ2 eaðkÞd ð1  R12 R23 Þ2 ðR12 þ R23  2R12 R23 Þ2 e2aðkÞd

ð5:48Þ

Inversely, the spectral absorption coefficient of the liquid can be written in terms ^ 1 ðkÞ of the measured spectral transmittance factor T sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 ð1  R12 R23 Þ2 aðkÞ ¼  log AðkÞ þ A2 ðkÞ þ ð5:49Þ d ðR12 þ R23  2R12 R23 Þ2 with Að kÞ ¼

ð1  R12 Þ2 ð1  R23 Þ2 ^ ð kÞ 2ðR12 þ R23  2R12 R23 Þ2 T

ð5:50Þ

The refractive index of the cuvette material, n2 ðkÞ, is easy to estimate by mea^ 0 ðkÞ of the empty cuvette: suring the spectral transmittance factor T qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 0 ð kÞ þ 1 þ 2T ^ 0 ðkÞ  3T ^ 2 ð kÞ 1þT 0 n2 ð k Þ ¼ ð5:51Þ ^ 0 ð kÞ 2T Estimating the refractive index of the liquid is possible by measuring both spectral transmittance and reflectance factors of the cuvette filled with the liquid. The matrix method provides a systems of two non-linear equations, functions of n3 ðkÞ and aðkÞ, which can be solve numerically for each wavelength.

5.3.4

Total Reflections

At normal incidence, light is perpendicular to all layers, i.e., h1 ¼ h2 ¼ h3 ¼    ¼ 0. As h1 moves away from the normal, all angles hi also move away from the normal. If one of the layers’ indices is such that n1 sin h1 =ni [ 1. Light is totally reflected from the ith interface. As long as n1 sin h1 is larger than each of the multilayer refractive indices ni , the light pencil can cross the multilayer; the incident rays that are not subject to total reflection are contained into a cone of half-angle   a ¼ arcsin minðni Þ=n1 i

ð5:52Þ

Figure 5.11 shows an example of clear multilayer made of glass, water and plastic, bordered by air at the back side. The light pencil comes from the glass.

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FIG. 5.11 – Transmission and total reflections of directional light in a transparent multilayer for different angles of incidence θ1. At small angles of incidence, below 41:8 ¼ arcsinð1=1:5Þ, some light can be transmitted. When h1 exceeds this value, n1 sin h1 exceeds n4 and total reflection occurs at the plastic-air interface. Then when h1 exceeds 62:5 ¼ arcsinð1:33=1:5Þ, n1 sin h1 exceeds n2 and total reflection occurs at the glass-water interface. Note that n1 sin h1 cannot exceed n3 and no total reflection occurs at the plastic-water interface. In case of total reflection at some interface, the Fresnel transmittance is zero and the transfer matrix attached to this interface, given by (5.38), is undefined, which generates a serious computational issue. In order to prevent it, one recommends using an alternative format for the transfer matrices where the forward transmittance is incorporated into the matrix in a third row and third column: 0 1 0 1 R0 M ¼ @ R T T0  R R0 0 A ð5:53Þ 0 0 T These 3 × 3 matrices can be multiplied exactly in the same way as the 2 × 2 transfer matrices without any chance of division by zero. The transfer factors are deduced from entries mij ði; j ¼ 1; 2; 3Þ of the 3 × 3 matrix in the following way: R ¼ m21 =m11

T ¼ m33 =m11

R0

2 T0 ¼ ½detðMÞ=m11 =m33

¼ m12 =m11

ð5:54Þ

Even though we will pursue with 2 × 2 transfer matrices throughout the rest of this book, by consistency with the previous chapters, one strongly recommends using these 3 × 3 transfer matrices in computational programming.

5.3.5

Nonpolarity of Directional Transmittance

One observes from equations (5.38) and (5.40) that the transfer matrices representing interfaces and transparent layers have a determinant equal to 1, as well as those representing transparent multilayers since they are the product of these

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elementary transfer matrices. Consequently, by denoting as m11 the top-left entry of M, one has the following equality: detðMÞ=m11 ¼ 1=m11

ð5:55Þ

This means that, in accordance with equation (5.14), the forward and backward transmittances are identical. This property echoes the “nonpolarity of transmittance” principle stated by Kubelka for diffusing multilayers [134]. It is true for directional light in the absence of total reflection. It is therefore always true at normal incidence. It can be seen as an expression of the Helmholtz’ reverse path principle, which asserts that the quantity of light flowing from any “point” A to any “point” B is the same as that flowing from B to A. Notice however that the non-polarity property does not apply with bi-hemispherical transmittances, which is not surprising since the bi-hemispherical geometry is not a situation where light flows between two “points”. We can remind from §4.2, for example, that the fraction of Lambertian light crossing an air-glass interface from air to glass is t12 ¼ 0:9 and the fraction crossing it from glass to air is t21 ¼ 0:4 (the refractive index of glass being assumed 1.5).

5.4

Examples

In order to illustrate the models previously presented in this chapter and observe the influence of the different parameters (e.g., refractive index, absorptance, thickness), we propose two examples based on piles of colored transparent films. The first example is based on piles of identical films, symmetric or not, where the films are separated from each other by layers of air. In the second example, we consider symmetric films separated by either air or other media.

5.4.1

Piles of Colored Films Separated by Air

To predict the reflectance or transmittance of a pile of films separated by layers of air, their respective refractive index and intrinsic spectral transmittance must be known. We have shown in §4.6.3 how to obtain these two parameters from the film reflectance and transmittance measured at normal incidence. Then we can compute the angular reflectance and transmittance of each film for polarizations s and p, then the ones for natural light by averaging the two polarized components, and finally deduce the bi-hemispherical reflectance and transmittance by integration over the hemisphere. If for any physical reason a given film is not symmetric, for example because of a thin coating on one side, the refractive index and intrinsic spectral transmittance do not suffice to characterize the optical properties of the film. Its angular reflectances and transmittance on both sides may be difficult to predict for the two polarizations from measurements at normal incidence. However, their prediction at normal incidence remains easy since polarization is not to be taken into consideration.

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Here, we propose to verify the prediction accuracy of the model through the example of a stacking of identical transparent films considered in reflection and transmission modes at normal incidence. The piles are made up of two different types of films: acetate color-impregnated films of cyan color (symmetric), and inkjet transparency films printed with green color (non-symmetric because of a shine effect on the printed side related to the refractive index of the inks, a phenomenon known as bronzing effect [101]). For each type of film, their reflectance and transmittance are measured on both sides at normal incidence (for example with a spectrophotometer based on the d:8° geometry, but the effective measurement geometry with non-scattering samples should be the 8°:8° geometry, which can be assimilated to the 0°:0° geometry, as explained in §4.6.3). Spectral reflectances and transmittances of piles of identical films are also measured and predicted wavelength by wavelength thanks to formulae (5.23) for the symmetric films, or (5.36) for the non-symmetric films. The measured and predicted spectral curves are plotted in figure 5.12. Their almost perfect superimposition for each pile shows that the model is fairly accurate, and their deviation from each other, expressed in terms of colorimetric distance in the CIE 1976 L*a*b* color space, is less than 0.4 unit of the ΔE94 metric. We clearly observe the general trends enounced in the introduction of this chapter: the reflectance of the pile increases each time a film is added. This is due to the light that is back-reflected from the two interfaces of the added film, even though absorption may attenuate this increase in some spectral domains, and even cancel it at certain wavelengths. The cyan acetate films, for example, are strongly absorbing around 600 nm and the reflectance does not increase in this spectral domain: it remains close to the floor value of 4%, which is the reflectance at normal incidence of the first air-film interface. On the contrary, the transmittance decreases each time a film is added because of the back-reflection of light at its interfaces and the attenuations by absorption within the polymer. In order to verify the invariance property of parameter a(λ), the spectral curves of aN ðkÞ have been plotted for all samples ð1  N  16Þ: they are almost perfectly superimposed. This indicates that the films satisfy rather well the assumptions of the model: the refractive index, absorption coefficient and thickness are homogeneous, and the interfaces are smooth. One also sees that parameter a is larger when the films are more absorbing, i.e., at wavelengths where the transmittance is the lowest.

5.4.2

Piles of Colored Films Separated by Different Media

It is interesting to observe how the spectral reflectance and transmittance of piles of identical colored films vary when the air layers between the films are replaced with other clear materials, for examples liquids. Let us study this situation with the cyan acetate films presented in the previous section (left of figure 5.12), by introducing between the films a liquid of refractive index n3 ¼ 1:4. The reflectances and transmittances of the piles, measured at normal incidence, are displayed in figure 5.13. Moreover, reflectances and transmittances have been simulated for a liquid having the same refractive index as that of the films, n3 ¼ 1:54.

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FIG. 5.12 – Spectral reflectance RN(λ) and transmittance TN(λ) at normal incidence of cyan acetate films (left column), and inkjet transparency films printed with a green color (right column). The colored numbers indicate the number N of stacked films. The spectral quantities measured on one film are plotted in black solid lines, those measured on N ≥ 2 films in red solid lines, and those predicted for N ≥ 2 in dashed lines. The spectral curves for parameter aN, computed and plotted for each number N of films, are all superimposed, which illustrates the invariance property of this parameter.

The flux transfer matrix model enables predicting easily the reflectance and transmittance of the different piles. First of all, refractive index n2 and intrinsic spectral transmittance t(λ) of one film are deduced from reflectance and transmittance measurements at normal incidence, by following the method explained in §4.6.3. Then, the flux transfer matrices of air-film interfaces F12 ðh1 Þ, the ones of

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air–liquid interfaces F23 ðh2 Þ, and the ones of acetate layer Lðh2 ; tk Þ are computed according to the considered incidence angle. Finally, the transfer matrix representing a piles of N films is given by one of the three equations below, according to whether the films are separated by air:

N 1 Lðh2 ; tk ÞF12 ðh1 Þ ð5:56Þ SN ðh1 ; kÞ ¼ F12 ðh1 Þ Lðh2 ; tk ÞF212 ðh2 Þ or by a liquid of refractive index n3 ¼ 1:4:

N 1 Lðh2 ; tk ÞF12 ðh1 Þ: SN ðh1 ; kÞ ¼ F12 ðh1 Þ Lðh2 ; tk ÞF223 ðh2 Þ

ð5:57Þ

or by a liquid with the same refractive index as the films, n2 ¼ 1:54:   SN ðh1 ; kÞ ¼ F12 ðh1 ÞLN ðh2 ; tk ÞF12 ðh1 Þ ¼ F12 ðh1 ÞL h2 ; tkN F12 ðh1 Þ

ð5:58Þ

The angular spectral reflectances and transmittances of the piles are deduced from the entries of the transfer matrices as indicated by equation (5.14). Their analytical expressions are too extensive to be reproduced here, but their numerical computation does not pose any difficulty. Since the surrounding air has a lower refractive index than the acetate and the liquid, no total reflection can occur, whatever the angle of incidence. The optical effect of the liquid relies on the fact that the relative refractive index of the interfaces is decreased, except for the two external interfaces since the pile remains surrounded by air. Hence, the back-reflections of light at these interfaces are strongly attenuated in comparison with what one obtains when the films are separated by air. The reflectance is lower and the corresponding color darker, as already noticed by comparing the pictures displayed in figures 5.4 and 5.10. In opposition, the transmittance is larger. In the case where the liquid has same refractive index as the films, the internal interfaces have no optical effect anymore: the multilayer is similar to a homogenous film of thickness multiplied by N. Let us analyze how the spectral reflectances and transmittance at normal incidence vary according to the different parameters of the model [99], by observing the spectra plotted in figure 5.13. The interaction between light and film piles may be summarized as the combination of an absorption phenomenon within the layers and a back-reflection phenomenon at the interfaces. The absorption phenomenon attenuates both the reflectance and the transmittance in a wavelength-dependent manner. The back-reflection phenomenon tends to increase the reflectance and decrease the transmittance in a proportion which depends on the reflectance of the interfaces but not on the wavelength. The two phenomena are amplified when one film is added to the pile. This yields spectral variations depending on wavelength. The transmittance of the piles decreases as the number of films is incremented. In the spectral domain where the acetate is the more absorbing (570–680 nm), it decreases rapidly towards zero. Outside this domain, the transmittance decreases more slowly, in proportion to the reflectance of the added acetate-medium interfaces. Regarding the reflectance, it is increased by adding films provided the reflectance of the acetate-medium interfaces is sufficiently high to compensate the loss due to absorption. When the medium between the films is air, the interface reflectance is

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FIG. 5.13 – Spectral reflectances and transmittances at normal incidence of piles of cyan acetate films (1, 2, 3, 4, 5, 6 and 10 films as well as, for reflectance, infinity) separated by layers of air (top row), a liquid with refractive index 1.4 (middle row) or a medium of same refractive index 1.54 as the acetate (bottom row). the highest, and the pile reflectance always increases when a film is added. With the liquid of refractive index 1.4, the reflectance increases only in the spectral bands where the acetate is weakly absorbing (440–510 nm) and decreases in the other spectral bands because absorption is stronger than back-reflection. One notices an invariance at 440 and 510 nm: absorption and back reflection effects perfectly compensate. When the refractive index of the medium matches the one of the films,

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FIG. 5.14 – Color variation in the piles of acetate films separated by layers of air (solid lines), liquid of refractive index 1.4 (dashed lines) and medium of refractive index 1.54 (dotted lines) viewed in reflectance and transmission mode, represented by the (L*, C*) diagram and the (a*, b*) plane of the CIE 1976 L*a*b* color space. The labeling numbers indicate the number of films stacked.

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the only reflecting interfaces are the front and the back interfaces. The front interface reflects an achromatic component (reflectance r0 = 4.5%) and the back interface reflects a chromatic component, which may be strongly attenuated due to absorption, depending on the layer’s thickness. One can also analyze these variations in reflectance and transmittance in terms of color attributes. The spectral reflectances and transmittances of the different piles are converted into CIE 1931 XYZ tristimulus values by considering a D65 illuminant, themselves converted into CIE 1976 L*a*b* color coordinates by considering the same illuminant, and a perfectly white object as white reference for the chromatic adaptation. These colors are represented in two graphs in figure 5.14, the (L*,C*) diagram showing lightness and chroma, and the (a*, b*) plane showing chroma C* and hue h*. When the number of films is incremented, the increase in spectral reflectance leads to similar increase in lightness. Saturation has a more complex evolution linked to the different spectral variations in the spectral domains where light is most and less absorbed. When adding one film increases the difference between the maximum and the minimum of the spectrum, chroma also increases (see for example the transmittances spectra of 1–6 films separated by air or by liquid, as well as the reflectance spectra of films separated by air). In contradistinction, chroma is decreased when this difference decreases (see the spectral transmittance of piles containing more than 6 films separated by air, or the spectral reflectance of pile containing more than 4 films pasted by the liquid with index 1.54). In every case, there exists an optimal number of films for which chroma is maximal.

5.5

Piles of Films on Top of a Specular Background

Let us now consider piles of plates or films deposited on a specular background, i.e., a mirror. As the background is opaque, their transmittance is zero and their flux transfer matrix is undefined. However, since one is only interested in predicting the angular reflectance of the multilayer reflector, on the ratio of entries p21 =p11 of the corresponding transfer matrix P, one can use a matrix of the following form for the mirror of angular spectral reflectance Rg ðh1 ; kÞ:   1 0 Mg ðh1 ; kÞ ¼ ð5:59Þ Rg ðh1 ; kÞ 0 N identical transparent plates with individual transfer matrix Q ðh1 ; kÞ are placed on top of this specular background. Flux transfer matrix PN ðh1 ; kÞ of the resulting reflector is given by PN ðh1 ; kÞ ¼ QN  ðh1 ; kÞMg ðh1 ; kÞ

ð5:60Þ

from which one deduces the angular spectral reflectance PN ðh1 ; kÞ ¼

0 p21 TN ðh1 ; kÞTN ðh1 ; kÞRg ðh1 ; kÞ ¼ RN ðh1 ; kÞ þ 0 1  RN ðh1 ; kÞRg ðh1 ; kÞ p11

ð5:61Þ

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This expression is actually similar to the expression for R1;2 in equation (5.2), where a pile of plates replaces plate 1, and a mirror replaces plate 2. It can be evaluated at each wavelength. Figure 5.15 shows two examples of reflectors of this kind, based on inkjet transparency films printed with some colors: cyan films are stacked on top of a red mirror, and green films (the same as the ones presented in figure 5.12) on top of a magenta mirror. As the number of films increases, the angular spectral reflectance of the multilayer reflector progressively, asymptotically transits from angular spectral reflectance Rg ðh1 ; kÞ of the background to the angular spectral reflectance R1 ðh1 ; kÞ of the infinite pile: It increases in the spectral domains where Rg ðh1 ; kÞ\R1 ðh1 ; kÞ, and decreases in the rest of the spectrum. Interestingly, at the wavelengths where the mirror and the infinite pile have the same reflectance, the angular reflectance of the reflector remains the same independently of the number of films.

FIG. 5.15 – Spectral normal reflectances and color pictures of piles of colored films on top of a specular reflector: cyan films on a red mirror (left) and green films on a magenta mirror (right). The spectral reflectances of the mirrors are plotted in dotted lines, and the limit reflectance corresponding to an infinite number of films in dashed lines. The numbers indicate the number of films placed on top of the mirror.

Chapter 6 Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light After two chapters dedicated to non-scattering materials, the present chapter addresses the case of scattering materials. Light scattering happens because of heterogeneities of the optical index of the medium at the microscopic scale. A variety of effects can happen depending on the size, density and organization of these heterogeneities, and the models able to predict them can be simple or very complex. In this chapter, we discard all types of scattering requiring a model based on wave optics, as well as very translucent materials for which the reorientation of light must be finely described, for example with the radiative transfer theory evocated in section 8.2.3. We focus here on strongly scattering media, like paints or papers, illuminated by Lambertian light, for which a two-flux approach similar to the introduced in the previous chapter applies. This two-flux approach includes the famous Kubelka–Munk model, named after Paul Kubelka and Franz Munk who published it in 1931 [133, 135], which describes the propagation of forward and backward Lambertian fluxes in a homogeneous scattering medium (homogenous at the macroscopic scale) and yields analytical formulae for the reflectance and transmittance of a layer of this medium with any thickness, possibly coated onto a diffusing background. The two-flux approach also includes the extension proposed by Saunderson in 1942 in order to take into account the reflections and refractions of light at the interfaces bordering the layers [199], as well as the extension proposed by Kubelka in 1954 for piles of different diffusing layers [134]. Thanks to a flux transfer matrix comparable to the one introduced for piles of non-scattering layers, one can extend these classical models to cope with diffusing multilayers having possibly different refractive indices. Each scattering medium is characterized by only two intrinsic parameters (generally wavelength-dependent), the absorption coefficient and the scattering coefficient. We will elaborate on the different methods permitting to obtain them from optical measurements. We will also show how, when scattering media are mixed, the individual coefficient can be combined in order to obtain those of the mixture.

DOI: 10.1051/978-2-7598-2647-6.c006 © Science Press, EDP Sciences, 2022

134

6.1

Optical Models for Material Appearance

The Kubelka–Munk Model

Despite its limitations due to the assumptions on which it relies, the Kubelka–Munk model is simple in both its theoretical development based on analytical formulae and its practical implementation based on affordable measurements. This explains why it is by far the most used scattering model in many application domains: since the paper by Kubelka and Munk was initially dedicated to paints, it has been widely used for papermaking [69], pigments [53, 185], photographic emulsions [243], spatial applications [145], Earth sciences (soils [12, 170], plants [5, 202], canopies [3], sea [86]), biology [75], medicine (skin [200] and other human tissues [252], dental materials [54, 117, 256], make-up [169], textile [171], etc. There is a good reason for such a success: this model yields a set of closed-form formulae relating the intrinsic properties to the material (absorption and scattering coefficients) and those of a layer of given thickness (reflectance, transmittance). Moreover, these formulae are compatible with measurements made with standard instruments (reflectance, possibly transmittance measurements). It is the only scattering model to offer this simplicity of calculation and practical implementation. However, in return for its simplicity, its validity is limited to strongly scattering materials illuminated by diffuse light and may give poor predictions with translucent, weakly scattering media or directional lighting. In this section, we propose to present the basis of the model and the analytical formulae that derive from it. We will then see its practical implementation, before returning to its limits.

6.1.1

The Kubelka–Munk Differential Equations

According to the Kubelka–Munk model, the attenuation undergone by a diffuse flux crossing a very thin layer of matter is similar to the one undergone by a directional flux in a thin non-diffusing layer, which gave rise to the law of Beer-Lambert in §4.5.1: the lost flux is proportional to the incoming flux and to the thickness of the layer crossed. The flux losses are therefore linear on a very small scale, and as always in this case, they become exponential on a larger scale. In a manner similar to that of the purely absorbent layer, part of the lost flux is absorbed, but only a part of it: another part is backscattered in the opposite direction. The proportionality coefficients for the lost fluxes by absorption and backscattering are usually denoted by K and S, respectively. They are assumed to be constant over the layer’s volume. One therefore has two opposite diffuse fluxes, the forward flux denoted by I and the backward flux denoted by J, which mutually contribute to each other at any level in the layer. Fluxes I and J are functions of depth, and possibly functions of wavelength since the absorption and scattering coefficients may be so. Figure 6.1 shows the flux losses and exchanges in a layer of infinitesimal thickness dz at depth z, for a given wavelength. From the figure, one can write the following two equations relating fluxes I and J at the depths z and z þ dz:

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light I ðz þ dz Þ ¼ I ðz Þ  KI ðz Þdz  SI ðz Þdz þ SJ ðz Þdz J ðz  dz Þ ¼ J ðz Þ  KJ ðz Þdz  SJ ðz Þdz þ SI ðz Þdz

135

ð6:1Þ

which can be written as a system of two differential equations of the first order: 8 dI > ¼ ðK þ S ÞI ðz Þ þ SJ ðz Þ < dz ð6:2Þ > : dJ ¼ SI ðz Þ þ ðK þ S ÞJ ðz Þ dz

FIG. 6.1 – Absorption and backscattering of forward and backward diffuse fluxes of a given wavelength at a depth z in a sublayer of thickness dz.

or as a vector differential equation of the first order:      d I  ðK þ S Þ S I ðz Þ ¼ : S K þS J ðz Þ dz J

ð6:3Þ

A convenient way to solve equation (6.3) relies on the matrix exponential, defined by a Taylor series similar to that of the classical exponential: expðMÞ ¼

1 X Mk k¼0

k!

¼ IþMþ

M2 M3 M4 þ þ  2 6 24

The solution of equation (6.3) is, for the considered wavelength [63],        I ðz Þ K þS S I ð0Þ ¼ exp z  J ðz Þ S  ðK þ S Þ J ð 0Þ

6.1.2

ð6:4Þ

ð6:5Þ

Reflectance and Transmittance Formulae

For each wavelength, the reflectance qh of a layer of thickness h is by definition the ratio J ð0Þ=I ð0Þ, and the transmittance sh is the ratio I ðh Þ=I ð0Þ, knowing that I ð0Þ ¼ I0 and J ðh Þ ¼ 0 when the layer is illuminated on the front side by the flux I0 :

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qh ¼

sinhðbSh Þ a sinhðbSh Þ þ b coshðbSh Þ

ð6:6Þ

sh ¼

b a sinhðbSh Þ þ b coshðbSh Þ

ð6:7Þ

and

where the definitions for the hyperbolic sine and cosine are recalled in equation (5.25) and parameters a and b are defined for each wavelength as K þS S

ð6:8Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi a2  1

ð6:9Þ

a¼ and b¼

The diffusing layer being assumed uniform, it is symmetric: it has same spectral reflectance and transmittance on both sides. As the thickness tends to infinity, the reflectance tends towards a limit value given by: 1

q1 ¼ lim

h!1 a þ b cothðbShÞ

¼

1 ¼ab aþb

ð6:10Þ

In the literature (see Ref. [249], p. 785), q1 is expressed as a function of K and S: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  K K 2 K ð6:11Þ þ2 q1 ¼ 1 þ  S S S The ratio K/S, also called remission function, can be expressed as a function of q1 : K ð1  q1 Þ2 ¼ S 2q1

ð6:12Þ

Equation (6.12) is the most popular result from the Kubelka–Munk theory. An observer perceives a layer of finite thickness h as infinitely thick if no light emerges on the back side. In practical terms, this is an opaque layer and yields the boundary condition iðhÞ ¼ 0. The invariance of parameter a that was noticed in §5.2.5 with piles of transparent plates is also true with a scattering layer. One may show that it is equivalently defined from the reflectance qh and transmittance sh of a layer of any thickness h: a¼

1 þ q2h  s2h 2qh

ð6:13Þ

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137

The remission function is therefore defined, in the general case of a layer with finite thickness, as K ð1  qh Þ2 s2h ¼a1¼ S 2qh

ð6:14Þ

When the layer thickness δ is very small, the reflectance is qd ¼ Sd and the transmittance sd ¼ 1  ðK þ S Þd ¼ 1  aSd. When δ tends to zero, expression (6.13) yields expression (6.8).

6.2

Layers in Optical Contact with a Background

Many practical cases, like ink on paper or paint on a substrate, can be seen as layers in optical contact with a diffusing background having a spectral reflectance factor qg ðkÞ. The boundary condition at z ¼ h is: J ðh; kÞ ¼ qg ðkÞI ðh; kÞ

ð6:15Þ

and the reflectance expression, still corresponding to the ratio J ð0; kÞ=I0 ðkÞ, becomes for each wavelength qh;g ¼

ð1  aqg Þ sinhðbSh Þ þ bqg coshðbSh Þ ða  qg Þ sinhðbSh Þ þ b coshðbSh Þ

ð6:16Þ

Assuming bSh 6¼ 0, equation (6.16) can be simplified as: qh;g ¼

1  qg ½a  b cothðbSh Þ a  qg þ b cothðbSh Þ

ð6:17Þ

Equation (6.17) is called the hyperbolic solution of the Kubelka–Munk equations. As bSh tends to 0, because the layer is either very thin or non-absorbing and non-scattering, ρ tends to qg . Two particular cases of interest for equation (6.17) are the cases where the layer is on an ideal black background (ρg = 0) and on an ideal white background (ρg = 1). In the case of the black background, the reflectance is similar to the one without background given by equation (6.6). In the case of the white background, it is, for each wavelength, qh;white ¼

ð1  aÞ sinhðbShÞ þ b coshðbShÞ ða  1Þ sinhðbShÞ þ b coshðbShÞ

ð6:18Þ

Paper boards with white and black areas as shown in figure 6.2, called “drawdown cards”, “application cards”, or “opacity charts” are frequently used in the paint and ink industries in order to compute the K and S values. For a more accurate result, it is preferable to consider the real reflectances of the white and black areas, which are neither 1 nor 0, as well as the Fresnel reflections and transmissions of light at the air-layer interface, as explained in §6.4.

138

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FIG. 6.2 – A drawdown card from the BYK Instruments company, with white and black areas, covered by a yellow paint [29].

6.3

Light Transfers at the Interfaces Bordering the Layer

The Kubelka–Munk formulae stand for a layer surrounded by a medium of the same refractive index as the layer. No reflection of light occurs at the layer’s interfaces in this case. However in practice, the layer is surrounded by air and the Fresnel reflectance and transmittance of the interface must be considered. This was the purpose of the equation proposed by Saunderson in 1951, often referred to as “Saunderson correction” [199].

6.3.1

Saunderson Correction and Inverse Formulae

Saunderson’s model considers a layer on a background, of intrinsic spectral reflectance (in the absence of interface) denoted here by qðkÞ. Spectral reflectance factor ^ ðkÞ that is measured necessarily incorporates the optical effect of the interface with R air, and depends upon the selected measurement geometry. The equation relating ^ ðkÞ and qðkÞ can be derived by various methods. One of them, chosen by R

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139

Saunderson himself, is used to describe the multiple reflections of light between the background and the interface. The multiple reflection process is similar to the processes that have been described in the previous chapter between the two interfaces of a transparent plate or between transparent plates; therefore, it is not surprising that similar formulae are obtained. The main difference lies on the angular distribution of light illuminating the interface, which is Lambertian on the layer’s side, and depends on the measuring geometry on the air side. The optical effect of the interface is described by four transfer factors, shown in figure 6.3: the external reflectance rs is the fraction of the incident flux reflected from the interface towards the detector (it may be 0 if the geometry excludes the specular reflection component); transmittance Tin corresponds to the amount of light that enters the medium; internal reflectance ri is the fraction of Lambertian flux issued from the medium which is reflected back to it; and factor Tout denotes the fraction of light issued from the medium and reaching the detector.

FIG. 6.3 – Reflections and transmissions of diffuse light at the air-medium interface. Let us denote by I0 ðkÞ incident spectral flux, I ð0; kÞ spectral flux entering into the layer, J ð0; kÞ spectral flux reflected from the layer and illuminating the interface from the back, and J0 ðkÞ spectral flux that is captured by the detector. One can write the two following equations: I ð0; kÞ ¼ Tin I0 ðkÞ þ ri J ð0; kÞ J0 ðkÞ ¼ rs I0 ðkÞ þ Tout J ð0; kÞ

ð6:19Þ

from which one deduces, knowing that J ð0; kÞ ¼ qðkÞI ð0; kÞ, that ^ ðkÞ ¼ rs þ Tin Tout qðkÞ R 1  ri qðkÞ

ð6:20Þ

The terms rs , ri , Tin , and Tout depend only on the refractive index of the medium. Assuming that the interface is flat, they are derived from the Fresnel formula with respect to the illumination and observation geometries. In the case of a bi-hemispherical geometry, the incident flux is Lambertian and the whole reflected light is captured by the detector. One thus has: rs ¼ r12 given by

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equation (4.8), ri ¼ r21 deduced from r12 according to equation (4.11), Tin ¼ t12 ¼ 1  r12 and Tout ¼ t21 ¼ 1  r21 . In the case of a hemispherical-directional geometry, where the incident light is Lambertian and only the spectral radiance along one direction is observed at an angle θ from the normal, one rather has rs ¼ R12 ðhÞ=p because only the spectral radiance coming at angle θ can be reflected towards the detector, R12 ðhÞ being the Fresnel reflectance of the interface. One also has Tout ¼ T12 ðhÞ=ðn 2 pÞ, where T12 ðhÞ denotes the Fresnel transmittance of the interface and the term 1=n 2 in Tout comes from the changing of solid angle due to the refraction of exiting rays through the interface, as explained in §4.1. Factor 1/π is the ratio between the spectral radiance going towards the detector and the total spectral exitance, according to Lambert’s law (see §1.5.1), but it is cancelled out if one considers the spectral reflectance factor with respect to a perfect white diffuser. Table 6.1 summarizes the expression for the four terms according to the different standard geometries recommended by the CIE presented in §2.6.1, see table 2.1 (factor 1/π is removed). TAB. 6.1 – Transfer factors attached to the interface for different standard measurement geometries. Geometry d:d di:8° de:8° 8°:d 45°:0°

rs r12 R12 ð8 Þ 0 R12 ð8 Þ 0

Tin 1  r12 1  r12 1  r12 T12 ð8 Þ T12 ð45 Þ

ri r21 r21 r21 r21 r21

Tout 1  r21 T12 ð8 Þ=n 2 T12 ð8 Þ=n 2 1  r21 T12 ð0Þ=n 2

Remark: When the interface is rough, rs , ri , Tin , and Tout must be computed from the spectral BRDF and BTDF of the interface, for example with the model presented in §8.1, by taking into account the precise geometry of the measurement device. However, the values for ri , Tin , and Tout remain nearly unchanged compared to the case of a smooth interface, as shown in [94]. ^ ðkÞ of the sample is measured, one can deduce When spectral reflectance factor R intrinsic reflectance qðkÞ of the layer (with background), as if there was no interface with the surrounding air, using the following formula: qðkÞ ¼

^ ð kÞ  r s R   ^ ðkÞ  rs Tin Tout þ ri R

ð6:21Þ

The Saunderson equation (6.20) will be generalized in the next chapter to other configurations where a non-scattering component is on top of a diffusing background. The non-scattering component can be a homogeneous transparent layer together with its interface with air (Williams–Clapper and Berns models), a heterogeneous transparent layer together with its interface with air (Clapper–Yule model), a pile of transparent layers with flat interfaces, etc.

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light

6.3.2

141

Saunderson Correction and Inverse Formulae for a Diffusing Plate

Let us now consider a diffusing plate, of refractive index n, having flat interfaces with ^ ðkÞ and transmittance factor T ^ ðkÞ are measured air. Its spectral reflectance factor R with a given geometry. One denotes by qðkÞ its spectral intrinsic reflectance and by sðkÞ is spectral intrinsic transmittance. The transfer factors at the interfaces, featured in figure 6.4, are similar to those defined in the previous section for one interface, and one assumes that they are similar on both sides. From figure 6.4, one can write the following equations: J0 ðkÞ ¼ rs I0 ðkÞ þ Tout J ð0; kÞ I ð0; kÞ ¼ Tin I0 ðkÞ þ ri J ð0; kÞ J ð0; kÞ ¼ qI ð0; kÞ þ sJ ðh; kÞ I ðh; kÞ ¼ sI ð0; kÞ þ qJ ðh; kÞ

ð6:22Þ

J ðh; kÞ ¼ ri ðh; kÞ þ Tin Jh ðkÞ Ih ðkÞ ¼ Tout I ðh; kÞ þ rs Jh ðkÞ One deduces from these equations the following expressions for the layer’s reflectance and transmittance factors: 2 2 ^ ðkÞ ¼ rs þ Tin Tout qðkÞ  ri ½q ðkÞ  s ðkÞ R 2 ½1  ri qðkÞ ri2 s2 ðkÞ

ð6:23Þ

FIG. 6.4 – Flux transfers in a diffusing plate having flat interfaces with the surrounding air. F, M and F′ denote the flux transfer matrices representing the front interface, the layer, and the back interface, respectively.

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142

and ^ ð kÞ ¼ T

Tin Tout sðkÞ ½1  ri qðkÞ2 ri2 s2 ðkÞ

ð6:24Þ

Notice that the flux transfer matrix method introduced for piles of plates in chapter 6 can be used here, by attributing to the front interface, the layer, and the back interface the respective matrices F, M and F′ featured in figure 6.4 and defined as follows, for each wavelength:   1 1 ri F¼ ð6:25Þ Tin rs Tin Tout  rs ri M¼

1 s



1 q

q s 2  q2

 ð6:26Þ

  1 1 rs F ¼ ð6:27Þ Tout ri Tin Tout  rs ri

Transfer matrix P ¼ pij representing the diffusing plate is thus given by matrix product FMF0 and the reflectance and transmittance are obtained using ^ ¼ p21 =p11 and T ^ ¼ 1=p11 . This yields of course same equation (5.14), i.e., R expressions as (6.23) and (6.24). From the measured spectral reflectance and transmittance factors of the diffusing plate, the spectral intrinsic reflectance and transmittance of the layer (without interfaces) can be computed thanks to the following equations, where ^ 0 ð kÞ ¼ R ^ ðkÞ  rs : R  02  ^ 2 ð kÞ ^ 0 ðkÞTin Tout þ ri R ^ ð kÞ  T R ð6:28Þ qðkÞ ¼   



^ ðkÞ þ Tin Tout ri R ^ ðkÞ þ Tin Tout ^ 0 ð kÞ  T ^ 0 ð kÞ þ T ri R 0

^ ð kÞ Tin Tout T s ð kÞ ¼ 



:   ^ ðkÞ þ Tin Tout ri R ^ ðkÞ þ Tin Tout ^ 0 ð kÞ  T ^ 0 ð kÞ þ T ri R

ð6:29Þ

These equations are obtained by inversing equations (6.23) and (6.24). They can also be obtained using the flux transfer matrix method, by building for each wavelength the transfer matrix representing the plate from the measured spectral reflectance and transmittance factors:   ^ 1 1 R P¼ ð6:30Þ ^ T ^2  R ^2 ^ R T

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light

143

and computing matrix product F1 PF0 1 based on the inverse of the matrices defined in equations (6.25) and (6.27). This model has been for example used to describe the interaction of light with plant leaves [4].

6.4

Deducing K and S from Measurements

When facing a new unknown material, and before being able to predict the reflectance or transmittance of a layer of any thickness, one must evaluate its spectral absorption and scattering coefficients, K ðkÞ and over the desired spectrum. Since one wants to evaluate two coefficients, two measurements are necessary. Two main methods are possible: the method based on a drawdown card with white and black backgrounds (see figure 6.2) needing two spectral reflectance measurements, and the method based on spectral reflectance and transmittance measurements. The drawdown card method is the most frequently used in application domains where the scattering medium is fluid, e.g., paints, inks, wax, etc. It can also be used with a solid plate of film, e.g., a polymer sheet, a white glass plate or a vegetable leaf, provided this latter is in optical contact with the card. The sample and the card must have similar refractive indices, around 1.5, and the must be waterproof to make sure that the index matching fluid of index 1.5 used to make the optical contact does not modify its optical properties. If these conditions cannot be satisfied, the reflectance-transmittance method is preferable. The two methods are presented below. It is important to be aware that they are not strictly equivalent if the material is not highly scattering [54].

6.4.1

Drawdown Card Method

A layer of scattering material is deposited onto the white and black areas of the drawdown card, by taking care that layer thickness h is the same on the white and black backgrounds. One wants to compute the spectral absorption and scattering coefficients of the medium, K ðkÞ and S ðkÞ. Actually, only unitless quantities K ðkÞh and S ðkÞh are accessible, and the K ðkÞ and S ðkÞ coefficient (in m−1) are obtained if the thickness h is known. When they are not covered by any material, the black and white areas of the card ^ k ðkÞ and R ^ w ðkÞ, which incorporate the have respective spectral reflectance factors R Fresnel reflections and transmissions at the air-card interface. Formulae (6.21) (inversed Saunderson correction) give spectral intrinsic reflectances qk ðkÞ and qw ðkÞ of the black and white materials: ^ k ðkÞ  rs R  ^ k ðkÞ  rs ri þ Tin Tout R ^ w ðkÞ  rs R qw ðkÞ ¼   ^ w ðkÞ  rs ri þ Tin Tout R qk ðkÞ ¼ 

ð6:31Þ

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144

Once coated with the diffusing material, the black and white areas have spectral ^ 0 ðkÞ and R ^ 0 ðkÞ which are also measured. One deduces from reflectance factors R k w these measured values the spectral intrinsic reflectances, respectively, q0k ðkÞ and q0w ðkÞ, thanks to an inversed Saunderson correction similar to the one applied on the uncoated areas, equation (6.31). One also knows, according to equation (6.17), that these intrinsic reflectances can be written: 1  qk ½a  b cothðbShÞ a  qk þ b cothðbShÞ 1  qw ½a  b cothðbShÞ q0w ¼ a  qw þ b cothðbShÞ q0k ¼

ð6:32Þ

By denoting parameter a by u, and expression b cothðbShÞ by v, one can derive from equation (6.32):   ðqw  qk Þ 1 þ q0w q0k  q0w  q0k ð1 þ qw qk Þ  u¼ 2 qw q0k  q0w qk   ð6:33Þ ðqw  qk Þ 1  q0w q0k þ q0w  q0k ð1  qw qk Þ  v¼ 2 qw q0k  q0w qk Then, for each wavelength,

  1 v Sh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi arccoth pffiffiffiffiffiffiffiffiffiffiffiffiffi and Kh ¼ ðu  1ÞSh u2  1 u2  1

ð6:34Þ

where function arccoth is also defined as

  1 x þ1 arccothðx Þ ¼ ln 2 x 1

The main issue with this method relies on the difficulty of producing good samples. With fluid materials, depositing a uniform layer on the card is hardly achievable manually. With some practice, manual film applicators with variable aperture can help; coating machines, when available, are preferable. With solid samples, the main issue is the optical contact: the matching index liquid used to paste the layer onto the background should have the same index as these, and any air bubbles, even microscopic, should absolutely be avoided since their optical effect may be significant. The method should be independent of thickness h, but it is crucial to avoid that the layer be opaque at any wavelength: spectral reflectances measured on the layer over white and black backgrounds must be noticeably different at every wavelength of the considered spectrum to prevent numerical divergence of equation (6.34). It is therefore recommended to check spectral reflectances measured on the two backgrounds.

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light

6.4.2

145

Reflectance and Transmittance Method

An alternative to the drawdown card method, applicable to solid diffusing plates, ^ and transmittance sheets or leaves, consists in measuring their reflectance factor R ^ factor T , then deducing their intrinsic reflectance ρ and transmittance τ thanks to equation (6.28). One can compute the terms Kh and Sh thanks to the following formulae:

q 1 Sh ¼ arcsinh b : ð6:35Þ b s and Kh ¼

q ð1  qÞ2 s2 arcsinh b s 2q

where b is a function of ρ and τ, derived from equations (6.9) and (6.13): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ih i 1 h ð1  qÞ2 s2 ð1 þ qÞ2 s2 b¼ 2q

ð6:36Þ

ð6:37Þ

Equation (6.36) directly comes from equations (6.14) and (6.35) is obtained by dividing the Kubelka–Munk reflectance formula with the transmittance formula, which gives: q sinhðbSh Þ ¼ b s

6.4.3

Choosing the Appropriate Method

When it is not possible to obtain a solid plate with parallel flat interfaces – which is often the case with paints, waxes, creams or foundations used in cosmetics – the drawdown card is the only possible method to obtain the absorption and scattering coefficients of the material. Otherwise, when a solid sample can be produced, both drawdown card method and reflectance-transmittance method are possible. In either case, it is recommended to verify the validity of the Kubelka–Munk model by applying the selected method with various samples with layers of different thicknesses: spectral values of the a parameter, defined by equation (6.13), should be nearly the same. Failing this, it is almost certain that the model will give poor reflectance predictions for thicknesses other than the one used for determining the K and S coefficients. The drawdown card method requires an optical contact which may be difficult to realize, but it provides better results if the sample is weakly scattering. The fact that the layer is bordered on the back side by a Lambertian background guarantees that the assumptions underlying the Kubelka–Munk model, i.e., the propagating fluxes, are perfectly diffused and the measurement geometry is bi-hemispherical, are well satisfied even when the material is weakly scattering. It is recalled that the reflectance factor is independent of the measuring geometry when the sample is a Lambertian reflector (see §2.1.1). Even though the samples here are not exactly

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Lambertian reflectors because of the angle-dependent flux transfers at the sample-air interface, these flux transfers are explicitly considered by the Saunderson correction according to the measuring geometry used, and the material itself can be considered as a Lambertian reflector. A d:8° or 8°:d geometry where the observation, respectively the illumination, is directional, can therefore be used. In conclusion, except for the problems related to the production of samples, the drawdown card method remains rather reliable. One must be aware, however, that since the K and S values have been determined with a strongly scattering background, the Kubelka– Munk model can only predict the reflectance of layers correctly on a strongly scattering background. This suits paints and other coatings, but it is problematic for layers of materials without background, such as plates or plants. The reflectance-transmittance method avoids the drawback mentioned just above. However, it has other drawbacks, especially when the sample is weakly scattering or highly translucent (this being the reason why we advocate its use only for strongly scattering media): the values for K and S may be erroneous, or even aberrant, because the light captured by the detector is not representative of the whole diffuse fluxes propagating within the material. This is especially problematic in the Saunderson correction which considers the internal reflections at the interfaces, crucial events being at the origin of the multiple reflection process occurring within the plate. Remember from §2.6.4 that in the case of a transparent plate measured with a d:8° or 8°:d geometry, the effective measurement geometry is 8°:8°, i.e., the light detected comes from one direction. If the plate is weakly scattering, a significant fraction of the light detected has not been deviated from this direction and thus remains directional. The corresponding internal reflectance at the interfaces is not r21 (around 0.6 for a refractive index 1.5) as considered by the classical Saunderson correction, but rather R21 ð8 Þ (around 0.04 for the same refractive index) as for directional light around the normal of the interface [54]. This makes a huge difference. In reality, depending on the angular paths followed by the rays reaching the detector, it is probable that the internal reflectance value lies between these two extreme values, and it can depend upon the layer thickness since light is more scattered into a thicker plate. Attempts to estimate this angular distribution, based on a 4-flux scattering model or a resolution of the radiative transfer equation, have been proposed recently [65]. Actually, to prevent any mismatch between the expected value and the measured one, a bi-hemispherical geometry should be used in both reflectance and transmittance modes, but the bi-hemispherical geometry for reflectance does not exist (the integrating sphere is used either to produce Lambertian illumination or to collect the scattered light over the hemisphere, but it cannot do both at the same time).

6.5

Example

The variation in reflectance and transmittance of a scattering medium according to the layer thickness is shown in figure 6.5 with a blue diffusing polymer. Measure^ and transmittance factor T ^ of a 200 µm thick sheet of ments of reflectance factor R this polymer have been made with a spectrophotometer. Intrinsic reflectance ρ and

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light

147

transmittance τ of this sheet, then the absorption and scattering coefficients of the material were deduced following the method presented in §6.4.2. The coefficient absolute values in mm−1 could be obtained since the layer thickness was known. They are plotted in figure 6.5a. Then, one could predict the intrinsic reflectances qh and transmittances sh for the different layer thicknesses indicated in the legend thanks to the Kubelka–Munk formulae (6.6) and (6.7). They are plotted in figure 6.5b and c, respectively. The ones associated with the sheet that has been measured, ρ and τ, are plotted in dashed lines. For a 2 mm thickness, the reflectance approaches the limit reflectance q1 , whereas the transmittance approaches 0.

FIG. 6.5 – (a) Spectral absorption coefficient K(λ) and scattering coefficient S(λ) of a blue diffusing polymer, and (b) spectral reflectance and (c) spectral transmittance of layers of this polymere with various thisknesses predicted by the Kubelka–Munk model. As the thickness increases, the reflectance increases and the transmittance decreases, in proportions that depend on the absorption and scattering coefficients. Similar trends were observed with piles of identical transparent films separated by air (see figure 5.12), predicted by the discrete two-flux model, valid for directional light and integer thicknesses (number of films). The influence of the scattering coefficient in the continuous two-flux model is similar to that of refractive index of the films, which determines the Fresnel reflectance of the interfaces, in the discrete

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two-flux model. The influence of the absorption coefficient in the continuous model is similar to the one of the intrinsic spectral transmittance of one film, itself related to the absorption coefficient of the transparent material.

6.6

Mixture of Scattering Materials

When various materials are mixed together to form a new homogenous medium, its spectral absorption and scattering coefficients, Km ðkÞ and Sm ðkÞ, are related to those of the initial materials, Ki ðkÞ and Si ðkÞ (i is a labelling number), by the following additive formulae, often referred to as the Duncan formulae [51]: X K m ð kÞ ¼ c i K i ð kÞ ð6:38Þ i

S m ð kÞ ¼

X

ci Si ðkÞ

ð6:39Þ

i

where ci are the volume fraction of the different materials, whose sum is 1. Spectral remission function K/SðkÞ of the mixture is therefore: P   K ci Ki ðkÞ ð6:40Þ ðkÞ ¼ Pi S m i ci Si ðkÞ Several studies showed that this method performs rather well for predicting the reflectance of mixed materials such as pigments in paints [188] or minerals in soils [12]. For each material introduced in the mixture, the determination of KðkÞ and SðkÞ requires two measurements: either two spectral reflectance measurements with white and black backgrounds, or spectral reflectance and transmittance measurements, as explained in §6.4. If the mixture is presented under the form of an opaque layer, i.e., it has an infinite optical thickness, it is not possible to determine KðkÞ and SðkÞ separately, only the remission function K/SðkÞ can be determined, by measuring spectral reflectance q1 ðkÞ and using equation (6.12). Duncan’s formulae are used in many application domains, such as textile or plastic dying, pigment mixtures in paints, etc. They are valid when the layer thickness is rather small, or when the colorant concentrations are not too high. It seems that a condition for their validity is that the invariance of parameter a ðkÞ with respect to the layer thickness [see §6.1.2, equation (6.13)] is verified for each of the mixed materials [30]. In particular, they are used in applications where a white substrate such as paper or textile, strongly scattering and weakly absorbing, is dyed with various colorants. One can assume that light scattering is mainly due to the substrate, and the scattering coefficient S ðkÞ remains nearly constant when impregnated by any dye [223]. Formula (6.40) thus becomes: P   Ks ðkÞ þ i ci Ki ðkÞ K ð6:41Þ ð kÞ ¼ S m S ð kÞ where the subscript s represents the substrate, and subscript i represents the different dyes.

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149

One can first determine spectral remission function ðK =S Þs ðkÞ of the bare substrate. The method depends on the kind of material under consideration. If the substrate is opaque (i.e., its optical thickness is infinite) and has no sharp interface with air, only its spectral reflectance q1 ðkÞ is measured. The remission function ðK =S Þs ðkÞ is then given directly by equation (6.12), for each wavelength. If it has a sharp interface with air, intrinsic reflectance q1 ðkÞ can be deduced from the measured spectral reflectance factor thanks to the inverse Saunderson formula, equation (6.21). When the substrate is a translucent layer, its reflectance ρ and transmittance τ are measured, and ðK =S Þs ðkÞ is given by equation (6.14). If its interfaces with air are sharp, the measured reflectance and transmittance factors are converted into intrinsic reflectance and transmittance thanks to the inverse Saunderson formulae, equation (6.28). Then, the substrate is impregnated with each dye i alone with the expected volume fractions ci . Its spectral reflectance qi ðkÞ is measured, as well as its spectral transmittance qi ðkÞ if the substrate is not opaque. The ratio ðK =S Þi ðkÞ is computed using the same method as for the bare substrate. The spectral remission function for the support impregnated with a dye mixture is given for each wavelength by:     X c 0 K  K K i ¼ þ ð6:42Þ S m S s S i c i i where ci0 is the concentration of dye i in the mixture, which may be different from the concentration used to produce the measured sample with this dye alone for the determination of the remission function ðK =S Þi . However, the additivity of equation (6.42) is not valid for all materials. It should be verified for each material by producing several various samples of different known dye concentrations, and computing the remission function ðK=S Þi for each sample at a certain wavelength, preferably the wavelength for which the dye is the most absorbing: this latter should vary proportionally to the concentration. If the remission function is a non-linear function of concentration, the Kubelka–Munk model might provide poor prediction accuracy of the reflectance of the mixture [223]. Many physical reasons can explain this non-linear function: for example, the dye may be too strongly absorbing and modify the scattering properties of the substrate. High concentrations in dying particles may also be an issue: even in a very thin sublayer, flux transfers between particles within the sublayer might happen, whereas the linearity of Duncan’s formulae relies on the assumption that light interact with each particle independently of each other.

6.7

Validity of the Kubelka–Munk Model

The Kubelka–Munk model has become very useful in many technical domains concerned with light scattering. It is mainly due to its reduced number of parameters (absorption coefficient, scattering coefficient and medium thickness) and to a set of simple formulae relating these parameters with measurements easily performable in practice. However, in some cases, large deviation between prediction and

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measurement is observed due to the fact that the optical properties of the medium and/or the measurement conditions are not compatible with the assumptions underlying the model. An abundant literature has been dedicated to the analysis of the Kubelka–Munk model limitations, as well as to improvements for specific applications [253–255]. As a special case of the radiative transfer theory (see §8.2.3), the Kubelka–Munk model relies on the following assumptions [198]: the average distance between diffuser and/or absorbers is very large compared to the wavelengths of light. The absorbing and scattering particles should therefore be relatively far from each other with respect to the wavelength of light, such as fibers in paper for example. The model assumes Lambertian illumination and is not adapted in case of collimated illumination. It also assumes strong, isotropic scattering in a homogenous semi-infinite medium. It does not apply with media whose scattering and absorption coefficients vary locally at mesoscopic scale, for example ink dots on paper in halftone printing (§7.4). The K and S values for which accurate predictions can be expected from the model are often discussed. In many cases, in particular when the concentration in either absorbing particles or scattering particles increases, while the other one remains constant, there happens that the predictions for both K and S vary [192]. This leads to the conclusion that the two coefficients are interdependent, and explains why, in many applications, one should consider the remission function K/S which is closely related to the reflectance of an infinitely thick medium according to equation (6.12), rather than the K and S parameters separately. Lastly, anisotropy of the propagating fluxes is a main concern of the Kubelka– Munk theory. Nobbs [177] noticed that the assumption of uniformly diffuse flux propagating in opposite directions is valid only for non-absorbing media, otherwise the fluxes become anisotropic and different absorption and scattering coefficients should be defined for each propagation direction. Anisotropic scattering is modeled by Monte-Carlo simulations or extensions of the Kubelka–Munk theory such as the multi-flux model developed by Mudgett and Richards [172], the four-flux model by Maheu et al. [150]. Some works have also extended the Kubelka–Munk model to fluorescent materials [2, 204, 217].

6.8

Diffusing Multilayers

The Kubelka–Munk model describes the reflection and transmission of light by a uniform diffusing layer whose scattering and absorption coefficients are constant over the whole thickness. In 1954, Kubelka [134] proposed for stacked diffusing layers an extension describing the flux transfers between the different layers. The layers may be non-symmetric, i.e., their scattering and absorption coefficients may vary along the normal of the layer, and their two faces may therefore have different reflectance values. Fresnel reflections at the interfaces are not considered in the model, but a correction similar to the Saunderson correction for the Kubelka–Munk model can be easily introduced.

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The mathematical methods for describing the flux transfers between diffusing layers is very similar to the ones introduced in chapter 5 for flux transfers between transparent plates. The only difference is that in this case all fluxes are perfectly diffuse and independent of polarization, and all transfer factors are bi-hemispherical ones, whereas in chapter 5, fluxes were directional and linearly polarized quantities, and transfer factors are angular quantities to be evaluated for each polarization state s or p. In the following sections, we will therefore summarize the main results by referring to the previous chapter for the detailed calculations.

6.8.1

Kubelka’s Compositional Formulae and Flux Transfer Matrix Model

Consider a non-symmetric diffusing layer labelled 1 on top of a second non-symmetric diffusing layer labelled 2. They are characterized by their respective transfer factors: front reflectance qi , forward transmittance si , back reflectance q0i , backward transmittance s0i , i ¼ 1; 2: This configuration is similar to the pile of two plates studied in §5.1.1 and §5.1.2. The transfer factors of the bi-layer are therefore similar to equation (5.5): s1 s01 q2 s1 s2 ; s12 ¼ 1  q01 q2 1  q01 q2 s2 s02 q01 s01 s02 ¼ q02 þ ; s012 ¼ 0 1  q1 q2 1  q01 q2

q12 ¼ q1 þ q012

ð6:43Þ

In his original paper [134], Kubelka derived these equations by adding the diffuse light components exiting at each side of the bi-layer in a similar way as featured in figure 5.2 (by replacing the directional fluxes with Lambertian ones and the angular transfer factors with be-hemispherical ones), and reducing the geometric series obtained. For additional layers, equation (6.43) can be used iteratively (see §5.1.3). Alternatively, one can use the flux transfer matrix model introduced in §5.1.4, where each layer is represented by a matrix of this form:   1 1 q0i Mi ¼ ð6:44Þ si qi si s0i  qi q0i Once again, all properties of the flux matrix model presented in chapter 6 apply here: a diffusing multilayer containing N layers labelled 1 at the front side up to N at the back side, is represented by the flux transfer matrix M ¼ M1 M2 M3 :::MN

ð6:45Þ

where matrices Mi , defined as in equation (6.44), represent each

individual layer. From the entries of the obtained transfer matrix M ¼ mij , one retrieves the multilayer transfer factors as follows:

Optical Models for Material Appearance

152 q ¼ m21 =m11 q0 ¼ m12 =m11

s ¼ 1=m11 s0 ¼ detðMÞ=m11

ð6:46Þ

Kubelka stated that strongly scattering multilayers have identical forward and backward transmittances even when the individual layers are non-symmetric, what he called ‘nonpolarity principle’: by assuming s1 ¼ s01 and s2 ¼ s02 , one gets s12 ¼ s012 . The same principle applies with transparent multilayers, as shown in §5.3.5.

6.8.2

Example: Piles of Identical Diffusing Sheets

In order to illustrate the model, spectral reflectances and transmittance of a symmetric sheet of blue diffusing polymer have been measured as well as the ones of two identical sheets without optical contact. The spectral reflectance and transmittance of the two sheets were predicted from the ones of one sheet, thanks to equation (6.43), and compared to the measurements in figure 6.6.

FIG. 6.6 – Spectral reflectances and transmittances of one sheet of blue diffusing polymer, and of a pile of two similar sheets, as measured and predicted by Kubelka’s equations.

The deviation between the spectra predicted and measured, assessed in equivalent color distance with the ΔE94 metric in the CIE 1976 L*a*b* color space (by considering a 2° standard observer, the D65 illuminant, and a perfect white diffuser as white reference for chromatic adaptation), is 0.14 unit for the reflectance, and 0.39 unit for the transmittance, which shows the good prediction accuracy of the

Diffusing Layers and Multilayers: Two-Flux Models for Diffuse Light

153

model for this strongly scattering material. As already observed with transparent films, adding one sheet increases the reflectance and decreases the transmittance.

6.8.3

Kubelka’s Formulae and Kubelka–Munk Model

One can verify that Kubelka’s formulae (6.43) are compatible with the Kubelka– Munk reflectance and transmittance formulae (6.6) and (6.7): when two layers of the same diffusing material (symmetrical in this case since the Kubelka–Munk model addresses only uniform media), of thicknesses h 0 and h 00 , are on top of each other, they form one symmetric layer of thickness h ¼ h 0 þ h 00 . One therefore gets, for each wavelength: q h ¼ qh 0 þ

sh ¼

s2h0 qh00 sinhðbSh Þ ¼ 1  qh0 qh00 a sinhðbSh Þ þ b coshðbSh Þ

sh0 sh00 b ¼ 1  qh0 qh00 a sinhðbSh Þ þ b coshðbSh Þ

ð6:47Þ

ð6:48Þ

Similarly, one may decompose the layer of thickness h into N identical sublayers of thickness h/N, the interesting case being when N tends to infinity: the sublayers have an infinitesimal thickness, and, by following the assumptions underlying the Kubelka–Munk model, their transfer factors are, for each wavelength: qh=N ¼ S

h h and sh=N ¼ 1  ðK þ S Þ N N

ð6:49Þ

As expected, the defining expressions (6.13) and (6.8) for parameter a are equivalent:   1 þ q2h=N  s2h=N K þ S K ðK þ 2S Þ h K þS a¼  ; ð6:50Þ ¼ ! S 2S N N !1 S 2qh=N pffiffiffiffiffiffiffiffiffiffiffiffiffi and parameter b remains defined as b ¼ a 2  1. By analogy with the reflectance and transmittance formulae presented in §5.2.1 for a stack of N identical plates, one can write for each wavelength: qh ¼

sinhðN log mÞ a sinhðN log mÞ þ b coshðN log mÞ

ð6:51Þ

sh ¼

b a sinhðN log mÞ þ b coshðN log mÞ

ð6:52Þ

where parameter ν is defined for each wavelength as in equation (5.19): m¼

i 1  ða  bÞS h 1 h N 1  ða  bÞqh=N ¼ sh=N 1  aS Nh

ð6:53Þ

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154

Using a classical limit of the exponential function,

x N lim 1  ¼ ex N !1 N one obtains:

2

1  ða  bÞS Nh lim N log m ¼ log4 N !1 1  aS Nh

!N 3  ðabÞSh  5 ¼ log e ¼ bSh; eaSh

ð6:54Þ

Finally, as expected by inserting equations (6.54) into (6.51), one retrieves the Kubelka–Munk transmittance and reflectance expressions (6.6) and (6.7).

6.8.4

Extended Saunderson Correction for Multilayers: The Matrix Method

In a multilayer, all layers do not have necessarily the same refractive index. The interfaces can therefore have an optical effect by reflecting part of the diffuse light. The interface between layer 1 (refractive index n1 ) and layer 2 (refractive index n2 ) receiving Lambertian light from layer 1 has bi-hemispherical reflectance r12 given by equation (4.7), or equivalently by equation (4.8) as a function of n ¼ n2 =n1 . Its transmittance is t12 ¼ 1  r12 . When it receives Lambertian light from layer 2, its reflectance r12 is given by equation (4.10) or equivalently by equation (4.13). Since r21 , t12 and t21 can all be expressed as functions of r12 (see equations (4.9) and (4.11), (4.12) in §4.2), only r12 needs to be computed. The transfer matrices attached to the interfaces are similarly defined as the one of those layers: f 12

1 ¼ t12



1 r12

r21 t12 t21  r12 r21



1 ¼ ð1  r12 Þ

1 r12

1  r12n1 2 1r12 ð1 þ n 2 Þ

! ð6:55Þ

n2

The transfer matrices representing the external interfaces depend on the illuminating and viewing geometry, as explained in §6.3.1 for the Saunderson correction. In the example featured in figure 6.7, a non-symmetric diffusing layer 1 of refractive index n1 is placed on top of a non-symmetric diffusing layer 2 of refractive index n2 . The bi-layer is surrounded by air (refractive index n0 ¼ 1), and illuminated by Lambertian light on both sides. It is observed at 0° from the normal of the sample (d:0° geometry). The transfer matrix representing the bi-layer is given by: M ¼ FM1 F12 M2 F0

ð6:56Þ

where M1 and M2 representing the layers are defined by equation (6.44), F12 representing the interface between the layers by equation (6.55), F representing the front interface by equation (6.25) in terms of the transfer factors rs , rs , Tin and Tout

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155

FIG. 6.7 – Kubelka’s model: diagram of flux reflections and transmissions by the diffusing layers (dashed arrows) and interfaces (solid arrows). On the right are featured the transfer matrices attached to the layers and interfaces.

(see table 6.1) and F0 representing the back interface by equation (6.27) in terms of the same factors. The reflectances and transmittances of the multilayer can be deduced from the entries of M according to the formulae (6.46).

Chapter 7 Nonscattering Layers on a Diffusing Background In the two previous chapters, we developed two-flux models adapted to either diffusing or non-scattering multilayers. We now propose that we deal with multilayers made of both non-scattering and diffusing layers, as it is the case for many objects: photographs, varnished supports, drawings protected by framing glass, printed surfaces, etc. The boom of color photography and printing industry in Rochester, USA, at the middle of the 20th century, led especially by the Kodak and Xerox companies, have generated an intense research activity in optics and radiometry for colored surfaces. Several base models published in the 1950s originate from there: the Williams–Clapper model for gelatin photo prints [247], the Yule–Nielsen and Clapper–Yule models for halftone ink prints [45, 258], and several extensions were then developed in the 1990s, at the beginning of the digital revolution in the color reproduction domain [17, 239]. Except for the Yule–Nielsen model, which relies on an empirical colorimetric approach initiated by Neugebauer in the 1930s, all models reported in this chapter – although not presented under this angle by their authors – can be viewed as extensions of the Saunderson model, presented in §6.3.1, where the flat interface considered by Saunderson is replaced with a non-scattering component incorporating non-scattering layers and interfaces. The equations attached to these models have therefore a similar structure of the type: ^ ðkÞ ¼ rs ðkÞ þ Tin ðkÞTout ðkÞqðkÞ R 1  ri ðkÞqðkÞ

ð7:1Þ

^ ðkÞ is the spectral reflectance factor of the sample, qðkÞ the spectral intrinsic where R reflectance of the diffusing background. The four terms rs ðkÞ, ri ðkÞ, Tin ðkÞ, and Tout ðkÞ are the transfer factors attached to the non-scattering component, derived from its angular reflectance or transmittance according to the measurement geometry used. They may all depend on wavelength due to the spectral absorptance of the layers. The detailed expressions for these terms will be given for each configuration set out in the following sections, by referring largely to the notions addressed in chapters 5 and 6. We will first consider a multilayer made up of uniform, continuous DOI: 10.1051/978-2-7598-2647-6.c007 © Science Press, EDP Sciences, 2022

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nonscattering layers, e.g., a gelatin layer for photographic prints (Williams–Clapper model) or a mixture of colored dyes (Berns model), then several nonscattering layers, e.g., a pile of colored films, and finally address discontinuous layers via the example of halftone ink layers used in printing. This chapter is also the opportunity to formalize the “generalized two-flux model”, which encompasses all these configurations.

7.1

Uniform Layer on Top of a Diffusing Background

In this section, the nonscattering component placed on top of the diffusing background is a uniform, nonscattering, absorbing layer. The air-layer interface and the layer are considered jointly as one component, of which angular spectral reflectances and transmittances on both sides are determined before deriving their spectral reflectance and transmittance for the angular distribution of light imposed by the measurement geometry used. The layer is in optical contact with the background, and it is assumed that the interface between the two is optically neutral, i.e., as if the layer and background had the same refractive index. The support is strongly diffusing, considered as a Lambertian reflector of spectral reflectance ρ(λ).

7.1.1

Williams–Clapper Model

The model introduced in 1953 by Williams and Clapper from Kodak company was initially dedicated to gelatin photographs [247], but it applies to any diffusing support coated with a uniform transparent layer, which also includes inks, varnishes or protection overlays as long as they are non-scattering. The transparent layer is characterized by its spectral absorption coefficient α(λ) or, equivalently, by its normal intrinsic transmittance t ðkÞ related to α(λ) according to Beer’s law, equation (4.31), and denoted tk in this section. The layer is also characterized by its refractive index n, assumed to be constant over the visible spectrum and which determines the amount of light reflected from and transmitted through the layer-air interface.

FIG. 7.1 – Angular spectral reflectances and transmittances of the non-scattering component formed by the ink-air interface and the transparent layer according to the Williams–Clapper model.

Nonscattering Layers on a Diffusing Background

159

At the air side, for directional light some angle of incidence θ, the angular spectral reflectance Rc ðh; kÞ and transmittance Tc ðh; kÞ of the nonscattering component are defined as follows (see figure 7.1): The front reflectance Rc ðhÞ is simply the Fresnel reflectance of the interface since the layer itself cannot reflect light in the bulk. Rc ðhÞ ¼ R12 ðhÞ

ð7:2Þ

The forward transmittance Tc ðh; kÞ includes the Fresnel transmittance T12 ðhÞ of the interface and the attenuation of light across the layer at the angle h0 ¼ arcsinðsin h=n Þ at which light is refracted [see §4.5.1 and §4.6.1, equation (4.36)]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin h=n Þ2 ð7:3Þ Tc ðh; kÞ ¼ T12 ðhÞtk At the back side, for a directional light making angle h0 with the normal of the sample, the backside reflectance R0c ðh0 ; kÞ includes the Fresnel reflectance of the interface, which is equal to 1 when h0 is larger than the critical angle arcsinð1=n Þ, as well as the attenuation along the double path across the transparent layer: 2=cos h0

R0c ðh0 ; kÞ ¼ R21 ðh0 Þtk

ð7:4Þ

The backward transmittance Tc0 ðh0 ; kÞ includes the attenuation of light across the transparent layer and the Fresnel transmittance of the interface. Thanks to the reciprocity of the Fresnel transmittance formulae, it is identical to Tc ðh; kÞ and can be expressed as a function of the angle in air h ¼ arcsinðn sin h0 Þ: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin h=n Þ2 1=cos h0 Tc0 ðh0 ; kÞ ¼ T21 ðh0 Þtk ¼ T12 ðhÞtk ð7:5Þ From the angular spectral transfer factors, one can now define the actual transfer factors rs ðkÞ, ri ðkÞ, Tin ðkÞ, and Tout ðkÞ attached to the non-scattering component according to the set up geometry. Consider an illumination being either directional at 8° or Lambertian, and a detection based on a detector at 8° or on an integrating sphere. The expressions for rs ðkÞ, ri ðkÞ, Tin ðkÞ, and Tout ðkÞ as functions of the refractive index n and of intrinsic spectral transmittance t are summarized in Table 7.1 for typical geometries recommended by the CIE. In their original paper, Williams and Clapper considered a 45°:0° geometry; the extension to a d:8° geometry has been introduced much later by Shore and Spoonhower [210]. Term rs is zero for the geometries excluding the specular reflection component. If the specular component is included, typically in the di:8° geometry, it is given by Rc ðhÞ evaluated at 8°, therefore by R12 ð8 Þ. It is generally independent of wavelength, unless the layer’s refractive index strongly depends upon wavelength. Term Tin depends only on the illumination geometry. When the incident light is directional and makes angle hi with the normal of the sample, it is given by Tc ðh; kÞ evaluated at angle hi : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin hi =n Þ2 : ð7:6Þ Tin ðkÞ ¼ T12 ðhi Þtk

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When the incident light is Lambertian, Tin is obtained by integrating Tc ðh; kÞ over the hemisphere in a similar manner as in equation (2.13), i.e., Z p=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin h=n Þ2 Tin ðkÞ ¼ T12 ðhÞtk sin 2hdh: ð7:7Þ h¼0

Term Tout depends only on the observation geometry. When the detector captures a spectral radiance along one direction at angle ho from the normal (as a human observer would actually do), the spectral transmittance is given by function Tc0 ðh0 ; kÞ evaluated at angle ho and multiplied by a factor 1=n 2 in order to account for the change of geometrical extent applied to the spectral radiance when transiting from the layer to air (see §4.1): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T12 ðho Þ 1= 1ðsin ho =nÞ2 tk ð7:8Þ Tout ðkÞ ¼ n2 When an integrating sphere collects all the flux emerging in air over the hemisphere, Tout is obtained by integrating Tc0 ðh0 ; kÞ as follows: Z p=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin ho =n Þ2 Tout ðkÞ ¼ T21 ðhÞtk sin 2hdh: ð7:9Þ h¼0

Term ri is independent of the measuring geometry since is depicts the reflection of Lambertian light from the diffusing support to itself. It is obtained by integrating R0c ðh0 ; kÞ as follows: Z p=2 2=cos h0 r i ð kÞ ¼ R21 ðh0 Þtk sin 2h0 dh0 ð7:10Þ h0 ¼0

Finally, the Williams–Clapper equation is given by equation (7.1). TAB. 7.1 – Transfer factors attached to the interface for different standard measurement geometries*. Geometry di:8° de:8°

0

8°:d

  n1 2 nþ1

45°:0° *

Tin

rs   n1 2 nþ1

0

Equation (7.7) Equation (7.7) "

  # n1 2 1 tk nþ1 

T12 ð45

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ffi

1= Þtk

Tout   # 1 n1 2 1 tk n2 nþ1 "   # 1 n1 2 tk 1  n2 nþ1 "

11=ð2n Þ

Equation (7.9) " 1 n2

1

  # n1 2 tk nþ1

The terms that should be evaluated at 8° are evaluated at 0°, which makes a negligible numerical difference.

Nonscattering Layers on a Diffusing Background

161

For example, if the measuring geometry is bi-directional with an illumination at angle hi and an observation at angle ho , the spectral reflectance factor is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1ðsin hi =n Þ2 þ 1= 1ðsin ho =n Þ2 T ð h ÞT ð h Þt qðkÞ 12 i 12 o ^ ðkÞ ¼ h R k  i R ð7:11Þ 0 p=2 2=cos h 0 0 0 n 2 1  h0 ¼0 R12 ðh Þtk sin 2h dh qðkÞ In the absence of a layer on top of the background, or when the layer is perfectly clear, the Williams–Clapper equation can be reduced to the Saunderson equation (see §6.3.1). Spectral intrinsic reflectance qðkÞ of the support can be deduced from its spectral reflectance factor by applying an inversed Saunderson correction, i.e., using equation (6.21). In the presence of a layer, its intrinsic spectral transmittance must be known or ^ ðkÞ of the experimentally determined. We can measure spectral reflectance factor R sample and deduce numerically the value for tk at each wavelength – unfortunately, ^ ðkÞ. Once tk is obtained, the there exists no analytical formula for tk as a function of R Williams–Clapper formula enables us to predict easily the reflectance factor that would be measured if the optical thickness of the transparent layer were changed, for example because of a higher concentration in absorbing substance or an increase of thickness. It suffices to replace tk with tkc , where γ denotes the ratio between the new and initial optical thicknesses.

7.1.2

Berns’ Model

Printing techniques rarely deposit continuous layers of colorants on the support, but rather small ink patterns as it will be shown later in this chapter. There exist, nonetheless, a few exceptions, among which the dye transfer technologies that are able to transfer continuous layers of dyes with controlled concentration (D2T2 technology) [124]. The surfaces printed in this way on a diffusing support, usually a white polymer, have a similar structure as the gelatin photograph prints and the Williams–Clapper applies. However, Berns proposed a simplified version where the orientation of light within the dye layer is not considered: intrinsic spectral transmittance t ðkÞ of the layer remains the same regardless of the angular distribution of the light. This means that t ðkÞ is taken out of the integrals in equations (7.7), (7.9) and (7.10), and no exponent is applied to it in case of oblique light path. Terms ri ðkÞ, Tin ðkÞ, and Tout ðkÞ – which depend on t ðkÞ – are therefore simplified, and can then be expressed in terms of ri , Tin , and Tout defined in the Saunderson model for an interface alone according to the measuring geometry (see §6.3.1): Tin ðkÞ ¼ t ðkÞTin Tout ðkÞ ¼ t ðkÞTout

ð7:12Þ

ri ðkÞ ¼ t ðkÞri 2

The Berns equation for the reflectance factor of the printed surface is therefore written:

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162

2 ^ ðkÞ ¼ rs þ Tin Tout qðkÞt ðkÞ R 1  ri qðkÞt 2 ðkÞ

ð7:13Þ

Notice that Berns did not use this formalism in his paper, but an equivalent one based on the Kubelka–Munk formalism, in particular equation (6.17) by considering that the layer placed on top of the diffusing background has a scattering coefficient equal to zero. A Saunderson correction is then applied. Moreover, Berns related intrinsic spectral transmittance t ðkÞ of the coloring layer with the ones attached to individual dyes, ti ðkÞ and their respective concentrations ci , according to Beer’s law (see §4.5.2): Y c t ð kÞ ¼ ti i ðkÞ ð7:14Þ i

7.1.3

Insurface and Subsurface Reflections According to the Lighting Geometry

The reflection of light by the air-material interface, represented by term rs in equation (7.13), is generally achromatic and related to the visual attribute of gloss. It is therefore often discarded from the measurements and the models, which aim rather at evaluating the object’s color from the light having penetrated into the materials. This is probably the reason why the respective magnitudes of the specular reflection component (or “insurface” reflection component) and the diffuse component are rarely analyzed. It should be noticed, however, that they strongly depend on the illumination geometry, in particular the solid angle under which the surface is lit. In order to show this fact, let us consider a surface made of a Lambertian background with intrinsic spectral reflectance ρ(λ) bordered by a smooth interface with air. One observes a small area Δs frontally, i.e., along its normal. This area is lit according to a conical geometry: the incident light spans a certain cone into which ð1Þ the spectral radiance is a constant, Lk , while it is zero outside. The geometrical extent attached to the cone is Z 2p Z a DG ¼ Ds cos h sin hdhdu ¼ pDs sin2 a ð7:15Þ u¼0

h¼0

ðs Þ

Spectral radiance Lk specularly reflected from the air-material interface along the direction of the observer is simply ðs Þ

ð1Þ

Lk ¼ R12 ð0 ÞLk ðd Þ

ð7:16Þ

whereas spectral radiance Lk issued from the material as a result of the multiple scattering process is (see §6.3.1):

Nonscattering Layers on a Diffusing Background

ðd Þ

Lk ¼

163

1 Tin Tout qðkÞ ð1Þ  E p 1  ri qðkÞ k

ð1Þ

ð7:17Þ ð1Þ

where Ek denotes the spectral irradiance, related to Lk by ð1Þ

Ek ¼

DG ð1Þ ð1Þ L ¼ p sin2 aLk Ds k

ð7:18Þ

Tin Tout qðkÞ ð1Þ L 1  ri qðkÞ k

ð7:19Þ

Equation (7.17) thus becomes ðd Þ

Lk ¼ sin2 a

and the total spectral radiance captured by a camera or an observer is   Tin Tout qðkÞ ð1Þ ðs Þ ðd Þ Lk ¼ Lk þ Lk ¼ R12 ð0 Þ þ sin2 a L 1  ri qðkÞ k ðd Þ

ðs Þ

ð7:20Þ

When α is small, Lk is much smaller than Lk . We understand why the optical system in glossmeters relies on collimated incident light (narrow solid angle) and why their response, when used on glassy samples, is almost independent of the color of the material behind the smooth surface. We also understand the importance of the solid angle of illumination and the refractive index of the material, reason why they are precisely specified by the standards (see §3.3.2). ðd Þ ðs Þ The relative levels between Lk and Lk also become interesting to analyze whenever an object reflects specularly the light source on only part of its surface, either because it is wide and the source is far away, or because it is curved. As shown by equation (7.20), the luminance contrast between the area where the source is specularly reflected and the area where it does not depends a lot on the solid angle subtended by the source at each point of the object, i.e., on the size of the source and its distance from the object. This can easily be experienced with a large uniform source, preferably Lambertian, which is placed near a colored plane object and gradually moved away from it in order to decrease the solid angle of illumination. If one takes pictures of the object during the experiment with a camera whose acquisition parameters (aperture, exposure time, gain, etc.) remain unchanged, one sees that the diffuse component disappears as the source moves away; from a certain distance, one sees only the color of the source in the area where it is specularly reflected, and the image is black elsewhere. However, a human observer attending this experiment will not have the same sensation: his visual system tends to adapt to the radiance level issued from the zone without specular reflection, by which the color sensation is produced. The phenomena of chromatic adaptation and color constancy reinforce this tendency. Thus, the color perceived in the zone without reflection will remain constant, whereas that of the area where the source is reflected will become dull. One can reproduce this impression by photography by adapting the acquisition parameters for each source-object distance so that the colors of the objects in the area without specular reflection of the source remain constant. This is what is shown in figure 7.2. As the

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164

FIG. 7.2 – Photographic print on glossy paper illuminated with a light table located at different distances from the print, being closer in (a) and the farthest in (f). The acquisition parameters of the camera are modified for each picture in order to obtain nearly the same color for the blank paper. The area where the source is specularly reflected becomes brighter as the source moves away from the surface and the solid angle of illumination becomes narrower. In pictures (e) and (f), this area is even completely white because the sensor has saturated.

source-object distance increases, the area where the source is reflected becomes brighter and brighter, or even white when the detector is saturated. The definition of a spectral radiance factor, in this case, is not trivial. Two options can make sense. The first option is to consider as reference surface a specular reflector, preferably achromatic, such as mirror whose spectral angular reflectance Rm ðhÞ is constant with respect to the wavelength. The spectral radiance reflected from the mirror is ðm Þ

Lk

ð1Þ

¼ Rm ð0 ÞLk ;

ð7:21Þ

and the spectral radiance factor is   ðd Þ ðs Þ 1 Tin Tout qðkÞ 2  ^ ð kÞ ¼ L k þ L k ¼ R sin R ð 0 Þ þ a 12 ðm Þ Rm ð0 Þ 1  ri qðkÞ Lk

ð7:22Þ

This option is consistent with the situation where a camera takes pictures of the object without changing any acquisition parameter. The second option is to consider as reference surface a perfectly white diffuser. ðw Þ Spectral radiance Lk issued from it is

Nonscattering Layers on a Diffusing Background

ðw Þ

Lk

¼

Ei ð k Þ ð1Þ ¼ sin2 aLk p

165

ð7:23Þ

and the spectral radiance factor of the sample is written ðs Þ

ðd Þ

 ^ ðkÞ ¼ Lk þ Lk ¼ R12 ð0 Þ þ Tin Tout qðkÞ R 2 ðw Þ 1  ri qðkÞ sin a Lk

ð7:24Þ

This option is more consistent with our visual experience: thanks to the chromatic adaptation, the diffuse reflection component tends to remain constant independently of the position of the light source. The specular component grows as the solid angle decreases.

7.2

Transparent Multilayers on Top of a Diffusing Background

The Williams–Clapper model only applies when the non-scattering layer is in optical contact with the support. When the layer and support are separated by an air layer (or any other clear medium with a different refractive index), two interfaces are introduced: the layer-air interface and the air-support interface. Since these two interfaces back-reflect some light, their effect is significant on the global reflectance of the sample. Figure 7.3 shows an example where cyan-colored acetate films are placed on top of a white and black background: one film is simply placed on the background and an air layer remains between them, whereas the other film is pasted onto the background with an index matching liquid. We can clearly see the difference visually: in the absence of optical contact, the sample has a lighter color because of

FIG. 7.3 – Cyan acetate films placed on top of a diffusing background of white and black color. The film on the top of the picture is simply placed on the background, and an air layer remains between them. The film on the bottom is pasted onto the background with an index matching liquid.

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166

the light back reflected from the two internal interfaces bordering the air layer. In the pasted film, some air bubbles below the film are made visible by displaying the same color as the film on the upper part of the picture. The picture was taken in specular reflection included conditions, excepts for some curved areas where the specular reflection component is absent, which therefore look darker. Depositing a colored transparent film on a diffusing support is the simplest case of a non-diffusing multilayer on a diffusing background. Here, the transparent component is formed by the air-film interface, the film layer, the film-air interface, the air layer (which is optically neutral) and the air-background interface. By extension, one can imagine cases comprising more layers of different refractive indices. All layers and their bordering interfaces participate in the overall reflectance of the sample. The methodology that we have adopted to present the Williams– Clapper model remains valid: it suffices to replace the angular spectral transfer factors attached to the non-diffusing component {interface + transparent layer}, i.e. terms Rc ðhÞ, Tc ðhÞ, R0c ðhÞ and Tc0 ðhÞ expressed in equations (7.2) – (7.5), with the ones of the new transparent component. This results in what has been called an “extended Williams–Clapper model” in Ref. [215]. In this paper, the angular transfer factors were calculated by following the iterative method described in §5.1.3, but the matrix formalism described in §5.1.4 can advantageously be used. Let us first consider a film on top of the diffusing support. The film is characterized by its refractive index n2 and its spectral intrinsic spectral transmittance t ðkÞ. The background is characterized by is refractive index n3 and its intrinsic reflectance ρ(λ), which is obtained by measuring the reflectance factor of the support and using equation (6.21). As written above, the transparent component topping the diffusing background is formed by the air-film interface, the film layer, the film-air interface, the air layer, and the air-background interface. By representing these constituting elements with their respective flux transfer matrices as defined in chapter 5, equations (5.38) and (5.40), flux transfer matrix M representing the whole transparent component is given by: M ðh1 ; kÞ ¼ F12 ðh1 ÞLðh; tk ÞF21 ðh2 ÞF13 ðh1 Þ

ð7:25Þ

where symbol * represents either the s or p polarization, h1 denotes the orientation of light in air and h2 ¼ arcsinðsin h1 =n2 Þ the orientation of light inside the film (figure 7.4). Notice that when entering the diffusing background, light is oriented according to angle h3 ¼ arcsinðsin h1 =n3 Þ. The angular spectral transfer factors attached to the transparent components, Rc ðh1 ; kÞ, Tc ðh1 ; kÞ, R0c ðh3 ; kÞ and Tc0 ðh3 ; kÞ, are deduced from the entries of M thanks to equation (5.14); they are computed for s and p polarizations, then averaged. Their analytical expressions – not reproduced here because of their length – are functions of Fresnel reflectances, thereby on the refractive indices, and of the intrinsic spectral transmittance of the layers, thereby on wavelength. Then, actual flux transfers rs , ri , Tin , and Tout are defined from the angular ones. rs , Tin , and Tout are computed as indicated in Table 7.1 according to the measuring geometry considered, while ri is always computed as follows: Z p=2 R0c ðh0 ; kÞ sin 2h0 dh0 ð7:26Þ ri ðkÞ ¼ h0 ¼0

Nonscattering Layers on a Diffusing Background

167

FIG. 7.4 – Transparent film placed on top of a diffusing background, with (a) the orientations of light in different media when directional light comes from air at angle θ1, and (b) the actual flux transfers between the transparent component (film and background’s interface). Once can extend the line of reasoning to a pile of k identical films instead of a single film. Transfer matrix Mk representing the whole transparent component is: Mk ðh1 Þ ¼ ½F12 ðh1 ÞLðh; t ðkÞÞF21 ðh2 Þk F13 ðh1 Þ

ð7:27Þ

and the procedure previously described can be repeated in a similar manner. One can also consider other materials than air between the films, by adapting the relative refractive indices of the internal interfaces accordingly. In figure 7.5, the spectral reflectance factors of two diffusing backgrounds topped by cyan acetate films are plotted. This complements the study proposed in §5.5 where the same cyan films were placed on top of a specular background. All reflectance factors correspond to a d:0° geometry. The two supports used here were a sheet of white polymer, and a sheet of green paper. The spectral reflectance factors of these two supports, denoted by Rp ðkÞ, are plotted in dotted line, and their spectral reflectance factors after adding incrementally 1–5 films are plotted in solid line. The spectral reflectance factor plotted in red dashed lines corresponds to the predictions given by the model, which coincide rather well with the measured ones. The blue dotted lines represent the predicted reflectance factor of an infinite pile. As it could be observed with specular reflectors, reflectance factors of samples transit gradually from one of the supports to one of the infinite piles, and the transition is faster and spectral domains are the most absorbing. When two extreme samples (no film, and infinite number of films) have the same reflectance factor value at a given wavelength, all samples have also this value independently of the number of films. We retrieve similar trends in oil painting, in particular art glazes, where a diffusing background (white or colored) is topped by layers of colored, very weakly scattering varnish [61, 212]. The back-reflection of light is not due to the air-film interfaces as in our example, but the slight back-scattering by the pigments. Drawings or photographs placed under a protective glass are of the same configuration as the one drawn in figure 7.4.

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Optical Models for Material Appearance

FIG. 7.5 – Measured (solid lines) and predicted (dashed lines) spectral reflectance factors of 1–5 cyan acetate films placed on top of (a) white PVC and (b) green paper with respective reflectance Rp plotted in dotted line. R∞ denotes the reflectance of an infinite pile of films.

7.3

Generalized Two-Flux Model

Although this chapter is dedicated to transparent layers placed on top of a diffusing background, we propose an aside comment allowing us to extend the methodology introduced in the previous section and to draw the limits of the two-flux approach for the propagation of light into materials.

7.3.1

Configurations Where a Two-Flux Model Applies

In essence, a two-flux model can be used in each of these three configurations: (1) A layer or a multilayer containing only nonscattering materials and smooth interfaces, illuminated by a directional flux. The reflected and transmitted fluxes are also directional and depend on the initial angle of incidence. The object is qualified as specular or transparent, and it can be fully characterized by its angular spectral transfer factors. (2) A layer or a multilayer containing only strongly scattering materials, which reflects or transmits a Lambertian flux no matter the way used to illuminate it. It is characterized by its geometry-independent spectral transfer factors. (3) A pile of non-scattering and strongly scattering layers under any kind of illumination geometry, observed along any direction, provided the following methodology is followed: each succession of nonscattering components (transparent layers and smooth interfaces) should be considered as a single nonscattering component. The angular spectral transfer factors of this component are to be determined using the two-flux model for directional light, for each polarization state s and p. Then, the angular spectral transfer factors are angularly integrated with respect to the angular distribution of the sources on their two sides, and for each transfer factor, the resulting expressions attached to the two

Nonscattering Layers on a Diffusing Background

169

polarizations s and p are averaged. One thus obtains specific spectral transfer factors of the non-scattering multilayer component, which can finally be combined with the ones of the surrounding strongly scattering layers. Flux transfer matrices can be used, by multiplying the transfer matrices of the non-scattering components (built from the specific transfer factors) and strongly scattering components. This methodology has already been illustrated by the case of a transparent multilayer placed on top of a diffusing background in §7.2. For example, the object featured in figure 7.6 belongs to the third category. At the upper side, there is a succession of four nonscattering components: the air-glass interface, the glass layer, the plate-air interface, and the air-paint interface (the air layer is optically neutral). They are considered together as one nonscattering component represented by transfer matrix M1. This matrix is based on the specific transfer factors of the nonscattering component determined with respect to the illumination and observation geometries selected at the upper side, and with respect to the Lambertian lighting issued from the paint on the lower side. At the middle of the multilayer, the paint-air interface and the air-paper interfaces are considered as one non-scattering component illuminated on both sides by Lambertian sources, represented by flux transfer matrix M3. At the lower side, the transparent coating bulk and the coating-air interface form one nonscattering component represented by matrix M5; this matrix is based on transfer factors computed with respect to the Lambertian lighting issued from the paper on the upper side, and the illumination and observation geometry used on the lower side. Finally, transfer matrix M of the whole multilayer is given by the following matrix product, to be evaluated for each wavelength: M ¼ M1 M2 M3 M4 M5

ð7:28Þ

FIG. 7.6 – Example of multilayer containing non-scattering and strongly scattering layers, and corresponding flux transfer matrices. All interfaces, represented by a bold black line, are assumed to be smooth.

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170

Let us insist on the fact that, in the generalized two-flux model, the multilayer must contain only non-scattering or strongly scattering components. Weakly scattering components such as an opalescent layer or a rough interface are generally not allowed, except in the special case where they are bordered by strongly scattering layers on their upper and lower sides, and their specific transfer factors can be evaluated. However, the reflectance or transmittance of a frosted glass plate, i.e., a transparent layer bordered by rough interfaces, cannot be predicted by a two-flux model since the rough interfaces are not bordered by any strongly scattering layer. The reason for this limitation of the two-flux approach is not obvious by reasoning from the flux transfer model. It appears more clearly by describing the multiple reflection process of light between the different components. Markov chains and their matrix formalism provide an excellent tool for that. This matrix formalism has no particular interest on a computational point of view: it leads exactly to the same formulae of transfer factors as the one derived with flux transfer matrices. However, it provides some interesting notions that deserve to be mentioned here.

7.3.2

Multiple Reflection Processes and Homogeneous Discrete-Time Markov Chains

A Markov chain is a convenient representation of the behavior of a physical system which describes the different states that the system may occupy and indicates how the system moves from one state to another in time [228]. It is particularly interesting with discrete time processes, where successive events happen with given probabilities. In our case, the successive events are reflections and transmissions of light by the different components of the multilayer, and the probabilities attached to these events are the corresponding reflectances and transmittances, which can be considered as probabilities since their values are between 0 and 1. The general definition of the Markov chain is the following. Consider a system that may occupy a finite number of states. Random variables Y0, Y1, Y2, …, Yk give the respective states y0, y1, y2, …, yk in which the system is at discrete “times” 0, 1, 2, …, k. A sequence y0, y1, y2, …, yk of states corresponds to a certain realization of a random walk. The probability m that this realization occurs is the conditional probability that the light is observed in state yn at time k, given that it was previously observed in states y0, y1, …, yk – 1, respectively, at times 0, 1, …, k – 1 m ¼ ProbfYk ¼ yk j Y0 ¼ y0 ; :::; Yk1 ¼ yk1 g

ð7:29Þ

This conditional probability m can also be interpreted as the probability of transition from state yk – 1 to state yk, given that it has previously moved from state y0 to state y1, from state y1 to state y2, etc. When transition probability yk1 ! yk is independent of previous transitions y0 ! y1 , y1 ! y2 , … yn2 ! yk1 , the stochastic process is said to be memoryless. This property may be written as ProbfYk ¼ yk jY0 ¼ y0 ; :::; Yk1 ¼ yk1 g ¼ ProbfYk ¼ yk jYk1 ¼ yk1 g Equality (7.30) is the defining equation of a discrete-time Markov chain.

ð7:30Þ

Nonscattering Layers on a Diffusing Background

171

FIG. 7.7 – Markov chain attached to a superposition of two optical components (layers or smooth interfaces).

Figure 7.7 shows an example of Markov chain corresponding to the superposition of two optical components, e.g., the two air-glass interfaces of a clear window (§4.4.1), or two glass plates (§5.1.1), or a pile of two strongly scattering layers (Kubelka’s model, §6.8.1), or an air-matter interface on top of a diffusing background (Saunderson model, §6.3.1), etc. Note that the graph is similar to the one drawn in figure 5.3. States 1–6 correspond to the forward and backward fluxes at different levels of the object. Arcs correspond to the transition probabilities of light between states, therefore to reflectances and transmittances. States representing the external sources are labelled 1 and 2; they are called ephemeral states since they are the starting point of a finite number of transitions (here equal to 1). Detectors are represented by absorbing states, labelled 5 and 6; the loop connecting each detector with itself indicates the fact that the light captured by the detectors no longer interacts with the components. For other states, called transient states, odd numbers (resp. even numbers) are attributed to the fluxes propagating forwards (resp. backwards). One can notice that transition probabilities yk1 ! yk are independent of time parameter k. The Markov chain is then said to be a homogenous discrete-time Markov chain (HDMC). If the system is in a given state i, it evolves towards a given state j with a same probability whatever the time and whatever the previous states in which the system has been through. It has been shown in [91] that every two-flux model can be represented by a HDMC, and that, reciprocally, the condition of validity of a two-flux model is that it can be represented by a HDMC. Every component of the studied object must have constant transfer factors during the multiple reflection process of light, i.e., time-independent transition probabilities. This condition is met with a glass plate illuminated by directional light: the light has the

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Optical Models for Material Appearance

same orientation each time it falls on an interface, and the interfaces have therefore always the same reflectance and transmittance during the multiple reflection process. It is also met with a strongly scattering background bounded by its interface with air (Saunderson model): the intrinsic reflectance of the strongly scattering layer is constant, and the internal reflectance of the interface is also constant since it receives at every time Lambertian light from the background. However, in the case of the frosted glass plate evoked in the previous section, the rough interfaces scatter light progressively, the angular distribution of light evolves during the process and the reflectances and transmittances vary accordingly. This latter case cannot be represented by a HDMC, and has said before, its reflectance or transmittance cannot be predicted from a two-flux model.

7.3.3

Transition Probability Matrices

The study of homogenous discrete-time Markov chains relies on a matrix formalism, based on transition probability matrices. Let mij be the probability of transition from state i to state

j, which is zero if the two flux states are not directly connected. Matrix M ¼ mij is called the single-step transition probability matrix of the Markov chain. The matrix attached to the Markov chain represented in figure 7.7 is 1 0 0 R1 0 0 T1 0 B 0 0 0 T20 R02 0 C C B B 0 0 0 R2 T2 0 C C B 0 ð7:31Þ M¼B 0 T10 C C B 0 0 R1 0 C B C B @0 0 0 0 1 0 A 0 0 0 0 0 1 For example, the probability of transition from state 1 (line 1 of M) to state 3 (column 3) is T1 . Since the Markov chain contains two absorbing states, it is an absorbing chain and the one-step transition matrix M can be decomposed into the following block matrix as follows [162]:   A B M¼ ð7:32Þ 0 I2 where A, transition matrix of transient states, is a square matrix whose entries are probabilities of transition from a nonabsorbing state to a nonabsorbing state, B a rectangular matrix whose entries are probabilities of transition from a nonabsorbing state to an absorbing state, 0 is a rectangular block of zeros, and I2 is the 2 × 2 identity matrix (the number 2 coming from the number of absorbing states). These blocks are featured by the lines drawn in the matrix in equation (7.31). The probabilities of transitions from state i to state j in k steps are given by entries of Mk , called k-step transition probability matrix. We are interested in the probabilities of transition “at the end” of the process, i.e., when the number of steps tends to infinity. One can show that as k tends to infinity, Mk converges towards the following matrix (see the details in [91]):

Nonscattering Layers on a Diffusing Background 0

k

M1 ¼ lim M ¼ k!1



0 0

0 B0 B  B0 1 B ðI  AÞ B ¼B B0 I2 B B @0 0

173

0 0 0 0

0 0 0 0

0 0 0 0

1 m15 1 m25 : :

0 0

0 0

0 0

1 0

1 1 m16 1C m26 C : C C : C C C C 0 A 1

ð7:33Þ

The fact that all entries in the first four columns are zero means that no light remains in the object: all the non-absorbed light has reached one of the two 1 1 1 1 detectors. The four entries m15 , m16 , m25 , m26 correspond to the transfer factors of 0 0 the object, i.e., respectively, T12 , R12 , R12 , and T12 (using similar notations as in the previous chapters). A straightforward matrix computation from equations (7.31) to (7.33) leads to the same formulae as those previously encountered whenever two components are superimposed (see §5.1.1, §6.8.1, or §6.3.1): T1 T10 R2 ; 1  R01 R2 T2 T20 R01 ¼ R02 þ ; 1  R01 R2

T1 T2 1  R01 R2 T10 T20 ¼ 1  R01 R2

R12 ¼ R1 þ

T12 ¼

R012

0 T12

ð7:34Þ

The method can be extended to any number of superimposed components. The Markov chain keeps the same structure as the one considered above: in the external positions, ephemeral and absorbing states represent, respectively, the sources and detectors; at each internal level, two states represent the forward and backward fluxes. The size of the transition probability matrices increases proportionally to the number of components, which is a drawback when the structures contain many layers (the flux transfer matrices are more convenient since their size is proportional to the number of fluxes, not to the number of components). However, the definition of the homogeneous discrete-time Markov chain is useful to remind us that only non-scattering and strongly scattering components can ensure a memoryless process (all reflectances and transmittances remain constant during the process), and the matrix formalism has interesting properties regarding the average number of transitions.

7.3.4

Average Number of Transfers

Let us pursue with the Markov chain shown in figure 7.7, associated with a superimposition of two components. From the one-step transition probability matrix M and the transition matrix of transient states A defined in equation (7.31), the fundamental matrix Q of the Markov chain is defined as 0 1 1 0 T1 =D R2 T1 =D B 0 1 R01 T20 =D T20 =D C C Q ¼ ðI  AÞ1 ¼ B ð7:35Þ @0 0 1=D R2 =D A 0 0 R01 =D 1=D

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174

with D ¼ 1  R01 R2 . Its entries qij give the average number of times the process is in state j if it has started in state i. Since states 1 and 2 are ephemeral, one logically has q11 ¼ q22 ¼ 1: the process occurs only once in these states. Entry q34 represents the average number of times light is reflected from the front side of component 2: q34 ¼

R2 R01 R2

ð7:36Þ

If one transposes this result to the case of a white diffusing background of intrinsic reflectance R2 ¼ 1 and refractive index 1.5, bordered by a smooth interface with air, R01 ¼ r21  0:6, one can conclude that light is reflected 1=ð1  0:6Þ ¼ 2:5 times on the background in average. If the background has an intrinsic reflectance of 0.7, the average number of reflections drops to 1.2.

7.4

Spectral Reflectance of Printed Surfaces

The last extension of the two-flux approach that we propose to address in this book concerns the case where some of the layers are not uniform, i.e., their absorptance varies with respect to the longitudinal position. This mainly concerns the printed surfaces, on which discontinuous ink layers are deposited. A specific technique, called halftoning, enable to generate many colors from a reduced number of colored inks, by printing the inks according to screens of small patterns. The variety of printing materials (papers or other supports, inks, varnishes or other overlays) is so wide, and the ink-support interaction so complex, that it is impossible to predict halftone colors without characterizing a few color patches produced with the considered printing materials and parameters. However, the number of color patches one needs to produce can be reduced to a reasonable limit by selecting them judiciously. Halftone colors can often be accurately predicted by combining the spectral reflectances of the color patches (spectral Neugebauer model, Yule–Nielsen model), or using a model based on a two-flux approach (Clapper–Yule model).

7.4.1

Halftone Colors

In contrast with art painting where the colors can be tuned continuously over the surface by selecting the appropriate material mixtures and concentrations, or the appropriate layer thickness, printing is based on a limited number of colored inks, whose quantity is generally fixed except for recent technologies (e.g., D2T2 or some professional inkjet printers [124]). Printing is therefore a binary process: in each point, a “dithering” method decides for each ink whether it is deposited or not, and the inked points are spatially arranged in order to form small patterns, the halftone screen, whose area varies according to the amount needed locally. The ink patterns are expected to be small enough to remain invisible by the observer at a common,

Nonscattering Layers on a Diffusing Background

175

natural, distance. Figure 7.8 shows two examples of halftone screens: a picture of a printed surface where a grey level image is reproduced by black ink patterns only, and a digital color halftone screen resulting from the superimposition of the cyan, magenta and yellow (CMY) ink dot screens.

FIG. 7.8 – Examples of periodical halftone screens: (a) Detail of a portrait in a magazine printed in black and white, (b) digital CMY halftone screen. There exist many types of halftone screens where the ink patterns can have various shapes, and can be organized either periodically or randomly [140]. In any case, the superimposition of the halftone screens attached to the N primary inks yields a mosaic of 2N areas whose colors are called Neugebauer primaries. For three primary inks (e.g., cyan, magenta and yellow), one obtains a set of eight Neugebauer primaries: white (no ink), cyan alone, magenta alone, yellow alone, red (magenta & yellow), green (cyan & yellow), blue (cyan & magenta) and black (cyan & magenta & yellow). In classical clustered-dot or error diffusion screenings, the fractional area occupied by each Neugebauer primary can be deduced from the surface coverages of the primary inks according to Demichel’s equations [48]. These equations are valid in all cases where the ink halftone dots are laid out independently, e.g. in stochastic screening, in error diffusion, or in mutually rotated clustered dot screens. For cyan, magenta and yellow primary inks with respective surface coverages c, m, and y, the surface coverages ak of the eight Neugebauer primaries are, respectively: aw ac am ay

¼ ð1  cÞð1  m Þð1  y Þ ¼ c ð1  m Þð1  y Þ ¼ ð1  cÞm ð1  y Þ ¼ ð1  cÞð1  m Þy

am þ y ¼ ð1  cÞmy ac þ y ¼ cð1  m Þy ac þ m ¼ cm ð1  y Þ ac þ m þ y ¼ cmy

ð7:37Þ

In the printing industry, color management mainly relies on look-up tables relating the digital colors and the printed ones, by printing hundreds or thousands of color patches and measuring them with a spectrocolorimeter [124]. This process is

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176

operational in quadrichromy (i.e., in four-ink printing) but long and cumbersome; and it must be repeated each time one printing material or one parameter in the printing process is modified. There is therefore an interest in predicting, thanks to an optical model, all these colors using a model calibrated from a small number of printed patches. Every prediction model exhibits a general equation involving spectral parameters describing the absorption and reflection of light by the inks and the paper, as well as the respective surface coverages of the ink dots. Below are presented three classical spectral reflectance prediction models developed for halftone prints.

7.4.2

Spectral Neugebauer Model

The model proposed by Neugebauer in 1937 intended to predict the colorimetric values of halftone prints by linear combination of the ones of the Neugebauer primaries measured on large uniform color patches [175]. This method can be translated into a spectral version by replacing the colorimetric values with spectral reflectances. For 3-ink halftones, the spectral Neugebauer equation is written: R ð kÞ ¼

8 X

ak Rk ðkÞ

ð7:38Þ

k¼1

where RðkÞ denotes the spectral reflectance of the halftone print, Rk ðkÞ; k ¼ 1; :::; 8 those of the Neugebauer primaries, and ak their respective surface coverages given by the Demichel equations as function of the ink surface coverages c, m, and y for the considered halftone color. Figure 7.9a shows an example of prediction given by the Neugebauer model in the case of a cyan ink halftone at nominal surface coverage 0.5. The halftone contains two primaries: paper white and cyan, whose respective spectral reflectances Rw ðkÞ and Rc ðkÞ are plotted in solid lines in the figure. The Neugebauer model in this case is written: RðkÞ ¼ ð1  ac ÞRw ðkÞ þ ac Rc ðkÞ

ð7:39Þ

Since each primary should occupy half the surface, the computed value of spectral reflectance of the halftone should be the average of Rw ðkÞ and Rc ðkÞ, plotted in dotted line, but one sees that it is far from its measured value (solid line). One may think that at the printing stage, the cyan ink spreads over the surface and therefore occupies a larger area than expected. Printers call this phenomenon “dot gain”. By fitting the surface coverage of the cyan ink so as to minimize the difference between predicted and measured spectra, one finds an optimal surface coverage ac = 0.59, which is not very plausible physically. The predicted spectral reflectance is plotted is dashed line; its deviation from the measured spectral reflectance remains important: the equivalent color distance would be well perceptible (2.2 units of CIELAB ΔE94).

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177

FIG. 7.9 – Measured spectral reflectances of blank paper, solid cyan and a halftone of cyan of nominal surface coverage 0.5 (solid lines, identical in the two graphs) and spectral reflectances of the halftone predicted (a) by the Neugebauer model with a surface coverage of 0.5 (dotted line) or a surface coverage of 0.59 (dashed line), (b) by the Yule–Nielsen model with a surface coverage of 0.52, with nYN = 4.

7.4.3

Yule–Nielsen Modified Spectral Neugebauer Model

The linear equation (7.38) does not predict accurately the reflectance of halftones printed on paper due to the scattering of light within the bulk paper and to the multiple reflections between the bulk paper and the print-air interface, which induce some lateral propagation of light from one primary to the surrounding ones. This phenomenon is known as the Yule–Nielsen effect [193, 258], or “optical dot gain”. In order to account for this effect, one may follow similar approach as Viggiano [239] by raising all transmittances in equation (7.38) to a power of 1=nYN . One obtains the Yule–Nielsen modified spectral Neugebauer equation (or simply Yule–Nielsen equation): " #nYN 8 X 1=nYN RðkÞ ¼ ak Rk ð kÞ ð7:40Þ k¼1

where RðkÞ and Rk ðkÞ have the same meaning as in the spectral Neugebauer equation (7.38). In the case of a halftone containing only cyan ink, one gets: h inYN YN YN RðkÞ ¼ ð1  ac ÞR1=n ðkÞ þ ac R1=n ð kÞ ð7:41Þ w c Once again, it is probable that the ink dots have spread over the support (“dot gain”). Their surface coverage ac is unknown and must be estimated. The best fit between the measured spectral reflectance and the one predicted by equation (7.40) is obtained for ac = 0.52, and nYN ¼ 4. This surface coverage is more plausible than

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the value 0.59 that fits the Neugebauer equation, and the agreement between the prediction and measurement is much better (0.24 unit of CIELAB ΔE94, see figure 7.9b). In order to better visualize the capacity of the Yule–Nielsen model to fit the actual reflectances of a halftone print, one may proceed to a graphical analysis based on equation (7.40), transformed as follows: " #nYN   RðkÞ Rc ðkÞ 1=nYN ¼ 1  ac þ ac ð7:42Þ Rw ðkÞ R w ð kÞ This relation of the form R=Rw ¼ f ðRc =Rw Þ can be verified experimentally from the values of Rw , R and Rc measured at different wavelengths, which provide as many points as wavebands contained in the spectral measurement [92]. Figure 7.10 shows an example of halftones of cyan ink (different from the one previously displayed) printed at 0.5 nominal surface coverage and measured with diffuse-8° geometry with specular reflection included. The best agreement between the measured and predicted spectral reflectances is obtained here for nYN ¼ 10. The theoretical curve R=Rw ¼ f ðRc =Rw Þ is plotted in solid black line for the fitted value of ac . The cyan squares indicating the 36 points (Rc =Rw ; R=Rw ) issued from the 36 wavelength of the spectral measurements are well-aligned along this curve. The red straight line shows how it would vary with the same value of ac and with nYN ¼ 1 (Spectral Neugebauer model).

FIG. 7.10 – Comparison between predicted and measured spectral reflectances of inkjet cyan ink printed at 0.5 nominal surface coverage: (a) diagram representing the curve predicted according to equation (7.40) (black solid curve) and the experimental points (Rc =Rw ; R=Rw ) issued from spectral measurements at 36 different wavelengths; (b) Measured spectral reflectances of the unprinted paper Rw ðkÞ, of the halftone RðkÞ and of the solid ink patch R1 ðkÞ (solid lines), and predicted spectral reflectance of the halftone (dashed line).

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179

This graphical analysis is very convenient to check whether the same non-linear law applies to all wavelengths. It may happen that some optical phenomena (for example a colored sheen due to a bronzing effect [101]) occur in a certain spectral domain, which could explain the fact that the experimental points are no longer aligned on the initial curve. The Yule–Nielsen equation is often accurate, but it remains empirical. Many studies have been trying to uncover the physical ground behind this simple equation, by describing the light transfers between the different primaries [6, 7, 102, 193]. The same nYN value applies for all halftone colors printed with the same printing setup, but it may vary every time a printing parameter is modified: support, inks or ink thickness, halftone screen frequency and halftone patterns, etc. The best nYN value for a printing setup is generally determined by printing and measuring the spectral reflectance of a set of colors, and finding the value which minimizes the deviation between the measured and predicted spectra. Recently, Mazauric showed that the nYN value could also be calculated from the measured reflectances and transmittances of the support printed with a few halftone colors, using a flux transfer matrix model [159]. The model can also be used in transmittance mode instead of the reflectance mode [96]. The nYN value usually differs in the two modes [159].

7.4.4

Clapper–Yule Model

The model proposed in 1953 by Clapper and Yule working at the Kodak company (Rochester, USA) [45] relies on a different approach, comparable to the Williams-Clapper and Berns models, by describing flux transfers between the diffusing support, the discontinuous ink layer and the print-air interface (figure 7.11).

FIG. 7.11 – Flux transfers between the diffusing support, the Neugebauer primaries and the print-air interface, according to the Clapper–Yule model.

The Clapper–Yule equation expresses spectral reflectance factor R(λ) of the halftone print as a function of intrinsic spectral reflectance qðkÞ of the support,

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intrinsic spectral transmittances tk ðkÞ ðk ¼ 1; :::; 8Þ of the ink layers corresponding to the Neugebauer primaries, the respective surface coverages ak of the primaries in the halftone, and terms Tin , Tex , rs and ri derived from the Fresnel reflectance of the print-air interface, already introduced in §6.3.1 for the Saunderson equation, depending on the measurement geometry: P Tin Tout qðkÞ½ ak tk ðkÞ2 P RðkÞ ¼ rs þ ð7:43Þ 1  ri qðkÞ ak tk2 ðkÞ We can notice the similarity between the Clapper–Yule equation and the Saunderson equation (6.20): both are equivalent when there is no ink: the white primary with intrinsic spectral transmittance t(λ) = 1 covers the whole surface, i.e., priaw = 1. In the presence of ink dots, the incident light crosses the Neugebauer P maries just after crossing the interface, which yields a term Tin ak tk ðkÞ. The outgoing light crosses the PNeugebauer primaries just before crossing the interface, which yields a term Tout ak tk ðkÞ. The light internally reflected on the interface at the support side crosses the Neugebauer primaries twice, each ray crossing one P primary; this yields the term: ri 8k¼1 ak tk2 ðkÞ. Notice that as in Berns’ model (§7.1.2), but in contrast with the Williams–Clapper model, the orientation of light within the ink layers is not considered. Clapper and Yule argued that inks are rarely perfectly transparent but rather slightly scattering, and the intrinsic transmittance corresponds to an average path length being similar for directional light or diffuse light. For the definitions of Tin , Tout , rs and ri according to the measuring geometry used, one can refer to §6.3.1. In their original paper, Clapper and Yule considered a 45°:0° geometry, and single-ink halftones. The model relies on the assumption that the Ppositions at whichPlight enters and exits the print are not correlated (terms Tin ak tk ðkÞ and Tout ak tk ðkÞ simply multiply, without considering the primary that each light ray has entered), because light travels laterally a distance much larger than the halftone period, due to scattering within the support. Since the lateral propagation distance within the support is limited (it is given by the point spread function of the support, see §2.7.2), the model should therefore be valid only with small halftone periods, i.e., with high halftone screen frequencies. However, the experience in printing on paper shows that the model already provides good predictive performances with middle halftone screen frequencies, beyond 50 lines per inch (lpi) [97].

7.5

Calibration of the Halftone Color Prediction Models

The Spectral Neugebauer, Yule–Nielsen and Clapper–Yule models are calibrated from a reduced number of color patches, displayed in figure 7.12 in the case of three inks (CMY), which need to be printed and measured. From the measured spectral reflectances, one can first deduce the spectral parameters used in each model, then estimate the spreading of the ink dots on the considered support.

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181

FIG. 7.12 – Color patches needed for the calibration of the prediction models for CMY colors.

7.5.1

Obtaining Spectral Parameters

For the Neugebauer and Yule–Nielsen models, the measurement with a spectrophotometer of solid layers of the eight Neugebauer primaries (row A of figure 7.12) suffices to have all required spectral parameters. For the Clapper–Yule model, the spectral parameters are spectral intrinsic reflectance qðkÞ of the support, and intrinsic spectral transmittances tk ðkÞ of the eight Neugebauer primaries. The same eight measurements as for the Yule–Nielsen model are needed. Reflectance factor Rw ðkÞ of the unprinted paper enables obtaining qðkÞ thanks to the following formula, derived from equation (7.43) with tk ðkÞ ¼ 1 since there is no ink: qðkÞ ¼

Rw ðkÞ  rs Tin Tout þ ri ðRw ðkÞ  rs Þ

ð7:44Þ

Then, by measuring spectral reflectance factor Rk ðkÞ of each solid Neugebauer primary patch (k ¼ 2; :::; 8), one deduces intrinsic spectral transmittance tk ðkÞ of the corresponding ink layer thanks to the following formula, again derived from equation (7.43): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rk ðkÞ  rs t k ð kÞ ¼ ð7:45Þ qðkÞ½Tin Tout þ ri ðRi ðkÞ  rs Þ

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7.5.2

Ink Spreading Assessment Methods

When considering a given printing setup, it is impossible to estimate a priori how much the different wavelengths are reflected from the paper or absorbed by the inks and how far the ink dots spread on the paper surface, until specific spectral measurements are performed. The spectral properties of the paper and inks are represented in each model by spectral reflectances or transmittances deduced from the spectral reflectance of full-tone colors. The growth of the ink dots, i.e. the dot gain, is assessed by establishing the correspondence between nominal and effective surface coverages for each ink, thanks to nominal-to-effective surface coverage functions, also called ink spreading functions. The spectral reflectances, respectively, transmittances, and the ink spreading functions are computed in a calibration procedure which ensures the ability of the model to account for the specific properties of the considered print. Once the spectral parameters are obtained, one can start assessing the dot gain. The correspondence between the effective and nominal surface coverages of the inks is represented by ink spreading functions as featured in figure 7.13. Although each model (except the Neugebauer model) already accounts for optical dot gain, the effective ink surface coverages may also compensate for a possible under- or overestimation of optical dot gain. Two dot gain assessment methods are possible: the basic ink spreading (BIS) method where the spreading of each ink is assessed independently of the others and the superimposition-dependent ink spreading (SDIS) method where ink superposition configurations are considered.

7.5.3

Basic Ink Spreading (BIS) Method

Each ink i is printed alone on paper at the nominal surface coverages ai = 0.25, 0.5 and 0.75, which corresponds to the 9 color patches represented in rows B, C and D of figure 7.12. Let us denote by Rmeas ða; kÞ their respective spectral reflectance. These i halftones contain two primaries: the ink which should occupy fractional area ai and the paper white which should occupy fractional area 1  ai . Applying the model’s equation with these two primaries and these surface coverages should predict ða; kÞ equal to the measured one. However, due to the fact spectral reflectance Rpred i that the effective ink surface coverage is different from the nominal one, these two reflectance values are not the same. One thus fits effective surface coverage ai0 as ai which minimizes the deviation between predicted and measured spectra, by quantifying the deviation expressed in three different ways: either as the sum of square differences of the components of the two spectra, i.e., ai0 ¼ arg min

730 nm X

0  a  1 k¼380 nm

h

i2 meas Rpred ð a; k Þ  R ð a; k Þ ; i i

or as the sum of square differences of their logarithm, i.e.,

ð7:46Þ

Nonscattering Layers on a Diffusing Background

ai0 ¼ arg min

730 nm X

0  a  1 k¼380 nm

h

i2 meas log Rpred ð a; k Þ  log R ð a; k Þ ; i i

183

ð7:47Þ

or as the corresponding color difference given e.g. by the CIELAB DE94 metric   meas ð a; k Þ; R ð a; k Þ ð7:48Þ ai0 ¼ arg min DE94 Rpred i i 0a1

Equation (7.46) is the most classical way of determining the effective surface coverage. Taking the logarithmic value of the spectra as in equation (7.47) has the advantage of providing a higher weight to lower reflectance values where the visual system is more sensitive to small spectral differences. Fitting ai0 from the color difference metric sometimes improves the prediction accuracy of the model in terms of color differences but complicates optimization. Even at optimal surface coverage ai0 , the difference between the two spectra is rarely zero and provides a first indication of the prediction accuracy achievable by the model for the corresponding printing setup. Once the 9 effective surface coverages are computed, assuming that the effective surface coverage is 0, respectively 1, when the nominal surface coverage is 0 (no ink), respectively 1 (full coverage), one obtains three sets of ai0 values which, by linear interpolation, yield the continuous ink spreading functions fi as plotted in figure 7.13.

FIG. 7.13 – Example of ink spreading curves fi obtained by linear interpolation of the effective 0 surface coverages ai=j which are deduced from measurement of patches with single-ink halftones (ink i) printed at nominal surface coverages 0.25, 0.5 and 0.75.

As an alternative, one can print halftones at nominal surface coverage 0.5 only and perform parabolic interpolation [28]. The number of patches needed for establishing the ink spreading curves is then reduced to 3 (row C in figure 7.12).

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7.5.4

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Superimposition-Dependent Ink Spreading (SDIS) Method

One often observes that the amount of ink spreading depends on whether the ink is alone on the support or superimposed with other inks. In addition to the effective surface coverages computed from single-ink halftones alone on paper, the SDIS method includes effective surface coverages computed from the single-ink halftones superimposed with a solid layer of either one or the two other inks. 36 color patches need to be printed, represented by the rows B to G in figure 7.12. The effective surface coverages of the halftones are obtained in the same way as in the BIS method, by considering the appropriate two colorants for each halftone. One obtains 12 sets of effective surface coverages, providing 12 ink spreading curves q 0 ¼ fi=j ðq Þ where the subscript i=j denotes ink halftone i superimposed with solid colorant j (see figure 7.14). By printing halftones at nominal surface coverage 0.5 only and performing parabolic interpolation, the number of patches to be printed and measured is thus reduced to 12 (rows C and F in figure 7.12). The superimposition-dependent ink spreading method (SDIS) often considerably improves the prediction accuracy of the models compared to the basic ink spreading method (BIS) [97].

7.5.5

Predicting the Spectral Reflectance of Halftones

Once the spectral parameters as well as the ink spreading functions are obtained, the model is calibrated. One can then predict the spectral reflectance of halftones with any set of nominal ink surface coverages c, m, and y. In case the dot gain has been calibrated by means of the BIS method, the ink spreading functions fi directly provide the effective surface coverages c0 , m 0 , and y 0 of the three inks: c0 ¼ fc ðcÞ; m 0 ¼ fm ðm Þ;

y 0 ¼ fy ðy Þ

These effective ink surface coverages are then “plugged” into the Demichel equations, which provide the effective surface coverages of the eight Neugebauer primaries. The general equation of the model finally predicts the spectral reflectance of the considered halftone. In case the dot gain has been calibrated using the SDIS method, the nominal ink surface coverages c, m, and y are converted into effective ink surface coverages c0 , m 0 and y 0 by accounting for the superimposition-dependent ink spreading. The effective surface coverage of each ink is obtained by a weighted average of the ink spreading curves. Their respective weights are expressed by the surface coverages of the respective primaries on which the ink halftone is superimposed. For example, the weight of ink spreading curve fc (cyan halftone over the white primary) is proportional to the surface of the underlying white primary, i.e. ð1  m Þð1  y Þ. In the case of three halftone inks, effective surface coverages are obtained by performing a few iterations with the following equations:

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185

FIG. 7.14 – Example of ink spreading curves fi=j obtained by linear interpolation of the

0 deduced from measurement of patches with single-ink halftones effective surface coverages ai=j

(ink i) printed at nominal surface coverages 0.25, 0.5 and 0.75 and superimposed with a solid layer of primary j.

c0 ¼ ð1  m 0 Þð1  y 0 Þfc ðcÞ þ m 0 ð1  y 0 Þfc=m ðcÞ þ ð1  m 0 Þ y 0 fc=y ðcÞ þ m 0 y 0 fc=m þ y ðcÞ

m 0 ¼ ð1  c0 Þð1  y 0 Þfm ðm Þ þ c0 ð1  y 0 Þfm=c ðm Þ þ ð1  c0 Þ y 0 fm=y ðm Þ þ c0 y 0 fm=c þ y ðm Þ y 0 ¼ ð1  c0 Þð1  m 0 Þfy ðy Þ þ c0 ð1  m 0 Þfy=c ðy Þ þ ð1  c0 Þ m 0 fy=m ðy Þ þ c0 m 0 fy=c þ m ðy Þ ð7:49Þ

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For the first iteration, c0 ¼ c, m 0 ¼ m and y 0 ¼ y are taken as initial values on the right side of the equations. The obtained values of c, m and y are then inserted again into the right side of the equations, which yields new values of c0 , m 0 , y 0 etc., until the values of c0 , m 0 , y 0 stabilize. The effective surface coverages of the primaries are calculated by “plugging” the obtained values for c0 , m 0 and y 0 into the Demichel equations. The spectral reflectance of the considered halftone is finally provided by the general equation of the model.

7.5.6

Four-Ink Halftone Colors

The models presented above can be extended to four ink halftones in a straightforward manner, by increasing accordingly the number of spectral parameters and of inks spreading functions. The number of colorants (Neugebauer primaries) becomes 24 = 16 instead of 23 = 8 for 4-ink halftones in quadrichromy. This yields 16 spectral parameters in the Spectral Neugebauer and Yule–Nielsen models. The number of ink spreading functions is 4 (one per ink) in the BIS method and 32 (all combination of 0, 1, 2 or 3 inks superimposed) in the SDIS method. The spectral parameters and the effective ink surface coverages which are necessary to build of the inks spreading functions are computed in the same manner as for 3-ink halftones. At the prediction stage, once the effective ink surface coverages of the four inks are computed, the surface coverages of the 16 Neugebauer primaries are provided by the 4-ink Demichel equations. They are then combined with the spectral parameters according to the model’s general equation, which is similar for 3 or 4 inks [28].

Chapter 8 Angle-Dependent Light Scattering We mentioned several common objects in the previous chapters for which the generalized two-flux models can apply: mirrors, colored filters, transparent films, glazing, paper sheets, highly diffusing paints, photographs, printed surfaces… However, these represent only a part of the objects that surround us, because all objects do not have a smooth and flat surface, nor are they all formed only by non-diffusing or highly diffusing layers. Many objects are neither transparent nor Lambertian: they reflect and transmit light in an angle-dependent manner, either due to slight scattering in one or more layers (volume scattering), or to the roughness of one or more interfaces (surface scattering). The use of bi-directional functions becomes necessary to describe the proportion of each directional flux which is reoriented into every direction with respect to its initial orientation. Optical phenomena responsible for this “weak” scattering are diverse because they depend on the characteristic size of the heterogeneities in the material or surface topology in comparison to the wavelength of the light. Approaches based on geometrical optics, such as ray-tracing algorithms, are valid if the heterogeneities are large, while electromagnetic models are needed if these heterogeneities are smaller. It is the same for surface heterogeneities. These various configurations have resulted in a profuse documentation of relevant literature since the 1960s, which we will not report here. We simply mention some of the main structural or photometric parameters needed to characterize radiometrically surface scattering and volume scattering, and present a few usual models.

8.1

Surface Scattering

An interface between two homogeneous media of different indices can reflect and transmit a beam of directional light in the regular directions well defined by the laws of Snell if and only if it is perfectly flat and smooth over the illuminated area. The meaning of “flat” and “smooth” is to be envisaged according to the consequences that the real surface topology has on an optical point of view. If the surface is curved, undulating, or deviates locally form the perfect mathematical plane with low spatial frequency, but remains smooth at the micrometric scale, it behaves like a distorting mirror. The Snell-Descartes laws and Fresnel formulae apply at each point DOI: 10.1051/978-2-7598-2647-6.c008 © Science Press, EDP Sciences, 2022

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188

of the interface depending on the local normal. The rays reflected and transmitted can concentrate in some areas or patterns, called caustics. If the scattering surface deviates from the perfect plane with higher spatial frequencies, these caustics are too small to be visible and light is more smoothly scattered. The surface is consequently said to be rough. The coherence of light plays a role: the scattering process is equivalent to diffraction, and it depends on wavelength, even though iridescences are not always perceptible because of the randomness of the surface structure. The frontier between smooth and rough surfaces depends on the characteristic magnitude Δh of elevation fluctuations, wavelength λ of light considered, and angle of incidence hi . One often refers to Rayleigh’s criterion reported by Beckmann in Ref. [14], which states that when Δh is lower than the critical value Dh\

k 8 cos hi

ð8:1Þ

the interface can be considered as smooth. Notice that this critical value is the lowest at normal incidence, and it tends to infinity as hi tends to π/2, which means that at grazing angles, all rough surfaces become smooth, therefore specular reflectors. The roughness concept generally discards periodical structures, which rather belong to the category of diffraction gratings. It therefore refers to topologies characterized by a random elevation distribution.

8.1.1

Surface Roughness Measurement

The relief of natural or manufactured surfaces generally varies according to multiple scales and it is often hard to measure it at high resolution over an area large enough to be statistically representative of the surface morphology. In a recent study on rough surfaces of different materials, Turbil reported the performances of Atomic Force Microscopy (AFM), chromatic confocal microscopy and stylus profilometry [235]. The AFM can achieve a nanometric resolution, but the scanned area is small (a few tens of micrometers); the chromatic confocal microscopy can scan millimetric surfaces with a micrometric resolution in position and height, while the stylus profilometer can scan the surface along one line with a resolution of around 100 nm. Several attempts have been made to simulate light scattering with optical models based either on geometrical optics or wave optics on these structures, but the agreement between the BRDF predicted according to this approach and the ones measured on the real surfaces is rather poor. Another approach consists in modeling the topology of the rough surface with a few statistical parameters, then using probabilistic optical models.

8.1.2

Modelling a Randomly Rough Surface

The topography of a rough interface has a random elevation as featured in figure 8.1. The elevation function is modeled by a probability distribution parameterized by a characteristic vertical length, the root-mean-square (r.m.s.) height σ, and by a characteristic horizontal length, the correlation length τ [225]. Another parameter is also commonly used: the r.m.s. slope m, corresponding to the ratio σ/τ [233].

Angle-Dependent Light Scattering

189

FIG. 8.1 – Elevation function representing the profile of a rough interface along an x-axis. The random pattern has a r.m.s. height σ and correlation length τ. The r.m.s. slope m of the interface is the ratio σ/τ.

Most models assume that the local slope along a rough surface follows a Gaussian distribution [14, 225]. In order to ease the application of optical laws, local slope is converted into local normal vector [77] denoted by angles ðhh ; uh Þ (figure 8.2). With an isotropic Gaussian slope distribution, the probability distribution function for the normal vector orientation is given by the Beckmann function [47, 173, 225]: tan2 ðhh Þ

e 2m2 D m ð hh Þ ¼ 2pm 2 cos4 ðhh Þ

ð8:2Þ

where m is the standard deviation of the slope (or rms slope). More recently, Walter et al. suggested the following distribution function, that they called GGX, tending to prevail now in computer graphics for the visual rendering of rough surfaces [241]: D m ð hh Þ ¼

m2 2

p½m 2 þ tan2 ðhh Þ cos4 ðhh Þ

ð8:3Þ

Both functions Dm ðhh Þ defined above depend only on polar angle θh due to the assumption of roughness azimuthal isotropy. We can verify that, as expected for distribution functions, their summation over the hemisphere is unity: Z 2p Z p=2 Dm ðhh Þ cos hh sin hh dhh duh ¼ 1 ð8:4Þ uh

8.1.3

hh ¼0

BRDF and BTDF Optical Models

The reflectance and transmittance of randomly rough interfaces can be deduced from their BRDF, respectively their BTDF using equation (2.12). BRDFs and BTDFs may be determined experimentally [59, 60, 225] or computed thanks to an optical model. The model is derived from equations relying on either physical or geometrical optics depending on the size of the roughness patterns [173].

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190

Physical optics models are directly based on the electromagnetic wave theory and Maxwell’s equations [22]. They should be used when the wavelength of light is larger or comparable to the r.m.s. height σ and correlation length τ. In such a case, the diffraction of the incident waves by the corrugations of the interface is dominant. It is assumed that the interface does not have any discontinuity or sharp arc compared to the wavelength of incident light. It may therefore be represented locally by its tangent plane, on which light is reflected according to Snell’s law and diffracted because of the small size of the plane element. This tangent plane approximation is the basis of Beckmann’s model [14], also known as Kirchhoff’s approximation [181].

8.1.4

Microfacet Model and Smith’s Shadowing-Masking Function

Models relying on geometrical optics explain the behavior of light when its wavelength is small compared to the roughness characteristic lengths. Slope distribution models, such as the well-known models developed by Torrance and Sparrow [233] and by Cook and Torrance [47] consider the rough interface as a set of randomly inclined microfacets reflecting and transmitting light like smooth interfaces. According to slope distribution models [47, 173, 233], the BRDF fR of a rough interface evaluated at the angles ðhi ; ui Þ for the incident irradiance (direction represented by vector L), and ðhr ; ur Þ for the reflected radiance (direction represented by vector V), is mainly given by the relative number of facets the normal H of which bisects the incident and reflected rays (see figure 8.2): H¼

LþV kL þ V k

ð8:5Þ

FIG. 8.2 – 2D representation of a rough interface. The directional incident light, coming from a direction denoted by vector L, illuminates a small portion of interface of normal vector H. It is reflected into the direction denoted by vector V.

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191

The polar angle of vector H is hh ¼ arccosðN  HÞ "

# cos hi þ cos hr ¼ arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ cos hi cos hr þ sin hr sin hi cosð/r  /i ÞÞ

ð8:6Þ

By assuming that the surface is isotropic, this relative number of facets is given by the function Dm ðhh Þ. two attenuation functions are to be considered: actual reflectance Moreover,  0 R12 hi of these facets given by the Fresnel reflectance formula, evaluated at angle h0i under which they receive the incident light: h0i ¼ arccosðL  HÞ 1 ¼ arccosðcos hr cos hi  sin hr sin hi cosð/r  /i ÞÞ; 2

ð8:7Þ

and attenuation function Gm, detailed below, accounting for the fact that some areas in the rough surface can be shadowed or masked by neighboring facets. The BRDF is thus given by:   Dm ðhh ÞGm ðhi ; hr ÞR12 h0i fR ðhi ; ui ; hr ; ur Þ ¼ ð8:8Þ 4 cos hi cos hr When the medium of transmission is non-metallic, the BTDF is given by [221]   Dm ðhh ÞGm ðhi ; ht ÞT12 h0i   fT ðhi ; ui ; ht ; ut Þ ¼ ð8:9Þ C h0i cos hi cos hr   where ðht ; ut Þ denotes the orientation of the transmitted ray and function C h0i expresses the change of solid angle due to the refraction by the interface: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2  cos h0i  n 2  1 þ cos h0i  0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C hi ¼ ð8:10Þ n cos h0i n 2  1 þ cos h0i The rough interface may comprise shadow areas, which increase with respect to the roughness and the incidence angle of light. Interface elements belonging to shadow areas do not contribute to the reflection nor to the transmission. This phenomenon, illustrated by figure 8.3, is called shadowing. Likewise, reflected and transmitted light may be partially blocked by neighboring corrugations. This phenomenon, called masking [233], is equivalent to shadowing but depends on the angle of observation instead of the angle of incidence.

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FIG. 8.3 – Shadowing: oblique incident light does not illuminate the whole surface. The fraction of facets that really contributes to the reflection of light from direction L to direction V is given by function Gm ðhi ; hr Þ, product of two similar functions gm, one for shadowing, and the other one for masking     Gm ðhi ; hr Þ ¼ gm hi ; h0i gm hr ; h0i ð8:11Þ Using a statistical model, Smith computed the following shadowing function gm [218] (   1  0 if cos h0i [ 0  ð8:12Þ gm h; hi ¼ Km ðhÞ þ 1  0 if cos h0i \0 where h0i is the local angle of incidence given by (8.7) and Λm is a function of angle θ which depends on the r.m.s. slope m: pffiffiffi  

  2m 1  cot2 h cot h pffiffiffi p ffiffi ffi K m ð hÞ ¼ exp  erfc 2 2m 2 p cot h 2m Function gm is comprised between 0 (facets completely shadowed or masked) and 1 (facets completely illuminated). At small and medium incidence angles, the illuminated fraction of the facet’s area is close to 1. The shadowing effect is thus small enough to be neglected. However, ignoring the shadowing at high incidence angles may yield an overestimation of the reflected and transmitted fluxes, and a subsequent violation of the energy conservation principle. According to Bruce [27] and Caron [32], shadowing should be taken into account when the incidence angle is higher than limit angle hshad depending on the r.m.s. slope m of the rough interface pffiffiffi  p ð8:13Þ hshad ¼  arctan 2m 2 Similar considerations apply for masking. The BRDF shown in figure 2.3 for illustrating Lambert’s equal-area projection for BRDF plane display was generated using this model, for a glass-air interface (relative refractive index 1.5) with roughness r.m.s. slope m = 0.1, and an incident angle of 30°. This model relying on geometrical optics has some limitations: firstly, the multiple reflections within the microcavities are ignored, whereas it has been recently

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shown that they can have a significant influence on the surface’s color, especially in the case of chromatic metals (gold, copper…) [104, 141, 197, 250]. Secondly, in order to be sure that diffraction has no significant effect, the source must be incoherent spatially and temporally. This means in practice that the facets must be significantly larger than the wavelength of light, and that the incoming light must come from several directions, for example from a large solid angle. The difference between the geometrical optics model and a wave model including diffraction has been well displayed for a brushed steel surface in Ref. [251].

8.1.5

Spherical Surfaces

Consider rough, facetted balls illuminated from direction L and observed in direction V identical for all pixels of the image, e.g. by a virtual camera located at infinity. Since in this configuration vectors L and V are fixed, all facets able to reflect some light from the source to the camera have the same orientation: their normal vector H is the vector bisecting L and V, and it is therefore fixed, as shown in figure 8.4.

FIG. 8.4 – Spherical rough surface illuminated from one direction (vector L) and viewed by a camera in one direction (vector V). According to the geometrical facet model, the facets able to reflect light towards the camera, drawn in orange, all have the same normal vector H, which bisects L and V, while normal N of the spherical average surface, thereby angle θh, is different in each point.

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However, normal N of the spherical average surface varies as a function of position on the sphere, as well as angle hh between N and H. In each point of the sphere, the probability that a facet is oriented according to normal vector H is given by function Dm ðhh Þ evaluated at the corresponding angle hh . By applying this model with metallic balls, one obtains the images shown in figure 8.5. Function Dm used is Beckmann’s distribution function given by equation (8.2) and the shadowing-masking function Gm is Smith’s function given by equations (8.11) and following. In this simulation, the SPD of the light source was assumed to be tri-chromatic, i.e., it was non-zero in three wavelengths 440 nm (blue), 510 nm (green) and 667 nm (red), with same power. The red (respectively green, blue) channel of the RGB image corresponds to the spectral radiance at 667 nm (respectively 510 nm, 440 nm) scattered towards the virtual camera, clipped at the pixel value 255; this generates the color rendering displayed. Row (a) shows balls of different metals with the same roughness (r.m.s. slope m = 0.3). Their different color rendering comes from the different complex refractive indices of the metals at the three selected wavelengths used in the computation of the Fresnel reflectance term. Row (b) shows the gold ball with different roughness values. For the smoothest balls, the clipping at 255 of the grey level values in the three channels of the images gives a white color; this artifact looks rather natural because it is also what happens in photography when the sensor saturates due to a too high radiance in the specularities of glossy objects. Of course, in the absence of diffuse ambient light, the areas that are not illuminated or poor in facets of normal H are black.

FIG. 8.5 – Color rendering of rough metallic balls under directional light (D65 illuminant) in the absence of diffuse ambient light, viewed from one direction, as simulated by the microfacet surface scattering model using Beckmann’s distribution function and Smith’s shadowing-masking function. (a) Balls of various metals with same roughness, (b) balls in gold with different roughness values. (Courtesy of Simon Pinault, Institut d’Optique Graduate School).

Angle-Dependent Light Scattering

8.2

195

Volume Scattering

As light encounters small fluctuations of refractive index within the medium, a portion of the incident light is scattered. In the atmosphere, scattering is responsible for the white color of clouds (Mie scattering [164]), the blue color of the sky and the redness of sunsets (Rayleigh scattering [190]). Scattering also occurs in liquids. Milk, for example, is composed of a suspension of almost transparent fat droplets which scatter light and give milk its white and opaque aspect. In the case of oceans, scattering is coupled to absorption, which produces the characteristic bluish color. Light is also scattered in solid heterogeneous media, such as paintings, papers, cotton and human tissues [189]. Different types of scattering are encountered according to the composition, shape, size and concentration of the heterogeneities, often considered as particles immerged into a binder. The polarization and the wavelength of the incident light may have a strong influence on scattering. We present here some commonly used parameters and models relative to volume scattering.

8.2.1

Scattering Description Parameters

A collimated beam traversing a path of length x into a scattering and absorbing medium undergoes an exponential attenuation T(λ) described by the Beer–Lambert law: T ðkÞ ¼ eKext ðkÞx

ð8:14Þ

where Kext is the linear spectral extinction coefficient (in m–1). The inverse of the extinction coefficient is the extinction free-mean-path length lext(λ), characterizing for each wavelength the distance along which a directional flux is attenuated by factor 1/e: lext ðkÞ ¼ 1=Kext ðkÞ

ð8:15Þ

The linear extinction coefficient may be decomposed into a component Ksca(λ) related to scattering and component Kabs(λ) related to absorption (the one used in Beer’s law, corresponding to the special case where Ksca(λ) = 0, see §4.5.1): Kext ðkÞ ¼ Ksca ðkÞ þ Kabs ðkÞ

ð8:16Þ

Free-mean-path lengths lsca ðkÞ and labs ðkÞ are also defined as: lsca ðkÞ ¼ 1=Ksca ðkÞ and labs ðkÞ ¼ 1=Kabs ðkÞ

ð8:17Þ

A scattering and absorbing medium is said to be homogenous when its spectral coefficients Ksca(λ) and Kabs(λ) are independent of position. As an effect of scattering, the trajectory of light is modified. The change of direction in an elementary volume of medium is specified by an angular volume scattering coefficient (VSF) [160], defined for every direction ðh; uÞ as

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196

bk ðh; uÞ ¼

d 2 Fk ðh; uÞ dIk ðh; uÞ ¼ Ek dXðh; uÞdV Ek dV

ð8:18Þ

where Eλ denotes the spectral irradiance on the front face of the volume, d 2 Fk ðh; uÞ the elementary spectral flux scattered out of volume dV into the elementary solid angle dx ¼ sin hdhdu, and dIk ðh; uÞ ¼ d 2 Fk ðh; uÞ=dXðh; uÞ the corresponding intensity along direction ðh; uÞ (see figure 8.6).

FIG. 8.6 – Irradiance on an elementary volume of diffusing material, and scattered intensity in one direction, from which is defined the angular volume scattering coefficient.

The VSF integrated over the whole sphere (4π sr) gives linear scattering coefficient Ksca ðkÞ (in m−1). Moreover, if it can be determined that volume dV of the scattering medium contains N similar particles, then the scattering coefficient is written as Ksca ðkÞ ¼ N rk

ð8:19Þ

where rk is the scattering cross section of each particle (in m2), defined as the ratio of ðpÞ ð pÞ flux Fs;k scattered by the particle to its irradiance Ek evaluated on a tangent plane perpendicular to the incoming light: ðpÞ

rk ¼

Fs;k

ðpÞ

Ek

:

ð8:20Þ

By dividing the VSF bk ðh; uÞ by Ksca ðkÞ, one obtains the angular scattering distribution function, fS ðk; h; uÞ, which satisfies the following normalization condition for each wavelength: Z fS ðk; h; uÞdx ¼ 1: ð8:21Þ ðh;uÞ24p

Even though fS is independent of the absorptance of the medium, it remains a function of wavelength as directions of scattering generally depend upon wavelength especially when scattering is due to diffraction (Rayleigh scattering, Mie scattering

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197

[22]). The rigorous definition for fS is the ratio of elementary scattered intensity dIk ¼ d 2 Fk =dx to total elementary scattered flux dFk : fS ðk; z; h; uÞ ¼

dIk ðz; h; uÞ dFk ðz Þ

ð8:22Þ

Equation (8.21) comes from the fact that the total scattered spectral flux dFk is the sum of all spectral intensities over the sphere. If scattering by the volume element is isotropic, equal intensity is emitted in every direction and fS is a constant equal to 1/(4π). The ratio of function fS of a given medium to the one it would have if it were isotropic is called the phase function, denoted by P: P ðk; h; uÞ ¼ 4p fS ðk; h; uÞ As a consequence of equations (8.21) and (8.23), we obtain Z 1 P ðh; uÞdx ¼ 1 4p ðh;uÞ24p sr

ð8:23Þ

ð8:24Þ

In the case of isotropic scattering, the phase function is 1 in all directions. In the opposite case, anisotropic scattering may be characterized by an anisotropy parameter g defined as the average cosine of the scattering angle: Z 1 g¼ P ðh; uÞcoshdx ¼ 1 ð8:25Þ 4p ðh;uÞ24p sr The incident light is mainly scattered backwards when g is close to −1 or forwards when g is close to 1. For isotropic scatterings, g = 0. Since for a given material the phase function is often unknown and difficult to determine experimentally, one often refers to a parametric phase function based on the anisotropy parameter g, referred to as Henyey–Greenstein phase function [106]. PHG ðhÞ ¼

1  g2 ð1 þ g 2  2g cos hÞ3=2

ð8:26Þ

Cross sections of Henyey–Greenstein functions with various g values are plotted in figure 8.7.

FIG. 8.7 – Examples of phase functions in the plane φ = 0 (mod. π) for one wavelength, with different values of anisotropy parameter g.

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198

Parameter g is also used for defining the transport free-mean-path length ltrans, corresponding to the distance from which one may consider that light has completely lost the memory of its original direction of incidence ltrans ¼

lsca 1  jg j

ð8:27Þ

The optical thickness sðkÞ of a scattering or/and absorbing layer having a thickness h and an extinction coefficient Kext ðkÞ is defined as sðkÞ ¼ Kext ðkÞh

ð8:28Þ

When τ is much larger than 1, a directional incident light is almost completely attenuated. When τ is small, the layer is translucent, i.e., one can distinguish an object located beneath the layer. After a certain number of scattering events, light propagates in an isotropic manner, i.e. it becomes Lambertian.

8.2.2

Types of Scattering

The notion of optical thickness defined above allows estimating the number of scattering events that a light ray undergoes across a given layer of the considered medium. In the particular case of a weakly absorbing medium (Kabs much smaller than Ksca), the optical thickness describes the strength of scattering. One may distinguish four scattering modes, according to the value of the optical thickness of the layer:  ballistic scattering, also called atmospheric absorption [81], in which light is almost not scattered: τ much smaller than 1 and h much smaller than lsca,  single scattering in which light is scattered once in the medium: τ ≈ 1 and h ≈ lsca. For particle sizes much smaller than the wavelength such as air molecules, smoke and dust, Rayleigh scattering [129, 190, 191] is applicable with the following phase function for unpolarized light: PR ðh; uÞ ¼

 3 1 þ cos2 h 8

ð8:29Þ

For larger particles with size comparable to the wavelength, Mie scattering [21, 22, 164] becomes applicable and is often represented by approximated phase functions such as the famous Henyey–Greenstein phase function parameterized by the anisotropy parameter g defined in equation (8.26). When particle sizes are much larger than the wavelength, geometrical optics models may be used [161, 216].  multiple scattering in which light is scattered a variable number of times [128]: τ > 1 and h > lsca,  diffusion where scattering events occur so many times that the resulting scattering is isotropic: τ  1 and h  lsca. According to equation (8.28), since parameter g defined by (8.25) is equal to 0, the transport length is given by the scattering length. The incident light has therefore completely lost the memory of its incident direction.

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199

For low concentrations of particles, it is assumed that they do not interact one with another. Scattering is said to be independent. Describing the scattering by one particle is sufficient to determine the scattering by the whole medium. For high concentrations of particles, scattering becomes dependent. In this case, geometrical optics may be used when the size of the particles are large compared to the wavelength of the incident light. However, when the particles are small compared to the wavelength, light is diffracted. In this case, scattering may be modeled by the Rayleigh scattering theory. The Mie scattering theory describes the diffraction of light by spherical particles of complex refractive index in a dielectric medium (real refractive index). Note that, except for some exceptional phenomena such as the Raman effect, scattering does not modify the wavelength of the incident light and is thus said to be elastic [73].

8.2.3

The Radiative Transfer Equation

In many applications, a simple phenomenological approach, based on the notion of directed light ray and conservation of energy, provides a realistic description of the scattering phenomenon. Considering a sufficiently large portion of the heterogeneous medium, the scattering process is described by a simple equation: the radiative transfer equation [34]. A priori, it is valid only when the scattering free-mean-path length lsca is large compared to the wavelength of the incident light and to the dimension of the heterogeneities responsible for the scattering, but specific studies have shown that its domain of validity can be enlarged to other cases. The radiative transfer equation expresses the conservation of the radiant flux in a given element of volume and a given direction. This energy balance shall be performed everywhere in the medium and in every direction. Consider a small cylinder of section dS and of length dl oriented according to the incident direction u. Radiance L(u) decreases along this direction due to absorption and scattering dLðuÞ ¼ ðKabs þ Ksca ÞLðuÞ dl

ð8:30Þ

At the same time, the cylinder receives radiances Lðu0 Þ from all directions u0 and scatters them partially towards direction u, which increases radiance L(u). The portion of radiance Lðu0 Þ that contributes to radiance L(u) is Ksca P ðu0 ; uÞLðu0 Þdx0 ; 4p where P ðu0 ; uÞ is the phase function of the considered cylindrical element of volume. By summing up the contributions of all directions u0 and adding the resulting global contribution to equation (8.30), one obtains the radiative transfer equation Z dLðuÞ Ksca ¼ ðKabs þ Ksca ÞLðuÞ þ P ðu0 ; uÞLðu0 Þdx0 ð8:31Þ dl 4p R

200

Optical Models for Material Appearance

This integro-differential equation has no general solution. An exact or approximated solution must be searched for every particular scattering medium. Various solutions have been developed. Let us mention a few:  The N-fluxes method [172] allows us to convert the integrodifferential equation (8.31) into a differential equation system thanks to an angular discretization. Solutions are obtained for azimuthally isotropic media, the discretization being performed only according to the zenithal angle [126]; the simplest particular case is the two-flux Kubelka–Munk model [133, 135]. Four-flux models have also been developed [151, 152].  The discrete ordinate method [222] is an exact but computationally costly method. The assumption of azimuthal isotropy is not necessary. Discretization according to the azimuthal angle is avoided thanks to a Fourier series development for the scattered fluxes and a spherical harmonic decomposition of the phase function.  The auxiliary function method [57] avoids angular discretization. An auxiliary function is introduced into the radiative transfer equation and decomposed into spherical harmonics. The radiative transfer equation is thus converted into an integral equation system, which can be solved numerically.  The adding-doubling method [189, 236] is based on an infinitesimal sublayer whose optical properties are described by a matrix. The (i, j)-entries of the matrix, deduced from the phase function, indicate the probability for a ray coming from direction labelled i to be scattered into direction labelled j. Then, the matrix is squared, or raised to a power k, to represent the optical properties of a sublayer with double thickness, respectively, with thickness multiplied by k. This doubling or adding operation is repeated until the thickness of the sublayer matches the total thickness of the layer.

8.2.4

Scattering in Lambertian Layers

The free-mean-path length lsca of a strongly scattering layer is very small compared to the layer thickness. Incident light loses the memory of its initial angular distribution as soon as it penetrates the layer. One can therefore assume that any illumination geometry leads to a same reflectance and a same transmittance, called intrinsic. Since light is scattered a large number of times within the layer, it is Lambertian at every point of the layer, especially at the layer’s bounding planes. We can also assume that light exiting the layer is unpolarized. The reflection and transmission by Lambertian layers can be modeled by the Kubelka–Munk two-flux model developed in §6.1, with a satisfying accuracy if the layer is rather weakly absorbing [126, 237].

Exercises

Exercise 1. Solid angles. An icosahedron is a regular polyhedron with 20 identical faces. Consider an isotropic light source located at the center of the polyhedron, emitting a flux of 4 W. What is the solid angle subtended by each face from the center of this polyhedron? What is the flux crossing each face? *** Exercise 2. Annular illumination geometry. An illumination geometry is said to be annular when the rays reaching illuminated area A on the sample come from all azimuths and form a polar angle with the normal of the sample lying in a certain range [α, α + Δα]. Compute the corresponding solid angle and geometrical extent.

*** Exercise 3. Spectral flux. A light source emits a uniformly distributed flux of 0.3 W between 432 and 437 nm and zero in the rest of the visible spectrum (this can be obtained, for example by filtering white light with a band-pass filter). (a) What is the spectral flux in the emission band of the source? (b) What is the visual spectral flux? *** Exercise 4. Irradiance by an isotropic light source. The minimum illuminance expected for reading a book is around 50 lux. One illuminates a book with an isotropic light bulb emitting flux F = 157 lm. What is maximal distance h between the bulb and the book giving the expected illuminance? ***

202

Exercises

Exercise 5. Luminous efficacy. In the graph below, the spectral power distributions of three light sources A, B and C are plotted. Order these light sources according to their luminous efficacy.

*** Exercise 6. Irradiance by a non-Lambertian extended light source. Consider a circular extended light source of radius R, non-Lambertian, emitting in every point the angular radiance given by function L(θ, φ) = L0 cos2(θ), where θ is the angle from the normal of the disk. The disk illuminates a flat panel parallel to it, located at distance h. Point C is the center of the disk, point D the nearest point of the panel from C (hence, h = CD). (a) Give the exitance of the light source, and the emitted flux. (b) Express the irradiance at point D as a function of L0, h and R. *** Exercise 7. Geometrical extents between areas on two planes. Inter-reflection is a phenomenon that occurs everywhere around us, each time a surface can reflect some light towards another surface and thus re-illuminate it (and reciprocally). To model this phenomenon, one needs to determine the geometrical extent between every pair of elementary areas in the scene. Let us consider here a simple case: two perpendicular planes, horizontal and vertical, of respective equations y = 0 and x = 0 in the (x, y, z) Cartesian coordinate system. Consider point AðxA ; 0; zA Þ on the horizontal plane, and point Bð0; yB ; zB Þ on the vertical plane (see left part of the figure). ♦ (a) Express the unit normal vectors u and v of the two planes. (b) Give the cosine of angle hA between u and (AB) and the cosine of angle hB between v and (AB). (c) Give the elementary geometrical extent between A and B. ♦ Then, consider the case where the vertical plane becomes oblique, forming angle α with the horizontal plane (see right part of the figure). Point B now has the coordinates ðxB ; xB tan a; zB Þ. (d) Determine the unit normal vector w of the oblique plane. (e) Determine the cosine of angle hA between u and (AB), and the cosine of angle hB between w and (AB). (f ) Determine the elementary geometrical extent between the elementary areas around any points A and B.

Exercises

203

*** Exercise 8. Relationship between BRDF and reflectance. Consider a surface illuminated through a certain solid angle Xi in which radiance is not uniform but given by a certain angular distribution function Li ðhi ; ui Þ. The light scattered by the surface can be captured over a certain solid angle Xr . The BRDF of the surface is fR ðhi ; ui ; hr ; ur Þ. Express the irradiance of the surface, then its exitance, and show that its reflectance in this geometry is given by equation (2.11). *** Exercise 9. Relationship between bi-hemispherical reflectance and angular reflectance. A surface with angular reflectance RðhÞ receives Lambertian irradiance E on finite area Δs. Express the incident radiance in a given direction ðh; uÞ of the hemisphere, the elementary flux flowing in that direction, and compute total incident flux Fi coming from the whole hemisphere. Then, compute total flux Fr reflected over the hemisphere. Finally show that bi-hemispherical reflectance r is given by the integral of equation (2.13). *** Exercise 10. Reflectance and reflectance factor in the case of Lambertian surfaces. The radiometric properties of a very matte surface, or Lambertian reflector, have been presented in §2.1.1. Let us show that with this kind of reflector, and with this one only, the reflectance and the radiance factor concepts are equivalent, by considering for example the so-called d:0° measurement geometry where the light is Lambertian over the hemisphere, and the reflected light captured along the normal of the sample (see §2.6.1). One considers that the surface receives flux Fi uniformly distributed over a certain small area DSi , and reflected flux Fr is also uniformly distributed over a certain area DS assumed to be the same as DSi . The radiance factor is defined with respect to a perfect white diffuser. (a) Determine the reflectance of the surface. (b) Determine radiance L issued from the surface and captured by the detector of the measuring device. (c) Determine radiance Lref issued from the perfect white diffuser illuminated in the same conditions and captured by the same ^ Conclude. ♦ NB: Except in the case of detector, then deduce radiance factor R. strongly scattering media like powders or paper, DS is often larger than DSi . If the detector does not capture light from whole area DS, some light is missed and

Exercises

204

the reflectance measured is under-evaluated. This is the edge-loss effect, evoked in §2.7.2. *** Exercise 11. Reflectance of a rough surface. One considers an opaque material with rough surface. Its BRDF at normal incidence is empirically modelled by the function: fR ðhr ; ur Þ ¼

mþ2 m cos ðhr Þ 2p

where hr denotes the polar observation angle (from the normal of the surface), ur the azimuthal observation angle, and m is a constant number. The incident light beam, collimated and perpendicular to the mean surface, illuminates small area A and produces irradiance E in this area. The radiance reflected in any direction ðhr ; ur Þ of the hemisphere by this area A is denoted by Lr ðhr ; ur Þ. (a) Which m value would correspond to a Lambertian reflector? Why? (b) Express Lr ðhr ; ur Þ as a function of E and fR ðhr ; ur Þ. (c) Express total exitance M over the hemisphere, as a function of m and E. What can be said about the absorptance of the material? ♦ The reflectance of the surface is measured using this light beam at normal incidence, and an integrating sphere to collect the reflected light. (d) What is the measuring geometry in this case, according to Nicodemus classification? (e) Determine the reflectance value of the surface for this measuring geometry. ♦ Then, the reflectance is measured using same light beam at normal incidence but collecting only light in one direction, thanks to a small detector of area B placed at distance x from illuminated area A. Area B is perpendicular to the line joining the centers of areas A and B. This line makes an angle ψ = 45° with the normal of A. The detector captures flux F. (f ) According to Nicodemus classification, what is the measuring geometry in this case? (g) What is the geometrical extent G between areas A and B? (h) Express the measured radiance Lm as a function of G and F. Express it also as a function of E and m. Finally, express F as a function of E, m, A, B and x. ♦ One performs similar measurement, with the same measuring geometry, on a perfect white diffuser. (i) Determine exitance Md from the diffuser, and radiance Ld ðhr ; ur Þ reflected in every direction ðhr ; ur Þ of the hemisphere. (j) Determine flux Fd captured by the detector ^ of the (express it as a function of E, A, B and x). (k) Calculate radiance factor R surface according to this geometry. ♦ Now, consider that m tends to infinity. (l) ^ Explain physically the Which type of reflector is it? (m) What is the value for R? result obtained. (n) If, instead of placing the detector at an angle of 45° with respect to the normal of area A, one places it at 0° (the detector area B is parallel to area A), ^ Is it possible for this quantity? what value would be obtained for R? *** Exercise 12. Calibration of a goniophotometer. Consider a goniophotometric system in which the flat reflector of interest is fixed, and the light source and the sensor are attached to independent rotating arms whose respective rotation axes are in the plane of the reflector. One focuses here on the case where the rotation axes of the two arms are confounded: both arms belong to the same plane which also contains the

Exercises

205

normal of the sample. The angle of incidence, θi, and the angle of reflection, θr, are measured from the normal of the sample. The detection system has a fixed geometrical extent: sensor area As is fixed, solid angle Xs over which it can receive light is fixed and perpendicular to the sensor’s surface, and distance x between the sensor and the center of the illuminated area on the sample is fixed. The sensor captures flux Fs ðhi ; hr Þ in every direction hr . One wants to deduce from it the BRDF fr ðhi ; hr Þ of the reflector in the plane of incidence. ♦ At normal incidence ðhi ¼ 0Þ, the thin incident light beam produces uniform irradiance E0 over a small circular area A0 on the sample. (a) What are the illuminated area Aðhi Þ and the irradiance E ðhi Þ at oblique incidence? ♦ At hr ¼ 0, the narrow solid angle of the detection system intercepts the same disk of area A0 . (b) Express solid angle Xs as a function of x and A0 . (c) What is the observed area at oblique angle θr? ♦ One performs a first measurement on a perfect white diffuser illuminated at normal incidence, with θr varying from 0 to π/2. (d) Determine the reflected radiance Lr ðhi ¼ 0; hr Þ, then flux Fs ð0; hr Þ captured by the sensor, as functions of θr (specify the geometrical extent) (e) Determine reflected radiance Lr ðhi ; hr Þ and captured flux Fs ðhi ; hr Þ at oblique incidence. Once again, specify the geometrical extent. (f ) What is the expected BRDF for the perfect white diffuser? ♦ One performs similar measurement, at oblique incidence, on a certain rough metallic surface. The flux measured is function Fs0 ðhi ; hr Þ of θr. (g) How can the BRDF of the sample be obtained? *** Exercise 13. Refractive index of a clear medium from reflectance measurement. Consider an infinitely thick nonabsorbing medium, whose refractive index is real number n that one wants to determine. One measures its reflectance R0 at normal incidence. (a) Remind yourselves of the Fresnel reflectance of the air-medium interface at normal incidence, then determine refractive index n as a function of R0 . (b) If one has an uncertainty of 1% on the measured reflectance value, what is the uncertainty on the obtained refractive index value? *** Exercise 14. Relationship between front and back Lambertian reflectances of a flat interface. Consider the interface between two clear media 1 and 2 of refractive indices n1 , respectively, n2 [ n1 (the relative index n ¼ n2 =n1 is therefore higher than one). When the interface is lit by Lambertian light from medium 1, its bi-hemispherical reflectance r12 is derived from the Fresnel angular reflectance according to equation (4.8). One considers here the case where it is lit by Lambertian light from the other side, i.e., from medium 2. Bi-hemispherical reflectance r21 is given by equation (4.10). One wants to show formula (4.11) relating r12 and r21 :
 r12 1 r21 ¼ 2 þ 1  2 n n (a) Which phenomenon occurs when the interface is lit from medium 2 and not from medium 1? Beyond which angle does it occur? (b) Transform the integral expressing r21 , given by equation (4.10), into two integrals on two different intervals, and compute them using the reciprocity of the Fresnel angular reflectance and Snell’s law. ***

Exercises

206

Exercise 15. Characterization of a clear plate. One wants to determine the refractive index n of a clear polycarbonate. One has two identical plates of thickness h = 1 mm with smooth plane interfaces. One measures (in %) reflectance R1 ¼ 9:58  0:01 and transmittance T1 ¼ 89:92  0:01 of one plate, as well as reflectance R2 ¼ 17:39  0:01 and transmittance T2 ¼ 81:61  0:01 of the (with optical contact) at  two plates  normal incidence. (a) Show that T2 ¼ T12 = 1  R21 and R2 ¼ R1 ð1 þ T2 Þ and verify that this is consistent with the values of R2 and T2 measured. ♦ In a first approximation, one considers that the polymer is not absorbing at all. (b) Express transmittance T1 as a function of n, the deduce the expression of n as a function of T1 and evaluate it with the uncertainty. (c) Express transmittance T2 as a function of n, then deduce the expression of n as a function of T2 and evaluate it with the uncertainty. (d) Which method is the most accurate? ♦ In order to gain in accuracy, one would like to take into account absorption within the material. (e) Express the absorptance A1 of one plate as a function of n and intrinsic spectral transmittance t of the plate and compute it numerically. (f ) Compute numerically absorptance A2 of two plates and comment this value in comparison with the absorptance of one plate. (g) Using the measured values of R1 and T1 and equations (4.46) and (4.47), compute both n and t. Is the value for n equal to the one obtained in question b)? (h) Deduce from t the absorption coefficient. *** Exercise 16. Piles of non-absorbing plates (Lambert’s formulae). One wishes to determine the formulae (5.28) expressing the angular reflectance and transmittance of piles of N identical non-absorbing plates (separated by air) at angle h1 as functions of N and reflectance R ðh1 Þ of one plate: RN ðh1 Þ ¼

NR ðh1 Þ 1  R  ð h1 Þ ; TN ðh1 Þ ¼ 1 þ ðN  1ÞR ðh1 Þ 1 þ ðN  1ÞR ðh1 Þ

(a) Use the compositional formulae (5.5) to express the reflectance and transmittance of two plates at the angle h1 for either the s or p polarization, and show that the formulae above are valid for N = 2. (b) Show recursively that they are valid for any integer N. (c) What are their limit values as N tends to infinity? *** Exercise 17. Non-symmetrical superimposed plates. Consider two non-symmetrical glass plates labelled 1 and 2, whose transfer factors are denoted by Ri, R0i , Ti and Ti0 , i ¼ 1; 2: Give their respective flux transfer matrices, and the flux transfer matrix of plate 1 on top of plate 2, with a layer of air between them. Then show that front-side reflectance R12 and forward transmittance T12 of the pile are given by equation (5.5). *** Exercise 18. Stacks of two plates. Consider a glass plate (plate A) of refractive index n = 1.5 and thickness h = 10 mm. It is perfectly non-absorbing, and its surfaces are flat and smooth. Its reflectance factor RA and transmittance factor TA are measured at normal incidence (0°:0° measuring geometry). (a) What kind of reflector is this

Exercises

207

glass plate? (b) How can one check that the plate is non-absorbing? (c) Can RA and TA be measured with an instrument based on the d:8° geometry? Why? (d) Determine the reflectance expression r0 of one air-glass interface at normal incidence as a function of n, then give its value. (e) Express reflectance RA and transmittance TA of plate A as functions of r0 , then calculate RA þ TA and comment the result. (f ) What is the degree of polarization of the light reflected from the plate at normal incidence? ♦ One has a second plate identical to plate A, and stacks the two plates together. A thin layer of air remains between them. (g) Is reflectance RA þ A of the stack higher or lower than RA ? Why? (h) Is transmittance TA þ A higher or lower than TA ? Why? (i) Express RA þ A and TA þ A as functions of RA and TA , then as functions of r0 . ♦ Then, one introduces a liquid between the two plates, whose refractive index is also n = 1.5. (j) Is reflectance RAA of the stack higher or lower than RA ? Is transmittance TAA higher or lower than TA ? ♦ One now considers a second plate B similar to plate A but slightly absorbing. The absorption coefficient of the glass is K. The plate’s reflectance RB and transmittance TB are measured at normal incidence. (k) Determine intrinsic normal transmittance tB of the plate as a function of K and h. (l) Express RB and TB as functions of r0 , K and h. ♦ Two plates B are stacked together and their reflectance RB þ B and transmittance TB þ B are measured. (m) Compare reflectances RB þ B and RB , and transmittances TB þ B and TB . (n) Express RB þ B and TB þ B as functions of RB and TB . ♦ Then a liquid of index n = 1.5 is introduced between the two plates B and reflectance RBB and transmittance TBB are measured. (o) Compare reflectances RBB and RB , and transmittances TBB and TB . Express RBB and TBB as functions of r0 , K and h. ♦ Plate A is placed on top of Plate B (with a layer of air in between). (p) Compare reflectances RA þ B and RB , and transmittances TA þ B and TB . (q) Express RA þ B and TA þ B as functions of RA , TA , RB and TB . ♦ The liquid of refractive index n = 1.5 is introduced between the two plates A and B. (r) Compare reflectances RAB and RB , and transmittances TAB and TB . ♦ A new plate C (without background) is made by the same absorbing glass than plate B, but its thickness tends to infinity (h ! 1). (s) What is the appearance of plate C? (t) Give expressions for intrinsic transmittance tC , and the plate’s reflectance RC and transmittance TC . (u) Express the bi-hemispherical reflectance of plate C (illuminated by a Lambertian flux over the hemisphere) and give its approximate value. ♦ Plate A is placed on top of this plate C (air remains between them). (v) Show that in this case, at normal incidence, RA þ C becomes RA þ C ¼ 3r0 =ð1 þ 2r0 Þ. *** Exercise 19. Small bottle of perfume. A small bottle of perfume is made of a clear glass of refractive index n1 = 1.52. Its absorption coefficient is close to zero. It has a cubic shape; all faces are flat. (a) Explain why the thickness of faces of the bottle has no influence on its appearance. (b) When illuminating the empty bottle with a pencil of light perpendicular to its larger face, express as a function of n1 the proportion of light that can cross it, and compute it numerically. (c) Compute again the proportion of light that can cross the bottle when it is filled with water (refractive index n2 = 1.33), first as a function of n1 and n2 , then numerically. Comment the result in comparison to the empty bottle. (d) Compute once more the proportion of light that can cross the bottle when it is filled with a perfume of

208

Exercises

refractive index n3 ¼ 1:36 and of absorption coefficient α = 0.003 mm−1, first as a function of n1 , n3 and the intrinsic normal transmittance t of the perfume layer (the thickness of perfume crossed by the light pencil is h = 35 mm), then numerically. Comment the result in comparison to the bottle filled with water. *** Exercise 20. White diffusing medium. One considers a diffusing slice of barium sulphate (BaSO4), a non-absorbing and strongly diffusing material, of thickness h. Its absorption coefficient K tends to 0 and its scattering coefficient is S, assumed to be constant over the visible spectrum of light. ♦ (a) What are the limit values for a, b, qh and sh as K tends to 0? (b) Calculate the limit value qh þ sh . Was this value expected? (c) What is the BRDF of a layer of this medium? (d) One  knows that  the following equality is true whatever thickness h of material a ¼ 1 þ q2h  s2h =2qh . Calculate a with the values for qh and sh obtained in question a), and verify that it is consistent with the value of parameter a also determined in this question a). ♦ The thickness of the slice is doubled. (e) Compare q2h and qh , then s2h and sh . (f ) To predict s2h , one can either consider a double thickness 2S in the Kubelka–Munk formulae obtained in question a), or consider the superimposition of two layers of thickness h using the Kubelka formulae. Show that the two methods are equivalent. (g) If the thickness of the slice tends to infinity, what is the limit value for its reflectance? (h) What is the relationship between the reflectance of the slice and the reflectance factor that would be measured using a spectrophotometer based on the di:8° geometry? *** Exercise 21. Piles of adhesive films. One wants to characterize white adhesive films optically. The material is homogeneous and strongly scattering, with index 1.5. One film has intrinsic reflectance ρ and transmittance τ. The films are not opaque; they have flat interfaces with air, and their thickness is denoted by h. All measurements are performed with an instrument based on the de:0° geometry. The reflectance and ^ 1 and T ^ 1 , are transmittance factors of one adhesive film, respectively, denoted by R measured, from which one deduces the intrinsic reflectance ρ and transmittance τ. Since the material is a homogeneous, strongly scattering medium, one assumes that the Kubelka–Munk model applies and it can therefore be characterized by scattering coefficient S and absorption coefficient K. ♦ (a) Express the respective flux transfer matrices representing the film, the layer of material, and the two interfaces. (b) Explain how ρ and τ are obtained. ♦ One observes experimentally that q þ s  1. (c) What does this mean? (d) Evaluate parameters a ¼ ð1 þ q2  s2 Þ=ð2qÞ and pffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ a 2  1 in this case. (e) What is the value for K in the present case? What, therefore, are the values for a and b as defined in the Kubelka–Munk model? Are they similar to the ones obtained in question d)? (f ) Express ρ and τ as functions of S and h. ♦ By pasting a second film on the first one, one obtains a double film with thickness 2 h. (g) Show, using Kubelka’s formulae, that in this case, the intrinsic reflectance q2 of the double film can be written q2 ¼ 2q=ð1 þ qÞ. (h) Determine the expression for s2 . (i) One film is added again. Express the intrinsic transfer factors q3 and s3 of the triple film as functions of ρ and τ. (j) Show that for any integer k  1, the reflectance qk and the transmittance sk of a stacking of k films are given by:

Exercises

209

qk ¼

kq 1q and sk ¼ 1 þ ðk  1Þq 1 þ ðk  1Þq

(k) According to the expressions for ρ and τ obtained in f), express transfer factors qk and sk as functions of S and h. Is it consistent with the result of question f)? (l) Determine the limit values for qk and sk when k tends to infinity. Comment. (m) Remember that for a pile of films with “infinite” thickness, the intrinsic reflectance is given by q1 ¼ a  b. Is it consistent with the value for q1 obtained in the previous question? ♦ Let us now consider an opaque dark board of refractive ^ g with index 1.5 and intrinsic reflectance qg . One measures its reflectance factor R ^ g as a function of respect to a perfect white diffuser. (n) Express reflectance factor R rs , Tin , Tout , ri and qg according to the Saunderson model. (o) Remind yourselves of the physical meaning for rs , Tin , Tout , ri , their specific expression for the considered measuring geometry, and their values for a refractive index 1.5. (p) Deduce intrinsic ^ g (inversed Saunderson forreflectance qg from the measured reflectance factor R mula). ♦ One film is pasted on the dark board. The film and the board are in optical contact. (q) Give intrinsic reflectance qgf of the system board + film. (r) Show that, according to Kubelka–Munk formulae, qgf is written as below and simplify this expression according to the result of question k):   1  aqg sinhðbSh Þ þ bqg coshðbSh Þ  qgf ¼  a  qg sinhðbSh Þ þ b coshðbSh Þ *** Exercise 22. Saunderson parameters in the d:8° and 8°:d geometries. A diffusing support bounded by a smooth interface with air is measured using two instruments, respectively, based on a d:8° geometry and a 8°:d geometry. Parameters rs , Tin , Tout , and ri in the Saunderson equation (6.20) are given in table 6.1. One focusses here on parameters Tin and Tout : show that their product is the same for the two geometries, therefore that the Saunderson equation is the same. *** Exercise 23. Color reproduction in publishing. A publisher wants to publish the work of a photographer in the format of an art book. The original photographs are black and white silver prints. For a better reproduction quality of pictures, he decides to print the book with two inks: a black ink and a grey ink. (a) What is the interest of using the grey ink in addition to the black one? (b) With two inks, how many Neugebauer primaries are there? ♦ The black ink is very strongly absorbing, its intrinsic transmittance is near zero. (c) How many Neugebauer primaries are there in this case? ♦ The surface coverages of the grey and black inks are, respectively, x and y. (d) What is the surface coverage of the Neugebauer primaries? ♦ Each Neugebauer primary is printed alone over a larger area on the paper, and the spectral reflectance factor is measured. (e) What is the spectral reflectance factor of the solid black color patch if the specular component is discarded from the measurement? and if it is included? (f ) Determine the spectral reflectance factor (specular component excluded) predicted by the Yule–Nielsen model with nYN = 2

210

Exercises

for a halftone color containing only black ink at 0.5 surface coverage (g) What is therefore the surface coverage of black ink to print if one desires to obtain a spectral reflectance being half of that of the bare paper? ♦ (h) Determine the spectral reflectance predicted by the Clapper–Yule model for the same halftone as in question f), if the measurements are made with an instrument based on the d:8° geometry (one can assume that the bare paper has a refractive index around 1.5) (i) What is therefore the surface coverage of black ink to print if one desires to obtain a spectral reflectance being half of that of the bare paper?

Corrections

Exercise 1. Twenty faces are identical and each of them subtends one twentieth of the sphere (solid angle 4π). The solid angle subtended by each face from the center of the volume is therefore 4π/20, and the flux crossing one face is 4/20 = 0.2 W. *** Exercise 2. The solid angle Ω associated with the annular geometry is given by the following integral: Z 2p Z a þ Da X¼ sin hdhdu ¼ 2p½ cos haa þ Da ¼ 2p½cosðaÞ  cosða þ DaÞ u¼0

h¼a

and geometrical extent G by the following one: Z 2p Z a þ Da Z sinða þ DaÞ  sinða þ DaÞ G¼ cos h sin hdhdu ¼ 2p udu ¼ p u 2 sinðaÞ u¼0

h¼a

u¼sinðaÞ

¼ p sinð2a þ DaÞ sinðDaÞ *** Exercise 3. (a) The spectral flux in the emission band is 0.3/(437‒432) = 0.06 W/nm. (b) The spectral flux, weighted by the standard sensitivity function V(λ) of the human visual system and integrated over the visible spectrum gives the visual flux: moreover, we note that “spectral visual flux” has no meaning. *** Exercise 4. The intensity emitted by the isotropic lamp in every direction is I = F/4π. According to Bouguer’s law (§1.6.1), the illuminance on the nearest point of the book from the source is E = I/h2. Therefore, the maximal distance is pffiffiffiffiffiffiffiffiffi h ¼ I =E = 0.5 m. *** Exercise 5. The power of source C is totally emitted in the UV, which is not visible by humans (the luminous efficacy is zero). The power of source A is mainly emitted

Corrections

212

in the infrared, which again is not visible by humans (the luminous efficacy is low). Source B emits only visible light, the luminous efficiency is the highest among the three sources. *** Exercise 6. (a) Exitance M of the light source, uniform over the source’s area, is obtained by integrating the radiance over the hemisphere: Z 2p Z p=2 Z p=2 M¼ Lðh; uÞ cos h sin hdhdu ¼ 2pL0 cos3 h sin hdh u¼0

¼ 2pL0

h¼0 1

Z

h¼0

pL0 u 3 du ¼ 2 u¼0

Flux F emitted is the exitance multiplied by the source area: F ¼ pR2 M ¼

p2 R2 L0 2

(b) Point D on the panel receives the radiance given by function L(θ, φ), θ ranging between 0 (radiance coming from C, the center of the disk) and b ¼ arctanðR=h Þ. The irradiance ED in D is obtained by integrating radiance L over the cone of apex half-angle β: Z b Z 2p Z b ED ¼ Lðh; uÞ cos h sin hdhdu ¼ 2pL0 cos3 h sin hdh u¼0 h¼0 h¼0 " # Z 1 pL0 h4 3 1 ¼ 2pL0 u du ¼ 2 ðh 2 þ R 2 Þ2 u¼cos b *** Exercise 7. (a) The unit vectors are: u ¼ ð0; 1; 0Þ and v ¼ ð1; 0; 0Þ. (b) The cosines of the angles are obtained by computing the following dot products: ! AB ð0; 1; 0Þ  ðxA ; yB ; zB  zA Þ yB ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos hA ¼ u  AB 2 xA2 þ yB2 þ ðzB  zA Þ xA2 þ yB2 þ ðzB  zA Þ2

cos hB ¼ v 

! BA ð1; 0; 0Þ  ðxA ; yB ; zA  zB Þ xA ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AB xA2 þ yB2 þ ðzB  zA Þ2 xA2 þ yB2 þ ðzB  zA Þ2

(c) By defining elementary area dxA dyA around point A and elementary area dxB dyB around point B, the elementary geometrical extent, as defined by equation (1.18), is given by: d 4G ¼

dxA dyA cos hA dxB dyB cos hB xA yB dxA dyA dxB dyB ¼h i2 2 AB xA2 þ yB2 þ ðzB  zA Þ2

Corrections

213

NB: This formula is not applicable when AB tends to zero, i.e., when A and B are too close from each other. (d) The unit vector is: w ¼ ðsin a;  cos a; 0Þ. (e) The dot products yield: ! AB ð0; 1; 0Þ  ðxB  xA ; xB tan a; zB  zA Þ xB tan a cos hA ¼ u  ¼ ¼ AB AB AB

! BA ðsin a;  cos a; 0Þ  ðxA  xB ; xB tan a; zA  zB Þ xA sin a ¼ ¼ AB AB AB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with AB ¼ ðxB  xA Þ2 þ ðxB tan aÞ2 þ ðzB  zA Þ2 . (f ) The elementary geometrical extent is: cos hB ¼ w 

d 4G ¼

xA yB sin a tan adxA dyA dxB dyB AB4

*** Exercise 8. According to equation (1.33), the light pencil incident from some direction ðhi ; ui Þ generates an elementary irradiance d 2 E ðhi ; ui Þ ¼ Li ðhi ; ui Þ cos hi sin hi dhi dui on an elementary area ds of the surface. The total irradiance originated from whole solid angle Xi is given by the following double integral: ZZ E¼ Li ðhi ; ui Þ cos hi sin hi dhi dui ðhi ;ui Þ2Xi

The contribution of incident radiance Li ðhi ; ui Þ to radiance Lr ðhr ; ur Þ is specified by the BRDF according to equation (2.6). The corresponding elementary exitance is d 4 M ðhi ; ui ; hr ; ur Þ ¼ fR ðhi ; ui ; hr ; ur ÞLi ðhi ; ui Þ cos hi sin hi dhi dui cos hr sin hr dhr dur By summing up the contributions of all radiances contained within Ωi, one obtains the elementary exitance in the direction ðhr ; ur Þ ZZ

d 2 M ðxi ; hr ; ur Þ ¼

ðhi ;ui Þ2xi

fR ðhi ; ui ; hr ; ur ÞLi ðhi ; ui Þ cos hi sin hi dhi dui cos hr sin hr dhr dur

Then, by summing up all the elementary exitances through the observation solid angle Ωr, one obtains the total exitance ZZ

M¼

ZZ

ðhi ;ui Þ2Xi

ðhr ;ur Þ2Xr

fR ðhi ; ui ; hr ; ur ÞLi ðhi ; ui Þ cos hi sin hi dhi dui cos hr sin hr dhr dur

By definition, reflectance is the ratio of reflected flux Mds to incident flux Eds. It is therefore the ratio M =E, which gives equation (2.11). ***

Corrections

214

Exercise 9. The incident flux is the product of the irradiance and the illuminated area: Fi ¼ EDs. According to Lambert’s law (see §1.5.1), since the illumination is Lambertian, incoming radiance Li in any direction ðh; uÞ of the hemisphere is E/π. Elementary flux d 3 F ðh; uÞ that illuminates area Δs from this direction is d 2 F ðh; uÞ ¼

E Ds cos h sin hdhdu p

Elementary flux d 2 Fr ðh; uÞ that is reflected is simply the product of the incident elementary flux and the angular reflectance: d 2 Fr ðh; uÞ ¼ RðhÞd 2 F ðh; uÞ ¼ RðhÞ

E Ds cos h sin hdhdu p

The total reflected flux is obtained by integrating the expression above over the hemisphere: Z p=2 Z 2p Z p=2 E E Fr ¼ RðhÞ Ds cos h sin hdhdu ¼ 2p Ds RðhÞ cos h sin hdh p p u¼0 h¼0 h¼0 Finally, the reflectance is by definition Fr =Fi , and one retrieves equation (2.13): Z p=2 Z p=2 Fr ¼2 RðhÞ cos h sin hdh ¼ RðhÞ sin 2hdh r¼ Fi h¼0 h¼0 *** Exercise 10. (a) By definition, reflectance is the ratio q ¼ Fi =Fr . (b) Reflected flux Fr ¼ qFi being uniformly distributed over area DS, the exitance is M ¼ Fr =DS. Since the outgoing light is Lambertian, the radiance in every direction of the hemisphere, thereby in direction of the detector, is L ¼ M =p ¼ qFi =ðpDS Þ. (c) The radiance captured from the perfectly white reflector, whose reflectance is 1, is similarly expressed: Lref ¼ Fi =ðpDS Þ. Consequently, the radiance factor is ^ ¼ L=Lref ¼ q. The radiance factor is equal to the reflectance. R *** Exercise 11. (a) A Lambertian reflector has an angle-independent BRDF proportional to 1/π. It corresponds to m = 1. (b) By definition of the BRDF, the reflected radiance Lr ðhr ; ur Þ is fR ðhr ; ur ÞE. (c) The exitance is obtained by integrating the reflected radiance over the hemisphere: Z 2p Z p=2 M¼ LR ðhr ; ur Þ cos hr sin hr dhr dur ur ¼0

hr ¼0

¼ ðm þ 2ÞE ¼ ðm þ 2ÞE

Z

p=2

hr ¼0 Z 1

cosm þ 1 ðhr Þ sin hr dhr

u m þ 1 du ¼ E

0

Corrections

215

One finds that for every m value, the exitance is equal to the irradiance, which means that the surface is non-absorbing. (d) The measuring geometry is directional-hemispherical (0°:d in the nomenclature recommended by the CIE). (e) On the one hand, incident flux is Fi ¼ EA. On the other hand, reflected flux is Fr ¼ MA ¼ EA, and one finds that the reflectance is 1, as expected since the surface is non-absorbing, and all the reflected light is collected by the integrating sphere. (f ) This corresponds to a bi-directional geometry (0°:45° according to the notation recommended by the CIE). (g) By assuming that areas A and B are small and distant pffiffiffi enough, the geometrical extent can be expressed G ¼ AB cos w=x 2 ¼ AB 2=ð2x 2 Þ. (h) The measured radiance is the ratio between of the measured flux and the geometrical extent: Lm ¼ F=G. According to the BRDF, it is also: Lm ¼ fR ð45 ; 0 ÞE ¼

m þ 2 m=2 2 E 2p

Therefore: F¼

m þ 2 m=2 m þ 2 ðm þ 3Þ=2 2 EG ¼ 2 EAB 2p px 2

(i) The perfect white diffuser reflects all incident light. The exitance is therefore equal to the irradiance: Md ¼ E. In every direction of the hemisphere, the reflected radiance is Ld ¼ M =p ¼ E=p. (j) The flux captured by the detector pffiffiffi is Ld G, therefore: Fd ¼ 2EAB=ð2px 2 Þ. (k) The reflectance factor is ^ ¼ F=Fd ¼ ðm þ 2Þ2ðm=2 þ 1Þ . (l) An m value tending to infinity corresponds to a R specular reflector, i.e., a perfect mirror (the BRDF tends to a Dirac delta function). (m) The reflectance factor tends to 0, which is consistent with the fact that all light is specularly reflected at 45° from the normal of the surface, and no light is reflected in other directions. The detector captures no light. (n) The detector is now located in the specular direction and captures all incident light. It thus receives a radiance equal to the incident radiance (which is not specified in the exercise), and therefore ^ is collects a flux much higher than the one captured with the perfect white diffuser. R therefore be much higher that 1 (but it cannot be infinite in practice). *** Exercise 12. (a) At oblique incidence, the illuminated area is an ellipse of area Aðhi Þ ¼ A0 =coshi . The irradiance can still be assumed uniform in first approximation. Since the same flux Fi ¼ EA0 reaches the sample on a larger area, the irradiance is decreased: E ðhi Þ ¼ Fi =Aðhi Þ ¼ E cos hi . (b) By assuming that the diameter of the illuminated disk is much smaller than the distance x, one can write: Xs ¼ A0 =x 2 . (c) As in question a), the observed area is an ellipse of area Aðhr Þ ¼ A0 =coshr . (d) Since the reflector is a perfect white diffuser, and since it is uniformly illuminated over the area A0 , the same radiance L = E/π stems from every elementary area contained in area A0 and in every direction of the hemisphere (thereby in the direction of the sensor). Therefore, Lr ðhi ¼ 0; hr Þ ¼ E=p. Notice that the observed area (i.e., the area intercepted by the solid angle Ωs of the detection system) is larger than the illuminated area, except for hr ¼ 0 where they are equal. The geometrical

Corrections

216

extent through which the light flows from the illuminated area to the sensor is defined by G ¼ ðA0 cos hr ÞAs =x 2. The captured flux is therefore proportional to cos hr : Fs ð0; hr Þ ¼ LG ¼

E ðA0 cos hr ÞAs px 2

(e) At oblique incidence, the illuminated area Aðhi Þ is larger than A0 . The radiance reflected in the direction of the sensor is Lr ðhi ; hr Þ ¼ E ðhi Þ=p ¼ E0 cos hi =p. At the sensor position hr ¼ 0, the observed area Aðhr Þ intercepted by the solid angle of the detection system is smaller than the illuminated area Aðhi Þ. This remains true until hr reaches an angle equal to hi . In this case, one can consider that the light reaching the sensor fulfills the solid angle ωs and the geometrical extent is defined by G ¼ Xs As . The flux captured by the sensor is independent of the angle hr : Fs ðhi ; hr Þ ¼ Lr ðhi ; hr ÞG ¼ E0 cos hi Xs As =p with Xs ¼ A0 =x 2 after question b). Then, when hr exceeds hi , the observed area becomes larger than the illuminated one. The reasoning line followed in question d) is still valid, and Fs ðhi ; hr Þ ¼ LG ¼

E ðA0 cos hr ÞAs px 2

Finally, the flux Fs ðhi ; hr Þ can be written:

E0 cos hi A0 As 1 Fs ðhi ; hr Þ ¼ cos hr px 2

if 0 hr hi if hi hr p=2

(f ) The expected BRDF for a perfect white diffuser is the constant 1/π. (g) The BRDF fr ðhi ; hr Þ of the metallic sample determines the radiance Lr ðhi ; hr Þ ¼ fr ðhi ; hr ÞE0 cos hi reflected in the direction of the sensor, which replaces the radiance Lr ðhi ; hr Þ ¼ E0 cos hi =p considered in question e). The flux measured is therefore given by:

E0 cos hi A0 As 1 if 0 hr hi 0 Fs ðhi ; hr Þ ¼ pfr ðhi ; hr ÞFs ðhi ; hr Þ ¼ fr ðhi ; hr Þ cos hr if hi hr p=2 x2 Hence, one can obtain the BRDF of the metallic mirror by simply dividing, for each angle, the flux measured on the metallic surface with the one measured on the perfect white diffuser and multiplying the result by π, which prevents the evaluation of the lighting and sensor parameters (A0 , E0 , xs , Ad , x): fr ðhi ; hr Þ ¼

Fs0 ðhi ; hr Þ pFs ðhi ; hr Þ

*** Exercise 13. (a) The Fresnel reflectance of the air-medium interface at normal incidence is R0 ¼ ðn  1Þ2 =ðn þ 1Þ2 , from which one deduces:

Corrections

217

n¼

pffiffiffiffiffiffi 1 þ R0 pffiffiffiffiffiffi 1  R0

(b) The uncertainty on the refractive index depends on the reflectance value measured, R0 . It is determined through the derivative of function n ðR0 Þ determined in the previous question: 1 n 0 ðR0 Þ ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi2 R0 1  R0 and with an uncertainty of the reflectance measured defined as DR0 ¼ 0:01R0 , one has: pffiffiffiffiffiffi 0:01 R0 0 Dn ¼ n ðR0 ÞDR0 ¼  pffiffiffiffiffiffi2 1  R0 with DR0 ¼ 0:01R0 . With a measured reflectance of 0.04, the refractive index is 1.5 and the uncertainty is 0.003 (or 0.2%). With a measured reflectance of 0.14, the refractive index is 2.2 and the uncertainty is 0.001 (or 0.45%). *** Exercise 14. (a) A ray coming from medium 2 is totally reflected if the angle of incidence is larger than the critical angle hc ¼ arcsinð1=n Þ. (b) The integral expressing the bi-hemispherical reflectance at the side of medium 1 can be written as Z hc Z p=2 r21 ¼ R21 ðh2 Þ sin 2h2 dh2 þ R21 ðh2 Þ sin 2h2 dh2 h2 ¼0

h2 ¼hc

In the first integral, according to the reciprocity of the Fresnel angular reflectance, one has R21 ðh2 Þ ¼ R12 ðh1 Þ where h1 satisfies Snell’s law: sin h1 ¼ n sin h2 . By differentiating Snell’s law, one also has cos h1 dh1 ¼ n cos h2 dh1 , and therefore sin 2h1 dh1 ¼ n 2 sin 2h2 dh1 . Hence, Z p=2 Z hc sin 2h1 r12 R21 ðh2 Þ sin 2h2 dh2 ¼ R12 ðh1 Þ dh1 ¼ 2 2 n n h2 ¼0 0 In the second integral, R21 ðh2 Þ ¼ 1 since light is totally reflected beyond the critical angle. Hence, Z p=2 Z 1 1 sin 2h2 dh2 ¼ 2 udu ¼ 1  2 n arcsinð1=n Þ 1=n The sum of the two integrals yields the expression for r21 given in equation (4.11). *** Exercise 15. (a) Thanks to composition equations (5.2) and (5.3), since here the two plates are identical and independent of polarization at normal incidence, one retrieves the measured values:

Corrections

218

T2 ¼

T12  0:8161  0:0002; and 1  R21

R2 ¼ R1 þ

T12 R1 ¼ R1 ð1 þ T2 Þ  0:1740  0:0002 1  R21

(b) The angular transmittance of a non-absorbing plate is given by equation (4.20) in §4.4.1. At normal incidence, the Fresnel reflectance of each interface, for any polarization, is r0 ¼ ðn  1Þ2 =ðn þ 1Þ2 . The transmittance is therefore: T1 ¼

1  r0 ðn þ 1Þ2 ðn  1Þ2 2n ¼ ¼ 1 þ r0 ðn þ 1Þ2 þ ðn  1Þ2 n 2 þ 1

from which one deduces n¼

1þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  T12  1:5987  0:0004 T1

NB: The uncertainty is estimated by evaluating n with T2 ¼ 0:8991 and T2 ¼ 0:8993. (c) The angular transmittance of a pile of N non-absorbing plates is given by equation (5.29) in §5.2.1. At normal incidence, one has T2 ¼

1  r0 ð n þ 1Þ 2  ð n  1Þ 2 n ¼ ¼ 1 þ 3r0 ðn þ 1Þ2 þ 3ðn  1Þ2 n 2  n þ 1

from which one deduces n¼

1 þ T2 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2T2  3T22  1:6005  0:0002 2T2

(d) The method using two plates looks more accurate: the uncertainty in the measurement has a weaker influence on the uncertainty on the refractive index, even though the error induced by the neglect of the absorption is higher. (e) The absorptance of one plate is ! ð1  r0 Þ2 r0 t 2 ð1  r 0 Þ2 t A1 ¼ 1  R1  T1 ¼ 1  r0 þ  1  r02 t 2 1  r02 t 2 ¼

ð1  r 0 Þð1  t Þ  0:005 1  r0 t

(f ) The absorptance of two plates is A1 ¼ 1  R2  T2  0:01. Since the absorptance of one plate is very low, the one of two plates is just the double. NB: One can show that the absorptance of the two plates is given by the following formula: A2 ¼

ð1  r0 Þð1  t 2 Þ 1  r0 t 2

Corrections

219

(g) Equations (4.46) and (4.47) yield n  1:5801  0:0004 and t  0:9950  0:0002. By neglecting absorption in question a), the refractive index was overestimated. (h) According to Beer’s law: t ¼ expðah Þ. Since thickness h is known, the absorption coefficient of the polycarbonate can be computed: it is a ¼  logðt Þ=h  5:3 mm1 . *** Exercise 16. (a) Since the plates are non-absorbing, their transmittance is 1  R ðh1 Þ. For two identical plates, the compositional formulae (5.5) yield: R2 ðh1 Þ ¼ R ðh1 Þ þ TN ðh1 Þ ¼

½ 1  R  ð h1 Þ  2 R  ð h1 Þ 2R ðh1 Þ ¼ 1  R2 ðh1 Þ 1 þ R  ð h1 Þ

½1  R ðh1 Þ2 1  R ðh1 Þ ¼ 1 þ R  ð h1 Þ 1  R2 ðh1 Þ

(b) By assuming that equation (5.28) are valid for N plates, and considering one additional plate placed on top of the pile of N plates, the compositional formulae (5.5) yield  R ðh1 Þ þ ½1  R ðh1 Þ2 R2 ðh1 Þ RN ðh1 Þ RN þ 1 ðh1 Þ ¼ 1  R ðh1 ÞRN ðh1 Þ R ðh1 Þ þ ðN  1ÞR2 ðh1 Þ þ ½1  2R ðh1 ÞNR ðh1 Þ ¼ 1 þ ðN  1ÞR ðh1 Þ  NR2 ðh1 Þ ðN þ 1ÞR ðh1 Þ½1  R ðh1 Þ ðN þ 1ÞR ðh1 Þ ¼ ¼ 1 þ NR ðh1 Þ ½1  R ðh1 Þ½1 þ NR ðh1 Þ and TN þ 1 ðh1 Þ ¼

½1  R ðh1 ÞTN ðh1 Þ ½1  R ðh1 Þ2 1  R ðh1 Þ ¼ ¼ 1  R ðh1 ÞRN ðh1 Þ ½1  R ðh1 Þ½1 þ NR ðh1 Þ 1 þ NR ðh1 Þ

which shows that equation (5.28) are also valid for N + 1 plates. (c) As N tends to infinity, RN ðh1 Þ tends to 1, which makes sense since the plates are non-absorbing and all the light is back-reflected. TN ðh1 Þ tends to 0 (see figures 5.5 and 5.6). However, there is one exception if h1 coincides with the Brewster angle: RpN ðhb Þ ¼ 0, for every N value. *** Exercise 17. For a certain angle of incidence and for each polarization (symbol  stands for either s or p polarization), the flux transfer matrices of plates 1 and 2 are of the form:
 1 1 R0i Mi ¼ ; i ¼ 1; 2 with Xi ¼ Ti Ti0  Ri R0i Ti Ri Xi

Corrections

220

For the same angle and polarization component, the flux transfer matrix of the pile is given by the product of the two individual matrices (the left-to-right product reproduced the front-to-back order of the plates):


 m11 m12 1 R02 1 R01 1 M12 ¼ ¼ M1 M2 ¼ T1 T2 R1 X1 m21 m22 R2 X2 !   0 0 0 1  R1 R2 R2 1  R01 R2  T2 T2 R01 1   ¼ 0 T1 T2 R1 1  R01 R2 þ T1 T1 R2 X1 X2  R1 R02 Using equation (5.14), one easily finds that R12 ¼ m21 =m11 ¼ R1 þ T12 ¼ 1=m11 ¼

0 T1 T1 R2 1  R01 R2

T1 T2 1  R01 R2

R012 ¼ m12 =m11 ¼ R02 þ

0 T2 T2 R01 0 1  R1 R2

0 The calculation of T12 needs more expansion: one must show that 0 0 T2 . detðM12 Þ ¼ T1 *** Exercise 18. (a) The glass plate is a specular reflector. (b) One can verify that it is non-absorbing by checking that RA þ TA ¼ 1. (c) Since the plate is a specular reflector, RA and TA can be measured with an instrument based on the d:8° geometry: the effective measurement geometry is a near-normal bi-directional geometry (precisely an 8°:8° geometry, which can be considered as a 0°:0° geometry without significant impact on the numerical values of reflectances and transmittances). (d) The reflectance of one air-glass interface at normal incidence is r0 ¼ ðn  1Þ2 =ðn þ 1Þ2 ¼ 0:04. (e) The angular reflectance and transmittance of the plate, at normal incidence, are

RA ¼ r0 þ

ð1  r0 Þ2 r0 2r0 ð1  r0 Þ2 1  r0 ¼ and TA ¼ ¼ 2 1 þ r0 1 þ r0 1  r0 1  r02

As expected, since the plate is non-absorbing, one has RA þ TA ¼

2r0 1  r0 þ ¼1 1 þ r0 1 þ r0

(f ) At normal incidence, the degree of polarization is zero if the incident light is unpolarized. (g) With four air-glass interface, the two plates reflect more light (and transmit less light) than one plate which has only two air-glass interfaces: RA þ A is higher than RA . (h) For the same reason, the transmittance TA þ A is lower than TA . (i) Thanks to the compositional formulae (5.2) and (5.3), written with two identical plates A, one has:

Corrections

221

RA þ A ¼ RA þ

TA2 RA TA2 2RA 4r0 1  RA 1  r0 ¼ ¼ and TA þ A ¼ ¼ ¼ 2 2 1  RA 1 þ RA 1 þ 3r0 1  RA 1  RA 1 þ 3r0

(j) The liquid has the same refractive index as the plates; its bordering interfaces become optically neutral and one obtains a single plate with double thickness. Since the glass is non-absorbing, the thickness has no effect on the reflectance and transmittance, which therefore remain unchanged: RAA ¼ RA and TAA ¼ TA . (k) The intrinsic normal transmittance is given by Beer’s law: tB ¼ expðKh Þ. (l) The angular reflectance and transmittance of an absorbing plate are given by equations (4.38) and (4.40). At normal incidence, for either polarization component and unpolarized light, one has: RB ¼ r0 þ

ð1  r0 Þ2 r0 e2Kh ð1  r0 Þ2 eKh and T ¼ B 1  r02 e2Kh 1  r02 e2Kh

(m) When stacking two plates B, one has four interfaces that back-reflect more light (and transmit less light) than the two interfaces of one plate. Reflectance RB þ B is therefore higher than RB , and TB þ B is lower than TB . (n) Once again, one may use the compositional formulae (5.2) and (5.3): RB þ B ¼ RB þ

TB2 RB TB2 and TB þ B ¼ 2 1  RB 1  R2B

(o) A for plates A, the liquid of refractive index 1.5 cancels the optical effect of the two central interfaces and the two plates form one plate with double thickness, which is therefore more absorbing. Hence, both reflectance and transmittance of the double plate are lower than the ones of one plate: RBB \RB and TBB \TB . The expressions for RBB and TBB are: RBB ¼ r0 þ

ð1  r0 Þ2 r0 e4Kh ð1  r0 Þ2 e2Kh and T ¼ BB 1  r02 e4Kh 1  r02 e4Kh

(p) With their four air-glass interfaces, two plates A and B reflect more light and transmit less light than plate B alone, which has only two air-glass interfaces. Hence, RA þ B [ RB , and TA þ B \TB . (q) Thanks to the compositional formulae (5.2) and (5.3), RA þ B ¼ RA þ

TA2 RB TA RB and TA þ B ¼ 1  RA RB 1  RA RB

(r) With a liquid of refractive index 1.5 between plates A and B, one obtains the equivalent of one plate, bordered by two air-glass interface. Between the interface, the clear glass layer (corresponding to plate A) is optically neutral. The normal transmittance of the plate is therefore the same as the one of plate B. Hence, RAB þ RB and TAB þ TB . (s) A very thick plate of absorbing glass has a black color since all the light that enters the plate is absorbed, and a glossy appearance due to the specular reflection of light by the front-side interface. (t) The intrinsic normal

Corrections

222

transmittance of the plate tends to 0. RC tends to r0, and TC tends to 0. (u) Since only the front-side interface can reflect some light, its bi-hemispherical reflectance is r12, given by equation (4.7): Z p=2 R12 ðhÞ sin 2hdh r12 ¼ h¼0

or by Duntley’s formula (4.8). (v) By placing plate A on top of the infinitely thick plate C, one has three consecutive air-glass interfaces placed on top of a black background. One may compute reflectance RA þ C by considering plate A on top of one interface (one can use the equality RA þ TA ¼ 1): RA þ C ¼ RA þ

TA2 r0 RA  r0 ðRA  TA Þ 3r0  3r02 3r0 ¼ ¼ ¼ 1  RA r0 1  RA r0 ð1 þ r0 Þð1  2r0 Þ 1 þ 2r0

*** Exercise 19. (a) The glass being non-absorbing, a slice of any thickness has the same reflectance and transmittance. This is no longer true when the glass is absorbing: increasing the thickness induces a decrease in both reflectance and transmittance. (b) For a light pencil perpendicular to the larger face, the empty bottle is a superimposition of two glass plates, separated by air. Its transmittance is therefore given by equation (5.29) with N = 2 and R12 ð0Þ ¼ r ¼ ðn1  1Þ2 =ðn1 þ 1Þ2 : T¼

ðn1 þ 1Þ2  ðn1  1Þ2 ðn1 þ 1Þ2 þ 3ðn1  1Þ2

¼

n1  0:857 n12  n1 þ 1

(c) As the bottle is filled with water, one has a superposition of four interfaces: air-glass, glass-water, water-glass and glass-air. The air-glass interfaces still have transmittance r given above; the glass-water interfaces have a relative refractive index n12 ¼ n1 =n2  1:14, and their reflectance at normal incidence is r 0 ¼ ðn12  1Þ2 =ðn12 þ 1Þ2  0:43%. To determine the transmittance of the bottle, one can use the transfer matrix method. The air-glass interfaces are represented by matrix F and the glass-water interfaces are represented by matrix F0 , expressed as in equation (5.38):

 1 1 1 r 1 r 0 and F0 F¼ 1  r r 1  2r 1  r 0 r 0 1  2r 0 Transfer matrix M representing the bottle filled with water is
 1 1 þ r þ r 0  3rr 0 2ðr þ r 0  2rr 0 Þ M ¼ FF0 FF0 ¼ ð1  rÞð1  r 0 Þ 2ðr þ r 0  2rr 0 Þ 1  3r  3r 0 þ 5rr 0 and the inverse of the top-left entry is the transmittance of the bottle fill with water is: T0 ¼

ð1  rÞð1  r 0 Þ 2n1 n2 ¼  0:917 1 þ r þ r 0  3rr 0 n12 þ ðn1  1Þ2 n2 n22

Corrections

223

Since the two internal interfaces have a lower relative refractive index than when the bottle is filled with air, their transmittance is higher, and the transmittance of the bottle is therefore higher. (d) The perfume is absorbing. One must determine the intrinsic normal transmittance t of the perfume “layer”, and introduce the flux transfer matrix. According to Beer’s law, t ¼ expðahÞ  0:90. At normal incidence, the transfer matrix representing the perfume “layer” given by equation (5.40) is written
 1=t 0 L¼ 0 t The transfer matrix of the bottle filled with perfume is M ¼ FF0 LF0 F, and the inverse of its top-left entry gives the transmittance of the bottle filled with perfume: T 00 ¼

ð1  rÞ2 ð1  r 00 Þ2 t

ð1  rr 00 Þ2  ðr  r 00  2rr 00 Þ2 t 2 16n12 n32 t ¼ h i2  0:827 ðn3 þ 1Þ2 ðn3 þ n12 Þ2  ðn1  n3 Þ2 þ n3 ðn1  1Þ2 t 2

Of course, the bottle filled with perfume is less transmissive than the bottle filled with water, due to the absorption of light within the perfume, even though the relative refractive index of the internal glass-perfume interfaces is slightly lower than the one of the glass-water interfaces (n1 =n3  1:10) and the interfaces are slightly more transmissive. However, despite the fact that the perfume is absorbing, the transmittance of the bottle filled with perfume is not much lower than the one of the empty bottle. *** Exercise 20. (a) As K tends to 0, the limit value for a is 1 and the one for b is 0. Kubelka–Munk functions qh and sh are undetermined, but thanks to the Taylor expansions of the hyperbolic trigonometric functions sinhðxÞ ¼ x þ Oðx 2 Þ and coshðxÞ ¼ 1 þ Oðx 2 Þ, one finds that their limits are: sinhðbShÞ Sh ! asinhðbShÞ þ bcoshðbShÞ K !0 Sh þ 1 b 1 ! sh ¼ asinhðbShÞ þ bcoshðbShÞ K !0 Sh þ 1

qh ¼

(b) The limit value for qh þ sh is 1 for any thickness, which is expected since K tending to zero means that the medium is non-absorbing and the energy is conserved. (c) The medium being strongly diffusing, one can assume that any layer is a Lambertian reflector: it BRDF is a constant equal to (see §2.4.2): fr ¼

Sh pðSh þ 1Þ

Corrections

224

(d) One retrieves from the expression for a that it is equal to 1: a¼

1 þ q2h  s2h ðSh þ 1Þ2 þ ðShÞ2  1 ¼1 ¼ 2ShðSh þ 1Þ 2qh

(e) As for any diffusing layer when the thickness is increased, one expects that q2h is higher than qh , and s2h is lower than sh (see §6.5). This is confirmed by: 2Sh Sh [ 2Sh þ 1 Sh þ 1 1 1 \ ¼ 2Sh þ 1 Sh þ 1

q2h ¼ s2h

(f ) One can also compute s2h using Kubelka’s formula (see §6.8.1): s2h ¼

s2h 1 1 ¼ ¼ 2 2 2 2Sh þ 1 1  qh ðSh þ 1Þ  ðShÞ

(g) When the thickness tends to infinity, the reflectance tends to 1. (h) Since the layer is a Lambertian reflector, its reflectance and the reflectance factor measured using a spectrophotometer based on the di:8° geometry are equal (see exercise 10). *** Exercise 21. (a) The transfer matrices are, respectively, !
 ^1 q 1 1 1 1 R P¼ ; M¼ ^1 R ^2 ^1 T ^2  R s q s 2  q2 T 1 1

 1 rs0 1 ri 1 1 F1 ¼ F2 ¼ 0 0 Tin rs Tin Tout  rs ri Tout rs0 Tin0 Tout  rs0 ri0 and one can write: P ¼ F1 MF2 .

 1 (b) One first computes matrix M ¼ F1 1 PF2 ¼ mij , and then q ¼ m21 =m11 and s ¼ 1=m11 . (c) q þ s  1 means that the material is almost non-absorbing. (d) Since q þ s  1, one has a  1 and b  0. (e) Remember that when x is very small, the sinh and cosh functions can be approximated by the following Taylor expansions sinhðxÞ ¼ x þ Oðx 2 Þ and coshðxÞ ¼ 1 þ Oðx 2 Þ. Since the material is almost non-absorbing, the absorption coefficient K is almost 0. (f ) With a  1 and b  0, the value for bSh is very small. One can write sinhðbShÞ  bSh and coshðbShÞ  1, and therefore q

bSh Sh b 1 ¼ and s  ¼ : bSh þ b Sh þ 1 bSh þ b Sh þ 1

(g) Using Kubelka’s formula, and reminding that q þ s ¼ 1, one has q2 ¼ q þ

s2 q qð1  q2 Þ þ ð1  q2 Þq 2qð1  qÞ 2q ¼ ¼ ¼ 1  q2 1  q2 1  q2 1þq

Corrections

225

(h) One can either use Kubelka’s formula, by reminding that q þ s ¼ 1, s2 ¼

s2 ð1  qÞ2 1  q ¼ ¼ 1þq 1  q2 1  q2

or notice that is one film is non-absorbing, two films are also non-absorbing, therefore: s 2 ¼ 1  q2 ¼ 1 

2q 1q ¼ 1þq 1þq

(i) One uses Kubelka’s formula for one film on top of the two films, for transmittance: s3 ¼

ð1  qÞ 11q ss2 ð1  qÞ2 ð1  qÞ2 1q þq ¼ ¼ ¼ ¼ 2 1 þ 2q 1  qq2 1 þ q  2q ð1  qÞð1 þ 2qÞ 1  q 1 2q þq

Then, q3 ¼ 1  s3 ¼

1 þ 2q  ð1  qÞ 3q ¼ 1 þ 2q 1 þ 2q

(j) One can follow the same reasoning line as above, using Kubelka’s formula for one layer on top of k – 1 layers: sk ¼

ð1  qÞ 1 þ1q ssk1 ð1  qÞ2 1q ðk2Þq ¼ ¼ ¼ 1  qqk1 ð1  qÞð1 þ ðk  1ÞqÞ 1 þ ðk  1Þq 1  q 1 þðk1Þq ðk2Þq

Then qk ¼ 1  sk ¼

1 þ ðk  1Þq  ð1  qÞ kq ¼ 1 þ ðk  1Þq 1 þ ðk  1Þq

(k) By writing qk ¼

k ShShþ 1

1Þ ShShþ 1

¼

kSh kSh ¼ Sh þ 1 þ ðk  1ÞSh 1 þ kSh

1 þ ðk  Sh þ 1  Sh 1 ¼ sk ¼ Sh þ 1 þ ðk  1ÞSh 1 þ kSh

one retrieves the equations obtained in question a) where h is replaced with kh. (l) As the material is non-absorbing, an infinitely thick layer reflects all light ðlimk!1 qk ¼ 1Þ and is opaque ðlimk!1 sk ¼ 0Þ. (m) With a = 1 and b = 0, one indeed has q1 ¼ a  b ¼ 1. (n) Saunderson’s equation: ^ g ¼ rs þ R

Tin Tout qg 1  ri qg

Corrections

226

(o) The geometry used is the de:0° geometry. Therefore: rs ¼ 0 Z Tin ¼

p=2

½1  R12 ðhÞ sin 2hdh  0:9 "
 # 1  R12 ð0Þ 1 1n 2 Tout ¼ ¼ 2 1  0:42 n2 n 1þn Z p=2 R21 ðhÞ sin 2hdh  0:6 ri ¼ h¼0

h¼0

  ^ g = Tin Tout þ ri qg . (p) Since rs is 0, one has qg ¼ R (q) According to Kubelka’s formula, qgf ¼ q þ s2 qg =ð1  qqg Þ. (r) Since q ¼ sinhðbShÞ=D and s ¼ b=D, with D ¼ asinhðbShÞ þ bcoshðbShÞ, then   s2 qg q þ ðs2  q2 Þqg sinhðbShÞD þ b2  sinh2 ðbShÞ qg   qgf ¼ q þ ¼ ¼ 1  qqg 1  qqg D D  sinhðbShÞqg ¼

ða  qg Þsinh2 þ bcosh sinh þ b2 qg   ðasinh þ bcoshÞ ða  qg Þsinh þ bcosh

By noticing that cosh2 ðxÞ  sinh2 ðxÞ ¼ 1 and a 2  b2 ¼ 1, then   ð1  aqg Þsinh þ bqg cosh ðasinh þ bcoshÞ ¼ asinh2  a 2 qg sinh2 þ abqg cosh sinh þ bcosh sinh  abqg sinh cosh þ b2 qg cosh2   ¼ asinh2  qg sinh2 þ bcosh sinh þ b2 qg cosh2  sinh2 ¼ ða  qg Þsinh2 þ bcosh sinh þ b2 qg Moreover: qgf 

ð1  qg ÞSh þ qg ð1  qg ÞSh þ 1

*** Exercise 22. As indicated in table 6.1 the Saunderson parameters for the d:8° geometry are Tin ¼ t12 ¼ 1  r12 with r12 given by Duntley’s formula (4.8) as a function of the refractive index n of the support’s material, and Tout ¼ T12 ð8 Þ=n 2 . For the 8°:d geometry, Tin ¼ T12 ð8 Þ since light is incoming at 8° from the normal, and Tout ¼ t21 ¼ t12 =n 2 according to relation (4.12). Hence, one obtains in both cases Tin Tout ¼ t12 T12 ð8 Þ=n 2 . Term ri remains the same in all cases since it is independent of the measuring geometry (it only requires that the flux diffused by the material is Lambertian). Term rs is R12 ð8 Þ in the 8°:d geometry. It is also R12 ð8 Þ in the d:8° geometry is the specular reflection component is included, since only the

Corrections

227

light pencil located in the regular direction with respect to the detector can be specularly reflected towards the detector and captured. Finally, the di:8° and the d:8° geometries are equivalent. This is an example of situation where Helmholtz’ reverse path principle applies. *** Exercise 23. (a) The grey ink can be selected in order to approach the particular color of silver prints more closely. Moreover, for a middle grey level, a full coverage of the paper by the grey ink is visually better than a halftone screen of black ink dots: the visual texture is smoother, and closer to the continuous-tone original print. (b) There are four Neugebauer primaries: paper white, grey, black, and grey + black. (c) If the black is strongly absorbing, the black and grey + black primaries have the same black color; in this case, one has only three primaries: white, grey and black. (d) The surface coverages of the four primaries white, grey, black and grey + black are given by the Demichel equations: aw ¼ ð1  xÞð1  yÞ; ag ¼ xð1  yÞ; ak ¼ ð1  xÞy ; and ag þ k ¼ xy. But since the black and grey + black primaries are gathered into a single one, the surface coverages are aw ¼ ð1  xÞð1  yÞ; ag ¼ xð1  yÞ; and ak ¼ y. (e) The spectral reflectance of the solid black color patch is Rk ðkÞ ¼ 0 if the specular component is discarded from the measurement because all the incident light is absorbed, except the light that is reflected by the air-ink interface. If this specular component is included into the reflectance measurement, it is equal to the Fresnel reflectance of the ink-air interface for the considered incidence angle. At normal incidence, Fresnel reflectance r depends on refractive index n of the ink, which is probably a complex number since the ink is strongly absorbing: r ¼ jð1  nÞ=ð1 þ nÞj2 (the modulus operator is important in this case). (f ) According to the Yule–Nielsen model with nYN = 2, the spectral reflectance of the halftone color containing only black with a surface coverage a is h i2 1=2 2 Ry ðkÞ ¼ ð1  yÞR1=2 w ðkÞ þ yRk ðkÞ ¼ ð1  yÞ Rw ðkÞ. In the special case where y = 0.5, R0:5 ðkÞ ¼ 0:25Rw ðkÞ. (g) The surface coverage giving a reflectance equal to pffiffiffi 0:5Rw ðkÞ is a ¼ 1  2=2  0:3. (h) According to the Clapper–Yule model, since the intrinsic transmittance of the black ink is zero, and the one of the white primary is 1, the spectral reflectance of the same halftone color is Ry ðkÞ ¼

Tin Tout qðkÞð1  yÞ2 1  ri qðkÞð1  yÞ

where qðkÞ is written in terms of the spectral reflectance factor of the bare paper, Rw ðkÞ, as qðkÞ ¼

Rw ðkÞ Tin Tex þ ri Rw ðkÞ

and where Tin  0:90, Tout  0:96=2:25  0:43, and ri  0:60 when the refractive index of the support is 1.5, see §6.3.1. In the special case where y = 0.5, Ry ðkÞ ¼

0:25Tin Tout qðkÞ 0:25Tin Tout Rw ðkÞ ¼ 1  0:5ri qðkÞ Tin Tout þ 0:5ri Rw ðkÞ

Corrections

228

and in the general case, Ry ðkÞ ¼

Tin Tout Rw ðkÞð1  yÞ2 Tin Tout þ yri Rw ðkÞ

(i) It transpires from the above equation that the surface coverage giving a reflectance equal to 0:5Rw ðkÞ actually depends on the spectral reflectance factor of the bare paper: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ri Rw ðkÞ þ 4Tin Tout  ½ri Rw ðkÞ þ 4Tin Tout 2 8Tin2 Tout y¼ 4Tin Tout but for typical values of Rw ðkÞ between 0.7 and 0.9, y is very close to 0.2.

Acknowledgements

This work is the result of multiple interactions with many people I was fortunate to meet during two decades of scientific activities. First of all I am grateful for interactions with individuals I consider as my masters: Roger Hersch, who initiated me into the optical study of colored materials and accompanied me throughout the year 2000, Pierre Chavel who nurtured me continuously for twelve years with his knowledge and culture in optics and I owe a lot to him for his considerable support, Jean-Marie Becker has accompanied me for even much longer, and Jean-Louis Meyzonnette who has been a main actor in my mastering of radiometry and has kindly accepted to write the preface. I express my deepest gratitude to them for their communicative passion, for their generous and unfailing help, and also for their reviews at various stages of the elaboration of this book, from its genesis to its very last versions. I also thank Lionel Simonot for the connivance maintained every day for almost twenty years, as well as all colleagues and students with whom I have had the opportunity to work with at the Ecole Polytechnique Fédérale de Lausanne (EPFL), at the Institut d’Optique Graduate School, at the Université Jean Monnet de Saint-Etienne, or elsewhere. Many thanks, in particular, go to those who substantially contributed to the improvement of the content of this book either with their comments or their experimental accomplishments: Raphael Clerc, Thierry Lépine, Thierry Fournel, Gaël Obein, Patrick Emmel, Romain Rossier, Jacques Machizaud, David Nébouy, Renée Charrière, Serge Mazauric, Nicolas Dalloz, Lou Gevaux, Alice Dupiau, Thomas Labardens, Julien Eymard, Arthur Gautheron, Vincent Duveiller, Fanny Dailliez, and Simon Pinault. Finally, I thank my wife, Valerie; her vision of the world brings me so much, and with her the appearance of things rises to dimensions that science alone cannot encompass.

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Index

B BRDF, 33 I Integrating sphere, 22 Irradiance, 16 M Monochromatic, 2 O Optical thickness, 198 R Radiance detector, 43 Radiance, 17 Radiative transfer equation, 199 Reflectance, 37 S Shadowing, 191 Spectral power distribution, 4, 68, 69, 71, 72, 74, 79