Wave optics in Infrared Spectroscopy Thomas G. Mayerhöfer ([email protected]) About the script Is your main intention to interpret IR spectra to find out which functional groups your sample has? To interpret your spectra quantitatively employing the Beer-Lambert law? To understand orientation in your samples using Linear Dichroism Theory? Then this script is not the one you will be interested in – you will be better off by any conventional text book dealing with IR spectroscopy. Do you know such textbooks by heart and are you interested to learn much more than such texts can offer you? You mastered them with ease and you are longing for a real challenge? To learn wave optics and dispersion theory to enlarge your knowledge and, ultimately, to extend advanced theory on infrared spectroscopy and expand the possibility to interpret IR spectra to yet unknown degrees? Then you have the right spirit to digest what this text offers you and to profit from it by being inspired to cross borders and extend the limits further. Take on the challenge and leave your own footprint… “Und was wir an gültigen Sätzen gefunden, Dran bleibt aller irdische Wandel gebunden…“ Chor der Toten, Conrad Ferdinand Meyer (1825-1898) Version: 1.04.17 2 Preface Gallia est omnis divisa in partes tres, quarum unam… no, this is not intended to be a book about the Gallic Wars, but the same is true for the field of infrared spectroscopy: There are in my opinion three different communities who apply infrared spectroscopy, namely one which is mostly interested in organic or biological matter, a second which wants to learn more about new inorganic compounds and a third which deals with determining the kind of matter in space and on stellar objects. In contrast to the three parts of Gallia at Caesar’s time, there is very few exchange between these communities (divide et impera!). Strangely, contemporary textbooks about infrared spectroscopy mainly originated from the first community and those are not really suitable to fit the needs of the other communities. This was already the case when I started my PhD in 1996. Being a chemist by training, I was raised to believe in the Beer-Lambert law (a scientist should never believe in anything as long as she or he plies her/his art!) and knew perfectly well that transmittance is correspondingly only depending on the absorption coefficient (sic!). Since the topic of my thesis was to investigate oriented glass ceramics, I had to familiarize myself with reflection spectroscopy, since the samples were by far too thick to investigate them by transmission measurements. Unfortunately, it seemed that reflection was much more complicated compared to transmission and reflectance was not only depending on the index of absorption, but also on the index of refraction, which, I learnt to my amazement, are two sides of the same coin. Luckily, it appeared that the theory about how orientation influences infrared spectra is a very simple one, as it is based completely on absorbance, a quantity I was perfectly familiar with. So, I studied the textbooks and understood that everything just depends on the angle between the polarization direction and the transition moment. Meanwhile, most of my fellow PhD students worked on completely different topics, like determining the structure of thin films by infrared spectroscopy. Strangely, for these films it seemed that the Beer-Lambert law is not applicable and they used a program called “SCOUT” to determine the dielectric function of these layers. Since the dielectric function is nothing else but the complex index of refraction function squared, it appeared that transmittance for thin films also depended on both, the real and the imaginary part of the index of refraction. Furthermore, some of their films consisted of two phases that were intimately mixed on a scale much smaller than the wavelength. The spectra consisted therefore not of a simple linear mixture of the spectra of the two phases, but had to be analyzed assuming a construct called effective medium. Fascinatingly, a polycrystalline material also seems to be such an effective medium and I learnt that, e.g., for a material with orthorhombic structure somehow every one of its three principal dielectric functions has to be mixed to give the averaged dielectric function of the polycrystalline medium. Oddly, a simple mix consisting of an arithmetic average of the dielectric functions did not work at all in practice and I got a first impression that the theory about linear dichroism that is elaborated in many textbooks of infrared spectroscopy might not be so fundamentally correct as it seemed. Indeed, one result of my PhD thesis was that it is not only the polarization direction relative to the transition moment that is important, but also the orientation of the transition moment relative to the surface of the sample (this result, although largely ignored, was actually known to Paul Drude some 130 years ago when he did his PhD). Another one was that there is a kind of magic angle, but it is not the one from the textbooks and it is not that magic as it was believed. Trying to publish this result was like running against walls and resulted in an epic failure. Probably it was this failure that really brought me into science, and, in particular, to study wave optics. In the following years I learnt to know the two formalisms that allow, strictly based on Maxwell’s equations, to calculate transmittance and reflectance of anisotropic media (a big thanks to my colleague Georg Peiter and his supervisor Hartmut Hobert for pointing those out to me). Key to a full understanding of these formalisms was 3 trying to program them, because this uncovers any ambiguity as all variables must be explicitly introduced and connected, otherwise the formalism does not produce correct, if any, results. Over the years I used these formalisms to understand the infrared spectra of polycrystalline materials with large crystallites, and to determine the dielectric tensor of many monoclinic (many thanks to my friend Vladimir Ivanovski for getting me addicted to those!) and, finally, triclinic crystals. Based on the idea of my, at that time, PhD-Student Sonja Höfer, we finally managed to determine the dielectric tensor of a crystal without previous knowledge about its symmetry and orientation. Back then I began to look a little bit in greater detail on the work that was performed on organic and biological material. Could it be that Linear Dichroism Theory is not the only fallacy that can be found in the textbooks of infrared spectroscopy? To be fair, it must be stated that the mostly comparably low oscillator strengths of the vibrations in organic and biological matter quite often disguise that wave optics is at play. I also have to admit that I was myself convinced for nearly twenty years that you don’t need to understand wave optics to interpret an infrared spectrum of such matter. However, after checking most of the many different techniques in infrared spectroscopy, I found only two combinations of techniques and samples where wave optics seems to play no bigger role, which are transmission measurements of gases and of pellets (the latter only if a proper reference spectrum is taken). This was a real shock to me. How could it be that most of the textbooks in infrared spectroscopy are centered around a quantity like absorbance which is practically incompatible with Maxwell’s equations? Very revealing in this respect was studying very old literature. It was quite odd for me to discover that, e.g., Paul Drude and his theoretical understanding of the correspondence between optical and material properties was much higher developed than that of most of the spectroscopists nowadays. A further example was the first use of dispersion analysis (the determination of the optical constants from spectra) to uncover the optical constants of NaCl in the beginning of the thirties of the last century without any computer, performed by Marianus Czerny! Also, these pioneers of infrared spectroscopy were fully aware that the Beer- Lambert law is just an approximation and that wave optics must be invoked to understand the spectra of NaCl plates. Somehow this knowledge got lost over the following years, probably because the corresponding manuscripts were written in the German language and later on infrared spectroscopists were more influenced by the school of Coblentz ("Experimental observations always have some value. This is not always true of theories which are built, more or less, upon hypotheses and must stand or fall with them."). Nevertheless, with the advent of the attenuated total reflection technique and some other developments, many spectroscopists in the seventies of the last century realized how important an understanding in wave optics is to evaluate and understand infrared spectra quantitatively. Weirdly, at the beginning of the eighties this knowledge vanished again. I can only speculate why this happened and my guess is that with the advent of the Fourier-Transform technology in infrared spectroscopy any instrument had to have a computer anyway and spectra could be transformed to absorbance very simply. This led to the very odd concept in infrared spectroscopy where you quite often here that absorbance spectra have been recorded – certainly not! Relative transmittance or reflectance was recorded and then transformed into quantities that are better called “transmittance absorbance” and “reflectance absorbance”, reflecting that they are not true absorbances but apparent one. Nowadays you find in the textbooks of infrared spectroscopy a sometimes-weird mixture of Maxwell-compatible and incompatible concepts and it is even hard for me sometimes to differentiate one from the other. This is what brought me to thinking about a textbook that concentrates on the aspects of wave optics in infrared spectroscopy, this and strange concepts in modern literature like the “electric field standing wave effect” in transfection infrared spectroscopy, the discussion of potential errors in infrared spectroscopy (see e.g. 1, 2) without even mentioning the principal shortcomings of the quantity absorbance or the interesting attempts to remove interference fringes from absorbance spectra by baseline corrections without citing the well- 4 established dispersion analysis related methods or understanding that those can be removed by Maxwell-compatible methods. I think that concepts like the quantum mechanical foundation of infrared spectroscopy or group theory, instrumental aspects etc. are well introduced in other textbooks, therefore this book will concentrate on introducing wave optics and dispersion theory to the interested reader (the latter, because I learnt in the last years that Beer’s law can actually be derived from dispersion theory). It should be therefore used as a kind of add-on. This is also how I understand the lecture series that I provide about this topic for master of photonics students at the Friedrich-Schiller-University from which this book is derived from. I hope it reflects somehow the spirit of Paul Drude from whom it is said that he was originally sceptic about the, at his times newly introduced, Maxwell equations, but then obviously learnt to value those highly. To be more precise, my hope is not only that it reflects this spirit but is also able to induce the same enthusiasm in the readers of this book. I will not end this section without thanking those that helped me to realize this work. For the moment, the only person who I am highly indebted to and unable to express my gratitude in an appropriate way, is my colleague Susanne Pahlow, not only for proofreading, but also for multiple suggestions to improve this manuscript (all remaining errors are certainly my own!). 5 Content 0 Introduction Part I – what is wrong with absorbance ................................................................... 10 1 The Calculus ................................................................................................................................... 19 1.1 Maxwell’s relations ................................................................................................................ 19 1.2 Boundary Conditions ............................................................................................................. 20 1.3 Energy density and flux ......................................................................................................... 22 1.4 The wave equation ................................................................................................................ 23 1.5 Polarized waves ..................................................................................................................... 26 1.6 Further reading ...................................................................................................................... 27 2 Reflection and Transmission of plane waves ................................................................................ 28 2.1 Reflection and Transmission at an interface separating two scalar media under normal incidence ........................................................................................................................................... 28 2.2 Reflection and Transmission at an interface separating two scalar semi-infinite media under non-normal incidence ....................................................................................................................... 31 2.2.1 s-polarized light ................................................................................................................. 33 2.2.2 p-polarized light ................................................................................................................. 36 2.2.3 Calculation of reflectance and transmittance ................................................................... 38 2.2.4 Example: Dependence of the reflectance from the angle of incidence .............................. 39 2.3 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – absorbing media ............................................................................................................. 40 2.4 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – total/internal reflection ................................................................................................. 42 2.5 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – matrix formalism ............................................................................................................ 45 2.5.1 Matrix formulation for s-polarized waves at a single interface ........................................ 46 2.5.2 Matrix formulation for p-polarized waves at a single interface ........................................ 47 2.5.3 Combined matrix formulation for waves at a single interface .......................................... 48 2.5.4 A layer sandwiched by two semi-infinite media ................................................................ 49 2.5.5 Arbitrary number of layers ................................................................................................ 52 2.5.6 Calculating the electric field strengths of a layered medium – coherent layers ............... 54 2.5.7 Incoherent layers ............................................................................................................... 55 2.5.8 Mixed coherent and incoherent layers .............................................................................. 57 2.5.9 Calculating the electric field strengths of a layered medium – mixed coherent-incoherent multilayers ......................................................................................................................................... 58 2.6 Further reading ...................................................................................................................... 61 6 3 Dispersion relations ....................................................................................................................... 63 3.1 Dispersion relation – uncoupled oscillator model ................................................................ 65 3.2 Excursus: Lorentz profile vs. Lorentz oscillator ..................................................................... 78 3.3 Excursus: Dispersion relations and Beer’s law ...................................................................... 81 3.4 Dispersion relation – coupled oscillator model ..................................................................... 91 3.5 Dispersion relation – semiempirical 4-parameter models .................................................... 96 3.5.1 Berreman-Unterwald model .............................................................................................. 96 3.5.2 Classical model with frequency-dependent damping constant .......................................... 99 3.5.3 Classical model with complex oscillator strength ........................................................... 100 3.5.4 Convolution model........................................................................................................... 102 3.6 Dispersion relation – Inverse dielectric function model ..................................................... 103 3.7 Dispersion relation – Drude model ..................................................................................... 111 3.8 Dispersion relation – Kramers-Kronig relations and sum rules ........................................... 113 3.9 Further reading .................................................................................................................... 122 4 Deviations from the (Bouguer-)Beer-Lambert law ...................................................................... 124 4.1 Transmission of a slab embedded in vacuum/air ................................................................ 126 4.2 Transmission of a free-standing film embedded in vacuum/air ......................................... 130 4.3 Reflection of a layer on a highly reflecting substrate - transflection .................................. 136 4.4 Transmission of a layer on a transparent substrate ............................................................ 140 4.5 Attenuated total reflection ................................................................................................. 146 4.6 Mixing rules ......................................................................................................................... 151 4.7 How to correct the deviations and obtain a wave-optics conform solution....................... 154 4.8 Further reading .................................................................................................................... 161 5 Introduction Part II – what is wrong with Linear Dichroism Theory ........................................... 163 6 Reflection and Transmission of plane waves from and through anisotropic media – generalized 4×4 matrix formalism .......................................................................................................................... 175 6.1 Berreman’s formalism: Maxwell equations and constitutive relations .............................. 176 6.2 Berreman’s formalism: Calculation of the refractive indices and the polarization directions . ............................................................................................................................................. 178 6.3 Yeh’s formalism: Maxwell equations and constitutive relations ........................................ 181 6.4 Yeh’s formalism: Calculation of the refractive indices and the polarization directions ..... 182 6.5 The Transfer Matrix ............................................................................................................. 183 6.6 The treatment of singularities ............................................................................................. 185 6.6.1 Degenerate Eigenvalues .................................................................................................. 185 6.6.2 Singular form of the Dynamical Matrix ........................................................................... 187 6.7 The calculation of reflectance and transmittance coefficients ........................................... 187 7 6.8 Simplifications for special cases .......................................................................................... 189 6.8.1 Non-magnetic ( = 1), dielectric anisotropic ( = ) material and normal incidence .. 191 ij ji 6.8.2 Non-magnetic ( = 1), dielectric ( = ) monoclinic material – a-c-plane. .................. 192 ij ji 6.8.3 Non-magnetic ( = 1), dielectric uniaxial ( = ,  = ) material. ............................. 195 ij ji a b 6.8.4 Non-magnetic ( = 1), dielectric ( =  ) uniaxial or orthorhombic material with principal ij ji orientations. .................................................................................................................................... 198 6.9 Further reading .................................................................................................................... 199 7 Dispersion relations - anisotropic oscillator models ................................................................... 200 7.1 Cubic crystal system ............................................................................................................ 201 7.2 Optically uniaxial: The Tetragonal, Hexagonal and Trigonal crystal systems ...................... 202 7.3 Orthorhombic crystals ......................................................................................................... 204 7.4 Monoclinic crystals .............................................................................................................. 205 7.5 Triclinic crystals ................................................................................................................... 206 7.6 Generalized oscillator models ............................................................................................. 207 7.7 Further reading .................................................................................................................... 208 8 Dispersion analysis of anisotropic crystals - examples ................................................................ 209 8.1 Optically uniaxial crystals .................................................................................................... 209 8.2 Orthorhombic crystals ......................................................................................................... 212 8.3 Monoclinic crystals .............................................................................................................. 213 8.4 Excursus: Perpendicular modes .......................................................................................... 215 8.5 Triclinic crystals ................................................................................................................... 221 8.6 Generalized dispersion analysis .......................................................................................... 225 8.7 Further reading .................................................................................................................... 227 9 Polycrystalline materials.............................................................................................................. 228 9.1 Random orientation ............................................................................................................ 228 9.2 Large crystallites and non-random orientation ................................................................... 241 9.3 Further reading .................................................................................................................... 244 8 Part 1: Scalar Theory 9 0 Introduction Part I – what is wrong with absorbance What is wrong with absorbance? Since every textbook in infrared spectroscopy sees it as the fundamental quantity it must be important! On the other hand, absorbance is not even mentioned once in the most important textbook of optics (“Principles of Optics”, Born and Wolf3). Who is right? I gave (and still give) this much consideration, since I was once a Saul myself. At present it seems to me that too much importance is placed on absorbance mostly (but not only) due to a fundamental misunderstanding that arose from a misinterpretation in connection with Fermi’s Golden Rule (a strong hint for the correctness of this hypothesis can be found in the review article by Matossi4).1 Accordingly, the intensity I of a light beam is decreased proportionally to the length of its way l through an absorbing medium which is characterized by a Napierian absorption coefficient () (is the wavenumber, the inverse of the wavelength): dI =−()Idl (0.1) Actually, it is not the light beam intensity I that can originally be found in this equation. Initially it is the electric field intensity E2: dI =−()E2dl (0.2) If we now focus on the part of the intensity that is absorbed I relative to the initial intensity of the A light beam I , the equation reads: 0 dI I dA=()E2dl (0.3) A 0 Here it is, the absorbance! Actually not. A is not the absorbance, it is the absorptance which is defined by 1-R-T (R and T are the specular reflectance and the transmittance, and we assume here that no scattering takes places), or in words, the part of the intensity that is absorbed. Accordingly, absorption is proportional to the local electric field intensity. It is local, because it can change not only by absorption! Every interface changes it, as does interference! At this point, the only thinkable situation where the electric field intensity remains unchanged by such optical nuisances is when we have a strongly diluted gas (which is the case Bouguer and later Lambert were investigating). In this case the local electric field intensity can be replaced by the intensity of the light beam since the only process that changes this intensity is absorption. In this case (and only in this case!) eqn. (0.1) can be integrated to yield absorbance A (to distinguish it from absorptance the symbol is written in non- italic style): 1 In this article, which appeared in a time when there was only one IR community, the Beer-Lambert law was explicitly mentioned and only applied for gases. In the same explicit way, it was stated that for solid samples multiple reflections must be taken into account and that the minima in transmittance spectra are to be found where the product of refractive index and index of absorption function (i.e. the imaginary part of the dielectric function) has its maximum. But only for very thin freestanding layers! 10