organic or biological matter, a second which wants to learn more about new inorganic compounds .... Since every textbook in infrared spectroscopy sees it as the.
Wave optics in Infrared Spectroscopy Thomas G. Mayerhöfer
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 other planets. In contrast to the three parts of Gallia at Ceasar’s time, there is very few exchange among these communities. Strangely, textbooks about infrared spectroscopy only originate 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 and knew perfectly well that infrared transmittance is correspondingly only depending on the absorption coefficient. 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 to thick to investigate them by transmittance measurements. Unfortunately, it seemed that reflection was much more complicated compared to transmission and the 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 strange 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 seemed 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 strange 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 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. Another was that there is a magic angle, but it is not the one from the textbooks. 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 programming them, because this uncovers any ambiguity as all variables must be explicitly introduced and connected, otherwise the formalism does not produce correct results. Over the years I used this formalism 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. At that time 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 epic failure that can be found in the textbooks of infrared spectroscopy? To be fair it must be stated that the comparably low oscillator strength 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 techniques and samples where wave optics seems to play no bigger role, which is the 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 very 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 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! 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 (if someone can shed more light on this, please provide me the corresponding information and references!). 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. 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 strange 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 transflectance infrared spectroscopy, the discussion of potential errors in infrared spectroscopy (see e.g. “Using Fourier transform IR spectroscopy to analyze biological materials”, MJ Nasse et al., Nature methods 8 (5), 413) 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-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 to the interested reader. 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 which 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 spirit in the readers of this book.
0. Introduction 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 Wolf). 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 it is a fundamental misunderstanding that arose from a misinterpretation in connection with Fermi’s Golden Rule. Accordingly, the intensity I of a light beam is decreased proportionally to the length of its way d through an absorbing medium which is characterized by a naperian 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. Originally it is the electric field intensity E2: dI = − ( ) E 2 dl
(0.2)
If we now focus on the part of the intensity that is absorbed IA relative to the initial intensity of the light beam I0, the equation reads: dI A I 0 dA = ( ) E 2 dl
(0.3)
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 we don’t have scattering), or in words, the part of the intensity that is absorbed. Accordingly, the process 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 very diluted gas. 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): dI = − ( ) dl → I I − ln = ( ) d → I0 I ( ) A = − log = d = ( ) d I 0 log e
(0.4)
Note that in this case we need not to take into account the wave properties of light. In any other context this is quite paradox, because in quantum mechanics we allow matter to have wave properties to understand absorption, but the use of absorbance means that we deny light to also have wave properties… Still, absorbance can certainly be used. As we will see later, the absorption coefficient is proportional to the imaginary part of the complex index of refraction function. Quantitative evaluation of spectra therefore means to determine the optical constants of the material investigated. Once these are evaluated we can certainly calculate the absorbance. However, the way it is usual done in infrared spectroscopy is to set the negative decadic logarithm of the transmittance or the reflectance equal to absorbance. In many cases this is pure nonsense as we will see later on when we calculate the
transmittance and the reflectance based on Maxwell’s equations. At this point the usual argument that follows is, come on what about spectrophotometry? Indeed, it can be shown (and, to my best knowledge we were the first to do this in 2015), that under certain circumstances the Bouguer-BeerLambert law is indeed (nearly) compatible with Maxwell’s equations. However, this is just because the measurement is not performed like suggested by eqn. (0.4), because it is not the negative decadic logarithm of the transmittance of a solution that is used but the ratio of this transmittance and the transmittance of the pure solvent. Moreover, as we will also see, in infrared spectroscopy the same trick will not work. Overall, I think the use of absorbance as the main quantity in infrared spectroscopy as this became established since the 80ies of the last century is highly misleading and dangerous and has strongly hindered the development of the field since then.
