textbooks, therefore this book will concentrate on introducing wave optics to the interested reader. It should be ... Since every textbook in infrared spectroscopy sees it as the fundamental ..... μ the permeability of free space. 1.2 Boundary ...
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 too 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 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 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 2
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 fallacy 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 are 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 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 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 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 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 transflectance 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 wellestablished 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 from 3
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!).
4
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.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 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 I0 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
(0.4)
I A log10 log10 e d d I0
1
A strong hint for the correctness of this hypothesis can be found in the review article by Matossi 3. Matossi, F., Ergebnisse der Ultrarotforschung. In Ergebnisse der Exakten Naturwissenschaften: Siebzehnter Band, Hund, F., Ed. Springer Berlin Heidelberg: Berlin, Heidelberg, 1938; pp 108-163.. 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!
5
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 2016 4), 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. One example where the ignorance of wave optics has greatly impaired the quantitative evaluation of spectra in the last years is the so-called electric field standing wave (EFSW) effect, which is nothing else but an interference phenomenon. This phenomenon can be fully understood with help of eqn. (0.2) as we will see in the course of this book. A very closely related effect is the occurrence of interference fringes in the spectra of thin films. It is unsettling to see that the removal of such fringes is sold nowadays as a mere baseline correction which is performed after improper conversion of transmittance or reflectance to absorbance. A proper wave optics-based correction which is known since the beginning of the 70ies also removes the sometimes “dispersion artifact” called influence of reflectance on transmittance spectra. Reflectance is precisely not constant around bands due to dispersion and cannot simply be removed by subtracting a constant baseline from “absorbance spectra”. This is already an example of a problem that is not related to interference, accordingly it also persists for films of organic or biological matter on a nearly indexmatched substrate like e.g. CaF2 crystals. Funnily enough, it is such films on CaF2 that show up to 30 % deviation between apparent absorbance and true or corrected absorbance. Such deviations are flanked by changes of the wavenumber positions. Generally, the wavenumber positions of the maxima in absorbance never reflect the oscillator positions. The deviations may be small, but can be as large as 25 cm-1 in transflectance. Overall, for materials that can be characterized by a scalar dielectric function, neglecting wave optics related effects may be to some extent possible, but to know when, a profound knowledge of this topic is essential. This becomes even more relevant when anisotropy related effects come into play. Usually in this case in the textbooks of infrared spectroscopy a theory is invoked that has its origin sometime around the 50ies of the last century 5. At that time there existed no full understanding of the optical properties of anisotropic materials, therefore this (very) approximate theory was developed, which could be called linear dichroism theory. Unfortunately, it is fully based on absorbance and therefore inherited all aforementioned shortcomings right from the start. However, on top of that comes a number of additional weaknesses. The denial of the effects of interfaces (again a reference to the assumption of a strongly diluted gas…) leads to the erroneous conclusion that it is only the angle of the transition moment with the polarization direction that dictates the intensity of a band in a spectrum. At interfaces, however, only the tangential component of the electric field is continuous and this results in band shifts, if the transition moment is not oriented 6
parallel to the surface, like the, for inorganic materials and layers, well-known Berreman effect which shares the reason for its existence with the so-called transversal-optical longitudinal-optical shift. These band shifts are also accompanied by changes of the band shapes. Those shifts are dependent from the oscillator strengths of the vibrations and may not be recognizable for weaker transitions in organic and biological material. This is not true, however, for the intensities. Depending on the angle between transition moment and interface, absorbance can easily be up to a 100 % stronger for the same angle between transition moment and polarization direction, if it is not parallel to the interface. While correction schemes for anisotropic materials may be extremely effortful (contra-intuitively, the most complicated situation is a randomly oriented polycrystalline material where the crystallites are not small compared to the resolution limit of light!), the spectra of homogenous and isotropic layers on substrates can, e.g. extremely fast and in an automatic manner be corrected, if we apply a little wave optics. So why apply sometimes extremely obscure chemometrics on apparent absorbance spectra, when a little bit of insight in wave optics can accomplish better results in shorter times? Before we go in medias res I do not want to conclude this section without mentioning two further problems that we will solve in the course of this book. First of all we do not stop – as books usually do – at materials with orthorhombic symmetry, i.e. we do not assume that the off-diagonal terms of the dielectric tensor function are zero. This means that monoclinic and triclinic crystals will be properly treated, so that their spectra can be understood and quantitatively evaluated. Furthermore, and this is something nearly nobody in the community is aware of, I will demonstrate that isotropy is just necessary, but not sufficient, for reducing a dielectric tensor to a scalar. In fact, as a consequence, at least three different types of optical isotropy exist. While this seems on the first view to be a philosophical insight with little practical relevance I think, it is everything else, because usually textbooks not only refrain from treating materials with lower symmetry than orthorhombic, but quite often stick to materials that can be characterized by a scalar dielectric function. Even then it is first important to understand the more general principle before the simpler one can be appropriately appreciated…
7
1 The Calculus 1.1 Maxwell’s relations After having studied the optics of anisotropic materials for quite some time, I once participated in the 8th International Conference of Advanced Vibrational Spectroscopy which took place in Vienna and listened to a talk about an improved method to take into account orientation in Raman spectroscopy. If I remember correctly, the talk also featured some infrared spectra for comparison purposes and after the talk I began a discussion with the PhD student who had given the talk. I must have been a pain in the neck with my persistence and with me trying to evoke Maxwell’s equations in the course of the discussion. Finally, the PhD-student somehow gave up and exclaimed something like “please consider, I am just a chemist!” I answered truthfully: “Yeah, me too…” This little anecdote just shall show you that I feel myself still not too comfortable face to face to Maxwell’s equations, in particular with the magnetic field H and the magnetic induction B for which I could still not develop a feeling (NMR spectroscopy was once interesting for me, but these times are long gone). Luckily, we will soon be able to get rid of both quantities and will be able to focus on the electric field E and the electric displacement D when we deduce the existence of propagating waves from Maxwell’s equations. We present them here in differential form, and since we will assume in the following that matter will be uncharged (volume charge density = 0) and that no net currents will flow in it (current density J = 0), we can write them here as: D 0 B 0 t B 0 E
H
(1.1)
D 0 t
Since as, at least I hope so, fellow chemists are among the readers, let me be a little bit more explicit about what we are seeing here. Actually all fields are written as vectorial quantities (therefore in bold), not least because we will also treat anisotropic materials. Overall, we see eight equations here, two that state that the divergence of the electric displacement and of the magnetic induction is zero,
W
Wx Wy Wz 0 x y z
W D, B ,
(1.2)
which means that we assume that there are no sources or sinks, where the field lines of D and B start or end. Furthermore each three more which state that from every temporal change of D and B a magnetic or electric field will be induced (or its shape will be altered if there was already one before):
8
U z U y y z x U x U x U z U U y y z x U z U U x y y z x
Vx t Vy t Vz t
.
U E, H;
V B, D
(1.3)
A very important point to consider is that we have overall eight equations, but we have twelve unknowns (three components per field). Accordingly, we cannot solve the equations unambiguously, if we do not evoke further relations. These further relations manifest themselves in the form of the socalled material equations, which are at the same time the defining equations for D and B: D 0E P H
B
0
M
.
(1.4)
In eqn. (1.4), P is the macroscopic polarization which incorporates the reaction of matter to the external field E and is, since we also want to allow anisotropic media, not necessarily co-linear to the generating electric field. For the magnetization M we will assume that it is zero, which is the case for all materials that we will encounter in this book (but, e.g., not for metamaterials etc.). 0 is the permittivity and 0 the permeability of free space.
1.2 Boundary Conditions The boundary conditions seemed to me, when I encountered them first, as not very important. How wrong I was! Actually, these conditions always come into my mind first, when I think about the failure of non-wave optics based infrared spectroscopy. As a chemist, it is for me the fundamental difference between a gas, which takes on every volume that is offered to it and liquids and solids, both of which have a very well-defined volume and, by that, also a surface. These two terms, volume and surface, are also the key terms to understand the derivation of the first boundary condition. As Klaus Barbey (my, in the meantime, retired mathematics lecturer) used to say, for us chemists the Gauss divergence theorem is easily to proof as it just states that everything that goes into a volume or out of it has to pass the surface that encloses it. Somewhat more scientifically formulated, if we have a vectorial property F, then the relation between the following volume integral and the surface integral is given by:
FdV FdS .
(1.5)
Its value becomes immediately clear, if we assume that the volume, we are interested in, is separated by an interface like in Scheme 1-1. We now reduce the height of the cylinder in a way that the interface is still contained in the cylinder until it reaches zero. The surface of this cylinder consists then only of the two circles normal to the interface. Accordingly, eqn. (1.5) is reduced to,
n F1 F2 0 , 9
(1.6)
Scheme 1-1. Scheme to illustrate a volume (left) which is intersected by an interface and the same volume after its height is reduced to zero.
since the volume becomes zero in the limit of zero height. If we now replace the left part of eqn. (1.5) by the first and the third Maxwell equation, the result is: n B1 B 2 0 M1 M 2 0 B1, B2,
H1, H 2,
.
(1.7)
n D1 D2 0 D1, D2,
1, E1, 2, E2, This means that the normal components of the B and the D field are continuous. Since we assumed that the magnetization is zero, also the normal components of the H field are continuous. This is different for the electric field. Since the polarization is non-zero, the electric field is not continuous. We obtain the relation between the electric field before (1) and after (2) the interface by the following equation, D 0 E P 0ε r E ,
(1.8)
ε
wherein ε and ε r are the dielectric tensor and the relative dielectric tensor, which are functions of frequency/wavelength/wavenumber (in eqn. (1.7) we have tacitly assumed that it can be diagonalized and has its principal component normal to the interface designated with the symbol ). What happens with the tangential components of the fields? We can derive a further boundary condition with help of Stoke’s theorem:
FdA F dl .
(1.9)
Again, when we do not try to find a mathematically strict proof, the statement of Stoke’s theorem is immediately clear, in particular, if we let the height of the rectangle approach zero as it is illustrated in Scheme 1-2: Obviously the tangential components of the vector field must be equal to compensate each other to zero:
n F1 F2 0 ,
(1.10)
Therefore, if we insert the second or the fourth of Maxwell’s equations into (1.10), we obtain the following boundary conditions:
10
E1,t E2,t H1,t H 2,t
.
(1.11)
Scheme 1-2. Scheme to illustrate an area (left) which is intersected by an interface and the same area after its height is reduced to zero.
These boundary conditions are not only pivotal for the calculation of reflectance and transmittance, but also immediately bring home the message that it is not only the orientation relative to the polarization direction that counts, but also the one relative to the interface.
1.3 Energy density and flux Originally, I was a little hesitant to include this section, in particular the part about the energy density, as it is not really needed for our purposes. On the other hand we need the energy flux, respectively Poynting’s vector, since it is needed to calculate transmittance into a second medium, if the angle of incidence is different from zero. On the other hand, for spectroscopists who want to measure transmittance the latter point is of no practical relevance, since the detector is usually located in the same medium as the source, so that it would be no practical problem to ignore the fact that the squared amplitude of the electric field in the last medium is not always the same as the energy flux inside the medium. On the other hand, even if I want to keep wave optics to the minimum, necessary to understand how strongly infrared spectra are affected by it, we are scientists after all, and the appearance of an otherwise unmotivated factor in the calculations of transmittance should somehow unsettle us. “Unfortunately”, the work performed by an electric field, which equals the energy dissipation, is the dot-product between charge density J and the electric Field E: J ∙ E. “Unfortunately”, because I withhold the charge density in Maxwell’s forth equation (1.1) which D actually reads: (1.12) H J. t For the moment, we use it in this form and replace the charge density by the left part to yield: D . (1.13) J E E H E t It is certainly not immediately clear how this could help. The same is true for the next step where we invoke a purely mathematical vector identity: D E H H E E H t . D J E E H H E E t J E E H E
11
(1.14)
Now it looks even more complex than before! But at this point we use Maxwell’s second equation to gain two terms (actually three, but the last two belong together): B D . E t t Energy flow Decrease of energy density
J E E H
H
(1.15)
If we assume that our medium is linear (not to high electric field strengths), we can write: J E E H U J E S t
12 E D H B t .
(1.16)
In eqn. (1.16), S is Poynting’s vector which gives the energy flux. Its amount is the power per unit area transported in the direction of the vector, and U is the energy density. Both quantities are usually formulated slightly differently: S 12 Re E H* U 14 Re E D* H B*
.
(1.17)
In eqn. (1.17), the asterisk denotes the complex conjugate. The latter representation is in particular important, when the complex number representation is used for the electric field.
1.4 The wave equation For all problems that we are going to tackle in the framework of this book we will assume that the light beams we are dealing with can be represented by plane waves. In the following we will see how Maxwell’s equations automatically lead to the existence of such waves. We start with the second and the fourth of Maxwell’s equations: B 0 t D H 0 t
I
E
II
.
(1.18)
Since we assumed that magnetization is zero, we can set B = H in (I) and apply the curl-operator: E
H 0 . t
(1.19)
We now take (II) and differentiate it with respect to time: 2εE H 2 0 . t t
(1.20)
Additionally, we have used the relation D εE . If we combine (1.19) and (1.20), the result is: E ε 2 E
12
2E 0. t 2
(1.21)
Solutions to this equation, which is called the wave equation, are given by E r, t E0 sin t k r and Er, t E0 cos t k r . Here, is the angular frequency given by 2 f , where f is the frequency and k is the wave vector that whose properties we will investigate further down. If we keep in mind, that electric fields are always real, we can, for convenience, also construct a solution to the wave equation from a linear combination according to: E r, t E0 cos t k r i E0 sin t k r E0 exp i t k r .
(1.22)
However, since we multiplied the second solution with the imaginary number i, only the real part has, strictly speaking, any physical relevance, so that eqn. (1.22) is the lazy variant of eqn. (1.23):
E r, t Re E0 exp i t k r .
(1.23)
When I first encountered eqn. (1.23), I had a very hard time to imagine how such a plane wave would look like (even when I additionally used “Optics” from Eugene Hecht, which really gives very basic and illustrative examples for the changes of the wave with time and position6). To make things a little bit simpler, let’s investigate how the polarization direction of E0 (the amplitude) and the change of position of locations of the same amplitude r(t) are connected. To that end let’s use that D 0 :
Dr, t ik Dr, t 0 k Dr, t 0 .
(1.24)
As a consequence, the direction of D r,t is perpendicular to the direction of the wave vector which is the direction of propagation. In general, this is not valid for E r,t , since for an anisotropic material the direction of D r,t and E r,t do not coincide! However, for the first part of the book we restrict ourselves to the case where the dielectric tensor is a scalar. In this case, in order to find out the relative orientation between the electric field and the magnetic field in the plane wave model, let’s investigate the rotation of E r,t : Ez E y z y Ez k y E y k z Ex Ez E E0 exp i t k r E i E x k z E z k x i k E . x z E y k x Ex k y E y E y y x
(1.25)
In eqs. (1.24) and (1.25), we see one big advantage of writing a plane wave in the exponential form, since we can easily evaluate derivatives and are able to replace calculating the derivative by multiplying it with either the negative wave vector (if it is a derivative with respect to location) or the angular frequency (if the derivation is with respect to time) multiplied with the imaginary number. If we use this in the second equation of (1.1), B M 0 0 . t ik E i H 0 k E H
E
(1.26)
Therefore the magnetic field is perpendicular to both E and k (for a material were the magnetization is non-zero and not co-oriented to the magnetic field, it is B that is perpendicular to the direction of propagation). 13
It may easier to grasp (1.26), if we simplify things a little bit. Throughout the book we will assume that the different media are stacked and we will assume the stack direction to be along the z-coordinate. Furthermore, since we are dealing with the idealization of plane waves, we introduce a further idealization and assume that our media have the same properties perpendicular to the stacking direction, i.e. along the x-y-plane (note that not only the results justify these assumptions, but also that the dimensions of our samples in the x-y-plane are very large compared to the wavelength). While our plane waves not necessarily travel along the z-direction (we will certainly allow non-normal incidence later-on), we will do so to illustrate eqn. (1.24): 0 E0, x 0 E r, t 0 E0 E0, y . k 0 z
(1.27)
We even simplify (1.27) further by assuming that E0, y 0 (as we will see down below this means that we assume that our plane wave does not only travel along z, but is additionally x-polarized. Therefore: Ez E y z y 0 0 Ex Ez B E E0 exp i t kz z E E i k z Ex i k z E0, x exp i t k z z t x z 0 0 E y Ex y x 0 k z E0, x exp i t k z z H 0, y exp i t k z z 0
.
(1.28)
Therefore, the magnetic field would be y-polarized. It may seem to some readers that I am too explicit at this point. Since we need these calculations again when we derive formulas for the reflectance and transmittance for layer stacks, I’ll take the risk of being too trivial to make things easier when we have to concentrate on other problems. To summarize this section, we can now formulate Maxwell’s equations in an equivalent form, but thanks to the plane wave assumption fully based on simple vector calculations: k D 0 k E B 0 k B 0
.
(1.29)
k H D 0
Furthermore, we can provide a simpler form of the wave equation:
k k E 2εE 0 .
(1.30)
And, last, but not least, Poynting’s vector can be simplified into the following form: k E H S 12 Re E H* S
14
Ek E k 2 E0 . 2 2
(1.31)
1.5 Polarized waves We have already worked with polarized waves in the last section without explicitly mentioning that. Since we plan to focus on anisotropic materials (to be more precise, only with materials of dielectric anisotropy), to develop a feeling for polarization is very important. But not only that: The calculations of reflectance and transmittance can become very complex for general anisotropy and non-normal incidence, therefore it is of advantage to separate two different cases where the light is polarized either perpendicular to the plane of incidence or parallel to the plane of incidence. In fact, the separation of these cases generally makes sense and simplifies the math. The reason for this simplification is based on the fact that perpendicular polarized light only has a component tangential to the interface between incidence medium and sample whereas parallel polarized light consists of a tangential as well as a normal component. In addition to so-called linear polarized light we also have to introduce elliptically polarized light, since media with dielectric anisotropy generally convert linear polarized light into elliptically polarized light. For the incidence medium we will generally assume that this medium can be described by a scalar dielectric function (like vacuum, air or materials that are used as crystals for attenuated total reflection like Ge, Si or ZnSe). Accordingly, E and H and k are mutually perpendicular in such a material and it is sufficient to focus on the direction of E to describe the polarization. For linear polarized light, the direction of E is perpendicular to k and confined to a line. In contrast, elliptically polarized light can be seen to consist of a superposition of two linear polarized waves along two mutually perpendicular axes that have a certain phase difference. In fact, this is exactly into what linear polarized light is transformed when it transmits through an anisotropic material. Assume a plane wave propagating in the z-direction: E z , t Re E0 exp i t kz .
(1.32)
The said superposition of two plane waves travelling along z and being polarized along x and y can then be formulated as: Ex z, t E0, x cos t kz
E y z, t E0, y cos t kz
.
(1.33)
E0 E0, x E0, y exp i y
The ellipsoidal nature of the polarization is easily verified. To do that, we normalize the x-polarized wave and square it: Ex E0, x cos t kz
Ex cos cos t kz cos E0, x
E x E 0, x
.
(1.34)
2
2 2 cos t kz 1 sin t kz
Furthermore, we also normalize the y-polarized wave: E y E0, y cos t kz Ey E0, y
cos t kz cos t kz cos sin t kz sin
15
.
(1.35)
We then subtract (1.34) from (1.35) and obtain an ellipse with semiaxes of length E0,x and E0, y which are at an angle relative to the axes of the coordinate system (see also Scheme 1-3): Ey E0, y
Ex
2
1 sin t kz E0 , x E x cos sin t kz sin E0, x 2
Ey E E x cos 1 x E 0, x E0, y E0, x Ex E0, x
2
Ey E0, y
2
2 sin
.
(1.36)
2
2cos Ex E y sin 2 E0, x E0, y
Scheme 1-3. Polarization ellipse
1.6 Further readings Here, I suggest in particular the book of Pocchi Yeh “Optical waves in layered media”, which adds many aspects that I thought less important for the course of the book (but I might be in error, and knowing more is always a good strategy!).7 Furthermore, I suggest the Born and Wolf “Principles of Optics”, which might be no surprise at all.8 Also, at somewhat more introductory level, Hecht’s “Optics” is a good supplement of the other two books, but certainly also an excellent read on its own.6
16
2 Reflection and Transmission of plane waves 2.1 Reflection and Transmission at an interface separating two scalar media under normal incidence We start with the simplest case which is given by a plane wave hitting interface that separates two semi-infinite scalar media under normal incidence. By a scalar medium we understand a medium that is homogenous and can be characterized wavenumber-independent (or, at least, in the interesting spectral range) by a scalar dielectric function. Of particular interest for us is when the incidence medium (the medium where the plane wave has its origin) consists of vacuum or air. The so-called exit medium would then be the sample. Why should these media be “semi-infinite”? Actually, we are simplifying things, if we assume that both media are semi-infinite, because this means that the part of the wave that is reflected at the interface and is travelling backwards in the incidence medium will never hit another interface. Therefore, an again-reflected wave that is travelling towards the interface will never exist. Therefore we do not have to take care of such multiple reflections and the consequences that result from a superposition of the waves. Actually, this is not true, since the original wave and the part that is reflected will be superposed. Since, as we will see, the phases of the original wave and its reflected self have the same phase at the interface and they are travelling in the same medium, their phases will have a fixed relationship over the whole incidence medium. Accordingly, they are said to be coherent. For the exit medium which is also assumed to be semi-infinite, the transmitted part of the wave is assumed to travel forward until the end of the universe. Because in this medium therefore only a forward travelling wave exists, a superposition will not take place. In practice, a sample will certainly never be semi-infinite. However, since we want to do spectroscopy, it most probably has absorption bands. Even somewhat away from the maximum of such a band, absorption is still not zero, so that a finite thickness will be sufficient to lead to semi-infiniteness. Absorption needs to be just high enough, that light from the backside of the sample does not reach the first interface again. If this happens, like e.g. for inorganic glass samples some 1000 cm-1 away from the highestwavenumber absorption, then a step of the reflectance can be seen, which is an indication that the model would have to be changed to a slightly more complex one. For the moment, however, we will assume that our media are non-absorbing and really semi-infinite. As a last assumption we will suppose that the interface is plan-parallel and smooth. We will in the following focus on the quantity I which is called the spectral irradiance (or, for a spectroscopist, the intensity), which is the irradiance of a surface per unit frequency, wavelength or wavenumber. In particular, irradiance is the radiant flux (power) that is received/reflected or transmitted by a surface per unit area. We will call: I0 the received radiant flux (r.f.) IR the reflected r.f. IT the transmitted r.f. Furthermore, we will define the reflectance R as the ratio of the reflected and the received r.f., R
IR , I0
and the transmittance T as the ratio of the transmitted and received r.f., 17
(2.1)
T
IT , I0
(2.2)
How do we get the reflected and the irradiate irradiances in our hands? This is what we introduced Poynting’s vector for! Remember the definition,
Sj
k Ej 2
2
j i, t , r ,
(2.3)
where i,t,r stands for “incoming”, “transmitted” and “reflected”, respectively. Eqn. (2.3) lets us calculate the flux when we know the corresponding electric field strengths. Therefore, what we need to do is to calculate the electric fields at the interface keeping in mind the continuity conditions that we derived in the last chapter. Scheme 2-1 illustrates the situation.
Scheme 2-1. A plane wave travelling through two semi-infinite media normal to their interface
The fluxes have the same direction as the waves. From Maxwell’s equations we know, as discussed in the last chapter, that the direction of the E field is tangential to the direction of propagation and, therefore, tangential to the interface. The recipe to calculate the reflectance and transmittance includes 3 steps: 1) Obtain two eqs. from the continuity of the tangential electric and magnetic fields. Convert H to E using Maxwell equations 2) Use the two eqs. to express a) Et in terms of Ei and Er to obtain the ratio between Er and Ei b) Er in terms of Ei and Et to obtain the ratio between Et and Ei 3) Calculate flux S in direction of z to obtain IR and IT. From that calculate R and T. Since these steps are the basic building blocks for the calculation of R and T up to layered media of arbitrary dielectric anisotropy (ok, admitted, it will become a tiny bit more complicated than promised here, but whoever keeps promises?), therefore I will guide you step by step through it. 1) Obtain two eqs. from the continuity of the tangential electric and magnetic fields. Convert H to E using Maxwell equations The first continuity relation tells us, that the tangential components of the electric fields are continuous. Accordingly: E1,tan E2,tan Ei E r Et
18
,
(2.4)
On the left side, i.e. in medium 1, we have a superposition of the incoming and the reflected field. This superposition must be equal to the transmitted electric field in medium 2. The form of eqn. (2.4) is of course of particular simple form, because there are only tangential components of the electric field. The same, however, is also true for the magnetic fields at the interface: k E H 0 H1,tan H 2,tan
k i Ei k r E r k t Et 0 0 0 k , 0 Ei 0 E r 0 Et k k k i,Z r ,Z t ,Z
(2.5)
1 Ei 1 E r 2 Et Ei E r
2 1
Et
The first transformation of (2.5) uses the second eqn. of (1.29) (noting that H = B since = 1). For the second we note, that the wave vector has only one component in the z-direction. For the third we have to keep in mind that the direction of the reflected wave is reversed (and therefore the sign changes). Now we have two relations between the incident, the transmitted and the reflected electric field:
I
E i E r Et 2 1
Ei E r
II
Et
,
(2.6)
2) Use the two eqs. to express a) Et in terms of Ei and Er to obtain the ratio between Er and Ei To that goal, we replace Et in the second equation in (2.6) by the left side of the first equation: 2 1
Ei E r Ei
2 1
Ei
Ei E r 2 1
Er Er
1 E 1 E , 2 1
2 1
i
E 1 r r Ei 1
r
(2.7)
2 1 2 1
By that we obtain the ratio between the reflected and the incoming electric field which is called the reflection coefficient r. b) Er in terms of Ei and Et to obtain the ratio between Et and Ei From the first equation of (2.6) we note that Er Et Ei . We insert this result into the second equation and obtain the transmission coefficient t:
19
Ei Et Ei 2E i E t 2E i
2 1
Et
Et
Et Et
2E i 1 t
2 1
2 1
E
2 1
Et 2 Ei 1
,
(2.8)
t
2 1
Now we are ready for the final step: 3) Calculate flux S in direction of z to obtain IR and IT. From that calculate R and T. Of general relevance is only the energy flow perpendicular to, i.e. through, the interface. This is a in this case trivial condition: 2 1
E 1 r r Ei 1 t
2 1
Et 2 Ei 1
2 1
kj
R
, 2
S r S j 2 E j Si k
2
j Sj Ej S 2 T t Si
R
k z ,r Er k z ,i Ei
k z , t Et T k z ,i Ei
(2.9)
2
r
2
2
2 1
t 2
n2 n1
t
2
Here, n is the index of refraction of the medium i. If the j are real numbers, then R +T = 1, which can easily be verified using the results of eqn. (2.9). Both quantities add up to unity of course not by accident. It is simply the law of energy conservation, since what is not reflected, must be transmitted (so far we have absorption excluded from the discussion by assuming that our dielectric function or index of refraction function is real).
20
2.2 Reflection and Transmission at an interface separating two scalar semiinfinite media under non-normal incidence Assuming normal incidence, as in the preceding section, polarization did not play a role, since the media were not anisotropic. In case of light that is non-normally incident, it makes sense to separate two particular cases, even for scalar media, which differ with regard to polarization. To understand the difference, we first have to define the so-called plane of incidence. The definition is, of course, absolutely straightforward, as the plane of incidence is defined on the one hand by the direction of the incoming plane wave, and, on the other hand, by the direction of the reflected wave (for normal incidence, both are certainly co-linear, and a plane of incidence cannot be defined). The polarization directions which we will now define, are s-polarized (“s” stands for the German “senkrecht”, which means “perpendicular”) and p-polarized (you might have guessed it already, “p” is short for parallel, which works in more than one language). Sometimes, you might also experience an alternative nomenclature, which I do not want to hide from you, even if it is much less common. In this nomenclature, s-polarized is called “transverse electric”, or, short, TE. This certainly means in both cases the same, namely that the electric field vector is perpendicular to the plane of incidence. Somewhat in contrast, but at the same time along with TE, the opposite is called TM, which stands for “transverse magnetic” and is in this case obviously referring to the direction of the magnetic field vector. As long as we are talking about scalar media, which are homogenous in the sense that the inhomogeneities are smaller than the resolution limit, both definitions obviously are fully synonymous. In any way, the big difference between normal and non-normal incidence is that p-polarized light has not only a component of the electric field tangential to the interface, but also a normal one and the latter increases with the angle of incidence (which is the angle between the normal to the interface and the direction of the incoming light). I have already mentioned that this normal component makes a big difference for the spectroscopist, since the larger this component, the more band shapes and peak positions will deviate from the those without this component. Let’s investigate the situation for non-normal incidence in some more detail (cf. Scheme 2-2). We assume that Z is the direction perpendicular to the interface (note that we are using capital letters for the Cartesian coordinates, since later-on, when we are discussing anisotropic materials, we will need small letters to denote the axes of coordinate systems fixed inside the material. Accordingly, capital letters belong to laboratory coordinate systems throughout the book). The incoming, reflected and transmitted wave vectors all lie in the plane of incidence, which we denote as the Y-Z plane (without loss of generality – as is usually stated at this point. Actually, this is not completely true, but this is not the right location within the book to discuss this point). Therefore, for s-polarized or TE light, the polarization direction is parallel to the X-axis. For p-polarized or TM light, the polarization direction is somewhere between the Z- and the Y-axis. Using the angle of incidence i, we can state that the component along Z is proportional to sin i.
Scheme 2-2. A plane wave travelling through two semi-infinite media non-normal to their interface
21
How can we determine the angle of reflection r and the angle of refraction t? Actually, there is even more to determine! If we take again a look at our plane wave, E r, t Re E0 exp t k r , and assume that at t = 0 s it hits the interface (Z = 0), then the continuity relations require that the phases are spatially equal:
k i r Z 0 k r r Z 0 k t r Z 0 ,
(2.10)
From (2.10) we can deduce that the following relations must hold (remember, kX = 0!): Yki ,Y Ykr ,Y Ykt ,Y ki ,Y kr ,Y kt ,Y , ki ,Y n1 sin i ,
,
(2.11)
kr ,Y n1 sin r , kt ,Y n2 sin t .
From that, we can deduce Fresnel’s law: ki ,Y kr ,Y kt ,Y , n1 sin i n1 sin r n2 sin t
.
(2.12)
This immediately results in the angle of incidence and the angle of reflectance being equal. Furthermore, if we assume that our incidence medium has a lower index of refraction that the exit medium, then n arcsin 1 sin i t i t . n2
(2.13)
Note that in order to match the Y-components of the wave vector at the interface, it is automatically required that the Z-components in the different media must differ. To summarize, all wave vectors, k i , k r , k t must lie in a plane, the plane of incidence. Furthermore, the tangential components kY must be equal. Our recipe to calculate reflectance and transmittance contains again three steps: 1) Calculate H from E using Maxwell’s equations 2) Obtain two equations from the continuity of the tangential components of the electric and magnetic fields. Calculate from these rs,p and ts,p 3) Calculate the flux S in direction of Z to obtain IR and IT. From that calculate Rs,p and Ts,p. Our plane wave can be written in the following way:
E j E0, j exp i t k i r E0, j exp i t YkY , j E j E0, j exp i
.
nj Zk Z , j E0, j exp i t Y sin j Z cos j c
j i, r , t
(2.14)
22
Here, i,r,t stand for the incoming, the reflected and the transmitted wave. The Maxwell equation that we need for the first step is again k i E 0 H (eqn. (1.26)).
2.2.1 s-polarized light To follow the above recipe we first have to calculate H from E. Remember, for s-polarization the electric field is polarized parallel to the X-axis. According to Scheme 2-3, which illustrates the situation, the polarization direction as indicated is actually anti-parallel to X (assuming a right-handed coordinate system, X points into the book (screen) plane whereas the electric field vector points toward you, the reader. Accordingly, the incident electric field can be written as:
Ei , X Ei expi .
(2.15)
Applying eqn. (1.26) leads to: k i E 0 H H k E 0 0 k i E E X k Z ,i , E k X Y ,i
0 k i 1 sin i c cos i
. H i ,Y 1 H i , Z 1
1
0 c 1
0 c
(2.16)
cos i Ei exp i
sin i Ei exp i
Accordingly, the incident magnetic field has two components, one that is antiparallel to the Y-axis and a second component in the Z-direction. Since the electric field is continuous at the interface, it keeps its direction upon reflection and transmission. Therefore, a change of direction of the magnetic field would be a sole consequence of a directional change of the wave vector. Since the Z-component of the wave changes upon reflection, the wave vector of the reflected wave indeed needs to be antiparallel to the Z-axis. Consequently, the Y-component of the magnetic field of the reflected wave H r ,Y is positive: H r ,Y 1
1
0 c
cosi Ei exp i .
Scheme 2-3. A s-polarized plane wave impinging non-normally on an interface of two semi-infinite and scalar media
23
(2.17)
The Z-component H r , Z does not change compared to H i , Z (eqn. (2.16)). For the transmitted waves the directions of both components of H ||t do not change. However, even when the signs do not change, since the value of the index of refraction is altered in the second medium, so are the values of the components of the magnetic field: H t ,Y 2 Ht ,Z
1 cos t Ei exp i 0 c
1 2 sin t Ei exp i 0 c
.
(2.18)
Now we have everything what we need to calculate the reflection and the transmission coefficient. First, we note that the exponential term in the Y-components of the magnetic fields is the same in the eqs. (2.15) - (2.18), so we can drop this term. From the continuity relation of the tangential components of the electric field we obtain the first equation: E X medium1 E X medium 2 Ei , X Er , X Et , X
.
(2.19)
As for normal incidence, the second equation is obtained from the continuity of the tangential components of the magnetic fields, H Y medium1 H Y medium 2 H i ,Y H r ,Y H t ,Y
,
(2.20)
by replacing the magnetic fields with the results obtained with help of eqs. (2.16) - (2.18): i r Ei , X 1 cos i Er , X 1 cos r Et , X 2 cos t
E
i, X
Er , y 1 cos i Et , X 2 cos t
,
(2.21)
Here, we have taken advantage of the fact that the angle of incidence and the angle of reflectance are equal. To get the reflection coefficient rs, we replace Et , X in (2.21) with the left side of the result from (2.19):
E
i, X
Ei , X rs
Er , X 1 cos i Ei , X Er , X 2 cos t
1 cos i 2 cos t Er , X
Er , X Ei , X
1 cos i 2 cos t ,
(2.22)
1 cos i 2 cos t
1 cos i 2 cos t
To derive the solution for the transmission coefficient, we replace Er , X from (2.21) by Et , X Ei , X :
E
i, X
Et , X Ei , X 1 cos i Et , X 2 cos t
2 Ei , X 1 cos i Et, X ts
Et, X Ei , X
1 cos i 2 cos t ,
(2.23)
2 1 cos i 1 cos i 2 cos t
I remember that, after having studied the corresponding equations I came back a couple of times and asked myself where I should get the angle of transmittance from. Maybe you don’t share my weakness 24
in this respect, then the following is just to jog my own memory. To solve this problem we first replace the cosine function by the sine function and then use Fresnel’s law to replace sin t : n1 sin i n2 sin t sin 2 t cos 2 t 1 cos t 1 sin 2 t 2
n cos t 1 1 sin i n2 cos t 1
,
(2.24)
1 sin 2 i 2
Employing eqn. (2.24) we arrive at the final expressions for the transmission and reflection coefficient which contain only known quantities: rs
Er , X
ts
Et, X
Ei , X Ei , X
1 cos i 2 1 sin 2 i 1 cos i 2 1 sin 2 i 2 1 cos i
,
(2.25)
1 cos i 2 1 sin 2 i
2.2.2 p-polarized light We again follow the same recipe, but this time we have to take into account that for p-polarization the electric field is polarized along both, the Y- as well as the Z-axis. According to Scheme 2-4, the Ycomponent is positive for the incident, the reflected and the transmitted wave, while the Z-component is positive only for the reflected wave. Since the incident and transmitted electric field can be written as, E j ,Y E ||j cos j exp i E j , Z E ||j sin j exp i
j i, t
,
(2.26)
the corresponding electric field of the reflected wave is given by: Er ,Y Er|| cos i exp i Er , Z Er|| sin i exp i
.
(2.27)
Applying eqn. (1.26) to the incident wave leads to: EZ kY EY k Z 0 k E EZ k X , k i 1 sin i c . (2.28) EY k X cos i 1 1 || sin i Ei|| sin i exp i cos i Ei|| cos i exp i 1 H i , X 1 Ei exp i 0 c 0 c
Therefore, the incident magnetic field has only one component, which is antiparallel to the X-axis. The transmitted wave keeps this direction for the magnetic field, while the direction is reversed for the reflected wave: 25
EZ kY EY k Z 0 k E EZ k X , k r 1 sin i c cos . (2.29) EY k X i 1 1 || sin i Er|| sin i exp i cos i Er|| cos i exp i 1 H r , X 1 Er exp i 0 c 0 c
Scheme 2-4. A p-polarized plane wave impinging non-normally on an interface of two semi-infinite and scalar media
The magnetic field for the transmitted wave is correspondingly given by: Ht , X 2
1 || E exp i . 0 c t
(2.30)
Again, we have everything what we need to calculate the reflection and the transmission coefficient. For a second time, we drop the exponential term, which is the same in eqs. (2.26) - (2.30). From the continuity relation of the tangential components of the electric field we obtain the first equation: EY medium1 EY medium 2 Ei|| cos i Er|| cos r Et|| cos t
.
(2.31)
Once more, the second equation is obtained from the continuity of the tangential components of the magnetic fields, H X medium1 H X medium 2 Hi, X H r , X Ht , X
,
(2.32)
by replacing the magnetic fields with the results obtained with help of eqs. (2.28) - (2.30): 1 Ei|| Er|| 2 Et|| ,
(2.33)
To calculate the reflection coefficient rp, we replace Et|| in (2.33) by employing (2.31): 1 Ei|| Er|| 2
Ei|| cos i Er|| cos r cos t
1 cos t 2 cos i Er||
rp
cos t 2 cos i Er|| 1 || Ei 1 cos t 2 cos i
Ei||
26
1 cos t 2 cos i ,
(2.34)
To derive the solution for the transmission coefficient, we replace Er|| with help of (2.31): 1 Ei|| 2 Et|| 1
Et|| cos t Ei|| cos i cos i
1 cos i Ei|| 2 cos i Et|| 1 cos t Et|| 2 cos i Ei|| 2 1 cos i Ei|| Et|| tp
1 cos t 2 cos i
,
(2.35)
2 1 cos i Et|| || Ei 1 cos t 2 cos i
Employing eqn. (2.24) we arrive at the final expressions for the transmission and reflection coefficient which contain only known quantities: 1 1
1 sin 2 i 2 cos i 2
1 1
1 sin 2 i 2 cos i 2 ,
rp
(2.36)
2 1 cos t
tp 1 1
1 sin 2 i 2 cos i 2
2.2.3 Calculation of reflectance and transmittance Finally, to obtain the reflectance and the transmittance for both cases, s- as well as p-polarized light, we have to calculate the flux S in direction of Z to obtain IR and IT. From that we can calculate Rs,p and Ts,p: Rs Ts
Z Sr ,s Z Si , s Z St , s Z Si , s
kZ ,r
k Z ,t
k Z ,i
k Z ,i
2
rs , R p 2
t s , Tp
Z Sr , p
kZ ,r
k Z ,t
Z Si , p Z St , p Z Si , p
rp
k Z ,i
k Z ,i
2
, tp
(2.37)
2
For the reflectance we have to evaluate the ratio of the Z-component of the wave vector of the incident and the reflected wave. Since incidence and reflection take part in the same medium and under the same angle, this ratio is simply unity: kZ ,r k Z ,i
1 cos r 1 cos i
i r
kZ ,r k Z ,i
1.
(2.38)
The corresponding ratio for the transmittance, in contrast, has to be taken explicitly into account:
k Z ,t k Z ,i
2 cos t 1 cos i
2 1
27
1 2 sin i 2
1 cos i
.
(2.39)
Accordingly, as for normal incidence, the reflectance is simply given by: R j rj
2
j s, p .
(2.40)
Whereas for the transmittance the ratio in eqn. (2.39) has to be accounted for: Tj
k Z ,t k Z ,i
tj
2
28
j s, p .
(2.41)
References 1. Baker, M. J.; Trevisan, J.; Bassan, P.; Bhargava, R.; Butler, H. J.; Dorling, K. M.; Fielden, P. R.; Fogarty, S. W.; Fullwood, N. J.; Heys, K. A.; Hughes, C.; Lasch, P.; Martin-Hirsch, P. L.; Obinaju, B.; Sockalingum, G. D.; Sule-Suso, J.; Strong, R. J.; Walsh, M. J.; Wood, B. R.; Gardner, P.; Martin, F. L., Using Fourier transform IR spectroscopy to analyze biological materials. Nat Protoc 2014, 9 (8), 1771-91. 2. Chalmers, J. M., Mid-Infrared Spectroscopy: Anomalies, Artifacts and Common Errors. In Handbook of Vibrational Spectroscopy, John Wiley & Sons, Ltd: 2006. 3. Matossi, F., Ergebnisse der Ultrarotforschung. In Ergebnisse der Exakten Naturwissenschaften: Siebzehnter Band, Hund, F., Ed. Springer Berlin Heidelberg: Berlin, Heidelberg, 1938; pp 108-163. 4. Mayerhöfer, T. G.; Mutschke, H.; Popp, J., Employing Theories Far beyond Their Limits—The Case of the (Boguer-) Beer–Lambert Law. Chemphyschem : a European journal of chemical physics and physical chemistry 2016, 17 (13), 1948-1955. 5. Zbinden, R., Infrared spectroscopy of high polymers. Academic Press: 1964. 6. Hecht, E., Optics,4/e. Pearson Education: 2002. 7. Yeh, P., Optical Waves in Layered Media. Wiley: 2005. 8. Born, M.; Wolf, E.; Bhatia, A. B., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press: 1999.
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