Wave optics in Infrared Spectroscopy Thomas G. Mayerhöfer (
[email protected])
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
Content 0
Introduction ..................................................................................................................................... 7
1
The Calculus ................................................................................................................................... 10
2
1.1
Maxwell’s relations................................................................................................................ 10
1.2
Boundary Conditions ............................................................................................................. 11
1.3
Energy density and flux ......................................................................................................... 13
1.4
The wave equation ................................................................................................................ 14
1.5
Polarized waves ..................................................................................................................... 17
1.6
Further reading...................................................................................................................... 18
Reflection and Transmission of plane waves ................................................................................ 19 2.1 Reflection and Transmission at an interface separating two scalar media under normal incidence ........................................................................................................................................... 19 2.2 Reflection and Transmission at an interface separating two scalar semiinfinite media under non-normal incidence ....................................................................................................................... 23 2.2.1
s-polarized light ................................................................................................................. 25
2.2.2
p-polarized light ................................................................................................................. 27
2.2.3
Calculation of reflectance and transmittance ................................................................... 29
2.2.4
Example: Dependence of the reflectance from the angle of incidence .............................. 30
2.3 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – absorbing media............................................................................................................. 31 2.4 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – total/internal reflection ................................................................................................. 33 2.5 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – matrix formalism ............................................................................................................ 36 2.5.1
Matrix formulation for s-polarized waves at a single interface ........................................ 37
2.5.2
Matrix formulation for p-polarized waves at a single interface ........................................ 38
2.5.3
Combined matrix formulation for waves at a single interface .......................................... 39
2.5.4
A layer sandwiched by two semi-infinite media ................................................................ 40
2.5.5
Arbitrary number of layers ................................................................................................ 43
2.5.6
Calculating the electric field strengths of a layered medium – coherent layers ............... 44
2.5.7
Incoherent layers ............................................................................................................... 45
2.5.8
Mixed coherent and incoherent layers .............................................................................. 47
2.5.9 Calculating the electric field strengths of a layered medium – mixed coherent-incoherent multilayers ......................................................................................................................................... 49 2.6 3
Further reading...................................................................................................................... 52
Dispersion relations ....................................................................................................................... 53
5
6
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 I A log10 I0
(0.4)
log10 e d d
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!
7
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 8
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…
9
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):
10
U z U y y z x U x U x U z U U y y z x U z U y U x 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 ,
11
(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:
12
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
13
(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 0. t t 2
(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
14
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 , where 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). 15
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 z x 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
16
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
17
.
(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 reading 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
18
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., 19
(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
20
,
(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:
21
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).
22
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
23
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)
24
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 cosi Ei exp i . 0 c
Scheme 2-3. A s-polarized plane wave impinging non-normally on an interface of two semi-infinite and scalar media
25
(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 26
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: 27
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||
28
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|| 1 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 i
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 Z Si , p Z St , p Z Si , p
kZ ,r
k Z ,t
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
29
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
j s, p .
(2.41)
2.2.4 Example: Dependence of the reflectance from the angle of incidence For the following example, we assume vacuum as incidence medium, i.e. 1 1 . The exit medium is characterized by 2 4 (accordingly, the indices of refraction are n1 1 and n2 2 ). For the angle of incidence 0 (normal incidence), the polarization leads to the fact that Rp (better called Ry in this case) has only a component parallel to the interface as has Rs.( better called Ry). Accordingly, in this case Ry = Rx and therefore Rs and Rp have the same starting point in Fig. 2-1. In contrast to Rp, Rs is a monotonically increasing function which ends at Rs = 1 at 90 , since then the plane wave is travelling perpendicular to the interface and, accordingly, is no longer reflected. Rp first decreases until it reaches zero at ≈ 63° in this particular example. The angle at which Rp becomes zero is called Brewster’s angle. Since at this angle the parallel polarized component is not reflected, natural polarized light becomes s-polarized upon reflection, which can be (and has been) exploited to construct polarizers (e.g. using Se9). For the transmitted light, a part of the s-polarized light is transmitted. Therefore it contains both, p-, and also s-polarized light the ratio of which depends on the index of refraction of the second medium. At least in our example, this has no practical relevance, as the assumption of a semi-infinite exit medium in any way excludes in principle any use of the transmitted light. This is also the reason why we do not show the dependence of the transmittance on the angle of incidence as there is no way to check the correctness of the calculation (to verify the theory, it would be, strictly speaking, necessary to implement a detector into the second medium with the same optical properties as the second medium. On the other hand, since R + T = 1, such a test would not be necessary, since if the reflectance adheres to the derived relations, so would transmittance). After the Brewster angle, Rp is also monotonically increasing and reaches unity together with Rs at 90°. 1.00 Rs Rp
Reflectance
0.75
0.50
0.25
0.00 0
15
30
45
60
75
/° Fig. 2-1: Dependence of Rs and Rp from the angle of incidence.
30
90
2.3 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – absorbing media On the first view, it seems that the relations that we derived in the preceding chapter are of little practical relevance. We are infrared spectroscopists and rely on the fact that materials absorb in the infrared. Certainly, there are spectral ranges where there are no bands, but this does not mean that absorption is zero (indeed, SiO2 bulk glass samples become, certainly depending on its thickness, not transparent below about 2000 cm-1, even when the highest fundamental seems to be at about 1200 cm-1. Why did I use the term “seems” in the last sentence? Because for me, the highest fundamental is at 1100 cm-1 and the shoulder at 1200 cm-1 is caused by orientational averaging,10 but it is too early in the book for this story). In any way, including absorption into our formulas is, as nature here is kind to us, trivial, at least nowadays, where we have computers to deal properly with imaginary numbers. Therefore, the good news is that all relations keep their form and we just have to replace the real index of refraction by its complex counterpart: n n ik .
(2.42)
Herein, n (non-italic) is the (real) index of refraction as we have discussed it so far.2 Analogously, k is the index of absorption (also non-italic to differentiate it from the wave vector), which will be a very important quantity as it is the bridge between wave optics-based and Beer-Lambert-based quantitative infrared spectroscopy. A first insight into what happens as a consequence of the introduction of the index of absorption can be gained, if we assume a plane wave travelling in the Zdirection being (for no particular reason, except to simplify things) X-polarized: Ex Z , t E0, X exp i t kZ Z .
Since kZ n
(2.43)
n ik , the wave can be written as: c c Ex Z , t E0, X exp i t n ik Z . c
(2.44)
And if we now separate real- and imaginary part of the wave, E X Z , t E0, X exp k Z exp i t n Z , c c
(2.45)
Exponential decay
we see that the electric field strength, and therefore also its intensity, decays exponentially. So this certainly must mean that the use of absorbance is justified and the Beer-Lambert law has rightfully its merits? If there were no interfaces to cross and the wave would be travelling in a homogenous medium forever, certainly, because then we can indeed assume that the electric field intensity at a certain Z is fully determined by the irradiance after exiting the light source:
2k I 0 exp Z I 0 exp Z . c A Z I Z
2
(2.46)
If the index of refraction is complex, we will sometimes stress this fact in the course of the book by adding an apostrophe: n’
31
This means that under these exceptional conditions using the absorbance A definitely makes sense! Under all other conditions, it is more meaningful to use the absorptance A, which we obtain from energy conservation: I R IT I A R T A 1. I0 I0 I0
(2.47)
When thinking about eqn. (2.47), keep in mind that our interfaces are assumed to be smooth (no diffuse reflection or transmission) and the media homogenous, which means that scattering is explicitly negligible/absent. Note that we have defined absorbance and absorptance along the IUPAC standards.3 In literature, these terms are often used differently or even synonymously and, sometimes, the definition that is used becomes only clear through the context. How does absorption alter the reflection from the interface between two media? In general, absorption increases reflection. On the first view, this seems to be paradox, but indeed, the strongest reflecting material in the infrared is gold due to having the highest index of absorption. As a consequence of this high reflection, not much radiation is actually absorbed (this completely changes in powdered form, which is why metal powders are usually black). For not too high indices of absorption, the dependence of reflection on the angle of incidence is depicted in Fig. 2-2. 1.00 Rs
Reflectance
0.75 0.50 0.25 0.00 Rp
k = 0 k= 0.5 k = 1 k= 1.5
0.75 0.50 0.25 0.00 0
15
30
45
60
75
90
/°
Fig. 2-2: Dependence of Rs and Rp from the angle of incidence for absorbing materials.
For Rs, the presence of absorption does not alter the curves substantially as can be seen in Fig. 2-2. Rs remains to be a monotonically increasing function of the angle of incidence, just the starting value for = 0° increases. In case of Rp, the changes are also, on the first view, not drastic. The function still possesses a minimum, which is, however, no longer a zero. Furthermore, this minimum is shifted to higher angles of incidence.
3
https://goldbook.iupac.org/
32
2.4 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – total/internal reflection In the preceding examples, we somewhat automatically assumed incidence from vacuum or air, but what if the incidence medium is the optically denser medium, meaning that it has the higher index of refraction? This seems to be a situation that gets us into trouble, since from Fresnel’s law we have deduced that for n1 < n2, i t . Consequently, if we reverse the situation, then the angle of refraction must be larger than the angle of incidence. However, we cannot arbitrarily increase the angle of refraction – once it is equal to 90°, the wave is actually no longer penetrating into the second medium, but travels parallel to the interface. To derive this, we start from eqn. (2.14) and focus on the transmitted wave: n Et E0,t exp i t 2 Y sin t Z cos t . c
(2.48)
Due to Fresnel’s law, eqn. (2.12), the Y-component of the wave vector stays the same and we can transform (2.48) with help of eqn. (2.24) into: n n Et E0,t exp i t 1 Y sin i 1 Z n22 n12 sin 2 i c c
.
(2.49)
For t 90 , sin t 1 , accordingly the corresponding angle of incidence, which is called the critical angle c , becomes: n c arcsin 2 . n1
(2.50)
Above c , n22 n12 sin 2 i 0 . Nevertheless, eqn. (2.49) is still meaningful. To see this, we replace
n22 n12 sin 2 i by i 2 n12 sin 2 i n22 and obtain,
n n Et E0,t exp i t 1 Y sin i i 1 Z n12 sin 2 i n22 , c c
(2.51)
which we can then transform into: n n Et E0,t exp 1 Z n12 sin 2 i n22 exp i t 1 Y sin i . c c
(2.52)
This result obviously is the cause of trouble, because our wave would grow exponentially with Z. Remembering that we actually deal with real waves and that the real part remains the same, even if we multiply the phase in the exponential function by -1 we have to repair our unphysical results by
33
introducing i 2 n12 sin 2 i n22 instead of i 2 n12 sin 2 i n22 to assure that the field decays exponentially with increasing Z:4 n Et E0,t exp Z n1 n12 sin 2 i n22 exp i t 1 Y sin i , c c
(2.53)
Eqn. (2.53) describes an electric wave, which has just a Y-component in the phase part, meaning that it travels along the interface between medium 1 and 2. Accordingly, there is no energy flux through the interface as long as n2 is real. The electric field intensity as a function of distance from the interface and wavenumber is illustrated in Fig. 2-3. The exponential decay of the field intensity with distance from the interface (black line in Fig. 2-3) is clearly visible, as is the scaling with the wavenumber (remember that c 2 ). The penetration depth is further influenced by the indices of both media and the angle of incidence according to eqn. (2.53).
Fig. 2-3: Electric field intensity at the interface for s-polarized incident light ( = 30°, n1 = 3.4, n2 = 1.4). The incidence medium extends from above down to the black line, which denotes the interface.
Eqn. (2.50) implies that the higher the index of refraction of the incidence medium, the smaller becomes c . The dependence of the reflectance from the angle of incidence still shows the same characteristics as can be seen in Fig. 2-4, but the reflectance already reaches unity at the critical angle and stays at this value. Why do we actually care about this particular situation? Since the wave cannot really enter the second medium, it cannot probe it, so this situation is useless for the spectroscopist, right? In contrast, this situation is highly interesting! Since we are interested in absorbing media, it is instructive to see how the situation changes, if we allow the second medium to be absorbing. If we assume a complex index of refraction, two changes can be realized in eqn. (2.53) concerning the first exponential function:
4
One or the other reader might think that changing the sign is an act of serious arbitrariness. To convince those, it might be enough to point out that there are two ways to introduce a complex index of refraction, namely in the form n n ik and n n ik . The use of one or the other form depends on the definition of the complex form of the wave and the fact that the form of the complex index of refraction must lead to exponential decay for non-zero k .
34
Z
c
n2 n 2 ik 2 n1 n12 sin 2 i n22 Z
c
n1 n12 sin 2 i n 22 k 22 2ink ,
(2.54)
Reflectance
c arcsin n2 n1
1.00 0.75
Rp
0.50
Rs
0.25 0.00 0
15
30
45
60
75
90
Angle of Incidence (°) Fig. 2-4: Dependence of Rs and Rp from the angle of incidence in case of total (internal) reflection.
Without evaluating (2.54) further, it is obvious that the wave would still be evanescent, but with faster decay as can also be seen in Fig. 2-5. From this figure, it is also obvious that absorption weakens the reflectance, which is no longer “total”, therefore this method is usually called attenuated total reflection (ATR) spectroscopy. In fact, for larger absorption indices, any resemblance with total reflection gets lost as is illustrated in Fig. 2-6.
Fig. 2-5: Electric field intensity at the interface for s-polarized incident light ( = 30°, n1 = 3.4, n2 = 1.4+i0.54).
The second change is that with increasing absorption index, an increasing part of the Z-component becomes real, that means the wave is no longer tangential to the interface and an increasing flux of energy into the second medium does occur where it is absorbed.
35
k=0 k = 0.06 k = 0.18 k = 0.54 k = 1.62
1.00 0.75 0.50
Reflectance
Rs
0.25 0.00 1.00 0.75 0.50 Rp
0.25 0.00 0
15
30
45
60
75
90
/°
Fig. 2-6: Dependence of Rs and Rp from the angle of incidence for n1 > n2 and absorbing materials.
2.5 Reflection and Transmission at an interface separating two scalar media under non-normal incidence – matrix formalism The formalisms that I presented in the last sections seemed to be of limited use for infrared spectroscopy, since, while measurements of reflection and total internal reflection play an important role, an understanding of the most important technique, which is the transmission technique, cannot be advanced. Indeed, to understand transmission spectroscopy from a wave optical point of view at least needs to introduce three media: The incidence medium, the sample and the exit medium, where the first and the last are considered semi-infinite and are usually air or vacuum. A typical example is a pressed tablet of an IR-transparent material containing a small amount of sample of finely grained consistency to form a medium that is homogenous to IR-radiation to avoid scattering. Further examples would be freestanding films (e.g. polymers) or sections. In case of inorganic materials it is, however, impossible to prepare sections thin enough so that they could be used for transmission measurements, at least not in the spectral range around their main bands (fundamentals) as thicknesses of the order of 100 nm would be necessary. In such cases layers could be prepared, e.g. by vapor deposition methods, on IR-transparent substrates. As a result, we have already two media of limited thickness that are transmitted by radiation! Furthermore, if liquids are to be investigated, we actually need one medium more, as liquids are usually contained in cuvettes, so that the cuvette material is transmitted twice, when light passes through the cuvette. Even if this is the same material with usually the same thickness before and after the sample, this adds another degree of complexity. Why would that be? Because in the infrared spectral range, absorption, even that of organic liquids, is strong enough to suggest that the thickness of a cuvette should be in the range of some 10 microns.5 In this case, as for freestanding films and films on a substrate, we have not only to take into account 5
Note that the use of solvents is usually not an option in infrared spectroscopy, as it is hard to find a proper solvent that is not so chemically similar and does not have absorption bands in the same spectral regions as the liquid to be investigated.
36
that we have multiple reflections, but also that we have interference. As a consequence, the measurement cannot be referenced to the measurement of an empty cuvette, since the index of refraction differences between cuvette material and content is usually much higher than if filled. Accordingly, interference effects will be enhanced, as we will see later. In thick substrates on the other hand, interference effects are often absent.6 Overall, this makes things somewhat complicated, as we would need to derive a formula for the freestanding film, another one for the layer on a substrate and another one for the liquid in the cuvette. Moreover, what do we do, if we have more than one layer on the substrate? Therefore, instead of deriving all these particular formulas, we go the inverse way and provide a general formalism, which contains all these special cases (and much more). Then, when we discuss particular cases, we go backwards and derive or state the explicit formulas. I hope that we then regain the vividness that we have to give up for the moment in order to encompass all possible cases. We first assume that we have exclusively coherent superposition of waves. In addition to the freestanding film or section, this would be the case for a layer stack of thin films with smooth and plane parallel interfaces. However, it could also be a film on a substrate as long as the substrate surfaces are plane parallel to each other and the spectral resolution is high enough (e.g. < 4 cm-1 for a 1 mm thick Si substrate). In the layer, or in each layer of the layer stack, there exist in general forward and backward travelling waves, since reflection occurs at each of the interfaces, which leads to multiple reflection of the incoming wave. It seems that this is a very confusing situation, but there is a recipe to find a consistent and general solution in a very concise way.
2.5.1 Matrix formulation for s-polarized waves at a single interface First, we extend the situation at an interface of the layer stack by assuming that we have an incoming and a reflected wave from both sides of the interface as it is depicted in Scheme 2-5.
Scheme 2-5. Forward and backward traveling s-polarized plane waves impinging non-normally on an interface of two semiinfinite and scalar media.
6
According to the textbooks, interference is absent, if the thickness of the medium is no longer in the same range as the wavelength of light and/or if the light comes from an incoherent light source. Actually, I think that this has nothing to do with the ratio of the wavelength to the thickness of the substrate, since we were confronted with interference fringes even in the spectra of a 1 mm thick Si substrate with light from an incoherent light source. It is my belief (yes, even in science there is room for belief!) that this is because every light wave interferes with its reflected self, which is of course perfectly coherent to the original wave. A better explanation for the absence of interference fringes in case of thick substrates is therefore that missing plane planarity destroys the interference effects.
37
Accordingly, the continuity of the transversal components of the electric and magnetic fields leads to the following two equations: E1, X E1, X E2, X E2, X E1, X n1 cos 1 E1, X n1 cos 1 E2, X n2 cos 2 E2, X n2 cos 2
.
(2.55)
These can be expressed in matrix form as: 1 n1 cos 1
1 E1, X n1 cos 1 E1, X
1 n2 cos 2
Ds 1
1 E2, X n2 cos 2 E2, X
.
(2.56)
Ds 2
Here, D j denotes the so-called dynamical matrix for s-polarized light and the medium j. For a s single interface, we obtain the reflection and transmission coefficients rs and ts in the following way: E1, X E2, X Ds 1 Ds 2 E1, X E2, X E1, X E2, X 1 Ds 1 Ds 2 E1, X E2, X
.
(2.57)
Ms
E1, X E1, X
E2, X Ms E2, X
M 21, s rs M 11, s
1 , ts M E2, X 0 11, s
E2, X 0
Certainly, eqn. (2.57) yields the same result as eqn. (2.25), since we assumed for the derivation that there is no incoming wave travelling in the –Z direction.
2.5.2 Matrix formulation for p-polarized waves at a single interface Equivalent to section 2.5.1 we now assume two p-polarized waves incoming from both sides of an interface. The situation is depicted in Scheme 2-6.
Scheme 2-6. Forward and backward traveling p-polarized plane waves impinging non-normally on an interface of two semiinfinite and scalar media.
The continuity of the transversal components of the electric and magnetic fields again lead to two equations, 38
E1, p cos 1 E1, p cos 1 E2, p cos 2 E2, p cos 2
,
(2.58)
cos 2 E2, p . n2 E2, p
(2.59)
E1, p n1 E1, p n1 E2, p n2 E2, p n2 which, expressed as dynamical matrices take on the following form: cos 1 n1
cos 1 E1, p cos 2 n1 E1, p n2
D p 1
Dp 2
In equivalence to its counterpart D j , D j denotes the dynamical matrix for p-polarized light s p and the medium j. Assuming that there is no incoming wave from the negative Z-direction, we recover the result from eqn. (2.36): E1, p E2, p D p 1 Dp 2 E1, p E2, p E1, p E2, p 1 D p 1 D p 2 E1, p E2, p
.
(2.60)
Mp
M 21, p E1, p E2, p M p rp E1, p E2, p M 11, p
1 ,tp M 11, p E2 , p 0
E2 , p 0
2.5.3 Combined matrix formulation for waves at a single interface Contrary to the trend to reduce dimensionality we here go the opposite way and increase it. To that end, we define a 4×4 matrix from the 2 2×2 matrices for s- and p-polarized light by placing the matrix for s-polarized light in the top-left quadrant while we put the matrix for p-polarized light in the bottomright quadrant: 1 n1 cos 1 0 0
1 0 0 E1, s 1 E n1 cos 1 0 0 1, s n2 cos 2 0 cos 1 cos 1 E1, p 0 0 n1 n1 E1, p 0 D1
1 n2 cos 2 0 0
0 0 cos 2 n2
0 E2, s 0 E2, s . cos 2 E2, p n2 E2, p
D2
(2.61) For the moment, this looks odd, in particular since the remaining quadrants are filled with zeros. On the other hand, it not too hard to imagine that by using a 4×4 matrix formalism we obtain the results for the reflection and transmission coefficients for s-and p-polarized light in one go:
E1, s E2, s E1, s E2, s E1, s D 1 D E2, s E1, s M E2, s 1 2 E1, p E2, p E E M 2, p 1, p E E E E 1, p 2, p 1, p 2, p
39
rs
M M 21 1 1 , ts , rp 43 , t p . (2.62) M 11 M 11 M 33 M 33
Here, D1 and D2 are the dynamical 4×4 matrices of medium 1 and 2. Since nowadays neither memory nor speed of computers pose a problem for the calculations at hand, the use of sparse matrix of large dimensionality should be ok. However, there is a further reason why I wanted to introduce eqn. (2.61) , which is we generally need 4×4 matrices when we talk about anisotropic media. For those media, the components, which are zero now, are a measure of s-polarized light that is converted to p-polarized light and vice versa. Since the 4×4 matrix formalism for anisotropic media becomes singular when it is used for scalar media (we will discuss this later in detail), it is good to know that we can directly use the matrices that we just introduced instead. Having introduced the 4×4 matrices, we make a step back to the 2×2 matrices to shorten the notation. We can do this, as we examine some properties of the matrices, which do not alter for the different polarization directions. We therefore suppress the subscript, which would indicate the polarization direction keeping in mind that the difference between s-polarized matrices and p-polarized matrices just surfaces once we have to specify the components of the dynamical matrices.
2.5.4 A layer sandwiched by two semi-infinite media It is now time for the next level of escalation. Actually, once you have mastered this one, you will be able to compile formulas for arbitrary layer stacks provided that they consist of thin, homogenous, scalar media with smooth interfaces (thin in this context always means that interference effects do exist). To simplify things, we paradoxically need to make them first a little bit more complex by introducing a new nomenclature. So far, we have just differentiated between fields on the two sides of an interface, but now we have to differentiate between fields in a layer directly on the right side of the left interface (the one with the lower value of Z) and those on the left side of the right interface. Those will in general be different, not only because of absorption, but also, as we will see later, because the field strength is not constant over the layer and might increase when increasing Z, even in absence of absorption! We therefore define E jL as the field strength of the wave that is travelling in positive Z-direction (therefore the “+”) immediately after the wave entered the medium j, that is, directly behind the left interface of the jth medium (the interface between the (j-1)th and the jth medium). Likewise, E jR is the field strength of the same wave immediately before it exits the jth medium, i.e. directly at the left side of the right interface (between the jth and the (j+1)th medium). For the wave travelling antiparallel to the Z-direction everything is analogous, the only change is that “+” is replaced by “-“ to indicate the change of the travelling direction. The situation just described is illustrated in Scheme 2-7 for the case that j = 2.
Scheme 2-7. Forward and backward traveling waves in a one layer system with incidence medium (medium 1) / layer / exit medium (medium 3).
40
The total field strength in medium j at the coordinate Z is then found by adding up the two fields belonging to the forward and the backward traveling wave: E j Z R j exp ik jZ Z L j exp ik jZ Z E j Z E j Z .
(2.63)
Rj and Lj are constants in each medium j. What has not been explained so far are the matrix products in Scheme 2-7. As we know from eqn. (2.62), the matrix product D j 11 D j generally links the fields before and after an interface, but what is the function of the matrix P2 ? Obviously, this matrix, which we call the propagation matrix further on, describes the alteration of the electric fields when the wave propagates from the left interface to the right interface. Therefore, we can write,
E2L E2R P2 , E2 L E2 R
(2.64)
exp i2 0 P2 . 0 exp i2
(2.65)
wherein the matrix P2 is given by:
Herein, 2 is the phase of the wave in medium 2: 2 k 2 Z d
2 n2 d cos 2 .
(2.66)
Who thinks that eqn. (2.66) appears somewhat “out of the blue”, is requested to go back to eqn. (2.43) , where eqn. (2.66) originates from (we have just lost the time dependence, because it is not relevant for the calculation of transmittance and reflectance). The difference of the sign in the arguments of the exponentials in eqn. (2.65) simply results from the fact that the element (1,1) belongs to the forward travelling wave (therefore, as a memory hook, the real part must show an exponential decay when the wave propagates in the positive Z-direction), whereas element (3,3) belongs to the backward travelling wave (and decays when propagating in the negative Z-direction). The only difference is that we have to consider also non-normal incidence and we do that by the factor cos2 (be reminded that you can easily calculate this factor, cf. (2.24)). It may be an unnecessary remark, but for scalar media 2 does not have a polarization dependence, so that P2 has the same form regardless of polarization. Accordingly, in the 4×4 matrix form, a propagation matrix for a scalar medium is of diagonal form with the same elements at (1,1) and (3,3) positions as well as at (2,2) and (4,4), respectively. If we put everything together, we can calculate the fields in medium 3 directly behind the second interface from the fields in medium 1 directly before the first interface according to, E3L E1R 1 1 D D P D D 1 2 2 2 3 , E1R E3 L M
(2.67)
and, from the matrix M, reflection and transmission coefficient, as well as reflectance and transmittance using M rl 21,l M 11,l
1 , tl M E3L 0 11,l
, E3L 0
l s, p .
Formally, we can also state the matrix product in the following form: 41
(2.68)
D j 11 D j D j 1 j .
(2.69)
1 1 t j 1 j r j 1 j
(2.70)
The matrix D j 1 j can be written as:
D j 1 j
r j 1 j . 1
Here, we have to distinguish between the different polarizations of the wave. For s-polarization we have, kj 1 Z k jZ
r j 1 j , s
kj 1 Z k jZ
t j 1 j ,s
,
2kj 1 Z kj 1 Z k jZ
,
(2.71)
where the prime emphasizes that the wavevector can be complex. For p-polarization we find: r j 1 j , p
nj 12 k jZ nj 2 kj 1 Z nj 12 k jZ nj 2 kj 1 Z
t j 1 j , p
,
2nj 1nj kj 1 Z nj 12 k jZ nj 2 kj 1 Z
.
(2.72)
Eqn. (2.68) can easily be generalized for an arbitrary number of layers. However, before we continue, let’s first put some flesh on the bones and take a look onto a simple example. This example shall consist of the calculation of the reflection and transmission of a slab that is suspended in vacuum with n1 n3 1 . Furthermore, we assume normal incidence so that kZj nj . Under the latter condition, the problem degenerates and the 4×4 matrix contains two identical 2×2 matrices M since there is no difference between s- and p-polarization (cf. section 2.1):
1 M 12 2
1 1 2 n2 1 2
D11
1 0 2 1 exp i2 0 exp i2 12 n2 D2
1 1 . 21n2 1 1 1 2 n2
D2
(2.73)
D1
With help of the matrix M we can calculate the reflectance and transmittance according to: t r
t t exp i 1 12 212 M 11 1 r21 exp 2i
t12
t t r exp 2i M 21 r12 12 21 212 M 11 1 r21 exp 2i
r12 Tt
2
R r
2
2n2 2 , t21 , 1 n2 1 n2
1 n2 n 1 , r21 2 r12 . 1 n2 1 n2
.
(2.74)
k3 Z 1 k1Z
When I look at the results, I have to admit that it would not be immediately clear to me that eqn. (2.74) indeed accounts for multiple reflections of light inside the slab that coherently interfere. As we will discuss this problem at length in chapter 4, it is one of the effects that can cause large deviations from the Beer-Lambert law, not only because of the multiple reflections that are not accounted for in this law, I will not go into the details. Instead, I will for the moment just provide a hint for those of you who cannot wait: Just calculate at every interface for every pass of the wave the electric fields of the waves being reflected and refracted and add them all up as this was done by G.B. Airy in 1833 to explain the 42
color of Newton’s rings.11 For those who can wait, do not worry, we will do this anyway and also provide the well-known short cut to the solution (which is inherently used whenever the matrix formalism is employed).
2.5.5 Arbitrary number of layers So far, we followed to a large extent pathways that have been influenced by 7. Yeh certainly also provided a formalism to calculate the reflectance, transmittance and absorbance for an arbitrary number of layers. We are, however, not only interested in the calculation of these quantities, but also in the calculation of the field strengths and the field intensities at an arbitrary Z, while, so far, we have calculated these only at interfaces. The interest in these calculations derives from the possibility to calculate field and intensity maps which I introduced already in section 2.4. These maps allow a very intuitive access to changes of absorption with layer thicknesses and wavenumber. They will be essential, or at least illustrative, for understanding the related interference effects and are worth the extra effort. To be able to perform the corresponding calculations, we will intermix the approaches from 7 to 12 in the following.
Scheme 2-8. Forward and backward traveling waves in a system with m layers. The incidence medium is medium 0 and the exit medium is medium m+1.
The situation for an arbitrary number of layers is depicted in Scheme 2-8. The calculation of the fields at the interface between layer m and exit medium (medium m+1) is straightforward. Looking at eqn. (2.67) we see that the product of matrices begins from the left side with the inverse of the dynamical matrix of the incidence medium (in this case medium 1) D11 . At the right side the product is completed by the dynamical matrix of the exit medium D3 . In-between we find the matrix product D2 P2 D21 . Accordingly, if we would have a second layer, its contribution to the matrix M would be D3 P3 D31 . Overall, if we rename the incidence medium as medium 0 and the exit medium as medium m+1, we can describe the forward and backward travelling wave in terms of the forward and backward travelling waves in medium 0 and the different dynamical and propagation matrices as: m Em 1 L E0R 1 1 . D0 Di PD i i Dm 1 E E i 1 0R m 1 L
(2.75)
M
Again, we use one 2x2 matrix for s-polarization and one 2x2 matrix for p-polarization. t and r as well as T and R are calculated in exactly the same way from the matrix M as in eqn. (2.68):
43
E M r 0R 21 E 0 R Em1L 0 M 11 Em 1 L t E0R
1 E m1L 0 M 11
Em 1 L r E m 1 L
M 12 M 11 E0 R 0
E t 0R E m 1 L
det M M 11 E0 R 0
.
(2.76)
We nevertheless repeat the formulas here, because we also want to introduce the reflectance from the backside of the layer stack, r , which is calculated under the assumption that light is incident from the backside (accordingly, medium 0 is then the exit medium and E0R 0 ) and the corresponding transmittance t . To make you acquainted with the notation of 12 we give again the formulas for the calculation of the reflectance (which is trivially the same) and the transmittance, Rl rl , l s, p 2
ts Re nm 1 cos m 1 2
Ts
n0 cos 0
,
(2.77)
t p Re nm*1 cos m 1 2
Tp
n0 cos 0
where nm*1 is the complex conjugate of nm1 . Note that now as we deal with absorbing media, we assume here that the incidence medium is real. For a spectroscopist, certainly nothing else would actually make sense, because otherwise light would usually not reach sample, let alone the detector.
2.5.6 Calculating the electric field strengths of a layered medium – coherent layers If you are not interested in being able to calculate field maps, this section can be skipped. If you are just care about calculating field maps for completely coherent systems, commercial software is available. On the other hand, this excludes e.g. the field map of a thin layer on a substrate (effective July 2018). So if you are interested in this situation you have to either ask me to calculate the field map for you, or, better, work through this and the following sections. To use Centurioni’s formalism12 we first have to subdivide the layer according to Scheme 2-8 in two parts ranging from medium 0 to medium j and from medium j to medium m+1. More precisely, we make the cut at the interface j/j+1. If we are want to calculate the fields directly on the left side of this interface from the fields left from the first interface 0/1, we have in analogy to (2.75): j 1 E jR E0R 1 1 D P D0 Di PD i i j j E . i 1 E0 R jR
44
(2.78)
There is one difference, though, which is that in eqn. (2.75), the final fields are those right from the last interface. This time, however, we are interested in those left from the next interface. This is the reason why we also need to include the propagation matrix Pj . The second part, conversely, agrees formally completely with (2.75): Em 1 L E jR m 1 1 . D D PD D j i i i m 1 E m 1 L i j 1 E jR
(2.79)
M j m1
On the other hand, instead of performing this calculation in two steps, we can do this in one step according to (2.75), and if we use the result for the transmission coefficient (eqn. (2.76)),
Em 1 L E0 R
M1 1 11 E0R Em 1 L , E0 R 0 M 11 0
(2.80)
we can reformulate eqn. (2.79) in the following way (note that we assume in the derivation of t in (2.76) that Em 1 L 0 since the exit medium is semiinfinite):
Em1 L E jR E jR M111 M M j m 1 j m 1 0 E0 R . Em1 L E jR E jR
(2.81)
This does not seem to be a big deal, and when I first saw this equation, I thought to myself, ok, what is this equation good for and from where do I get E0R ? It is good, that I kept this question to myself. For field maps, the calculated fields are always relative to E0R , so E0R is simply unity. If we now take into account, that we can calculate the fields E j Z and E j Z in the jth layer at a position Z by “propagating backwards”, i.e., d j Z E jR dj Z E j Z M111 P P M E , j d E jR j d j m1 0 0 R j j Ej Z
(2.82)
we have everything we need to calculate the field maps. Note that what stands in parentheses after P is in this case the argument, i.e. j in the exponential functions has to be replaced by this argument. The total field that we are interested in is then simply the sum of E j Z and E j Z .
2.5.7 Incoherent layers We have already talked about cases when layers or layer stacks can be considered as incoherent, which has nothing to do with using a non-coherent light source. If the thickness of the medium is homogenous, then it will nevertheless seem to be incoherent when either the thickness is increased (provided certainly, that the spectral resolution is limited) or the spectral resolution is correspondingly decreased. In any case, typical examples for incoherent layers are pressed pellets (KBr or CsI technique), substrates or the walls of a cuvette, when, as discussed above, no interference fringes are visible (still, interference fringes from layers on a substrate or the inside of the cuvette can appear; 45
those cases will be subject of the next section). How can we treat these cases? This is actually simple: We treat the waves as beams, i.e. we remove interference through superposition by squaring the fields (actually multiplying the field with its complex conjugate) and looking at intensities instead. This still does not mean that we now assume the Beer-Lambert law as valid (cf. eqs. (0.3) and (0.4)), since we can still have multiple reflections inside our layers, but, as we will see later, Beer-Lambert is a much better approximation in such cases. Another good news is that our matrix formalism for the treatment of coherent layers can (nearly) without changes be taken over, the only difference is that we are no longer dealing with fields, but with intensities, which we will denote with the capital letter U ( U E
2
). The corresponding situation for an arbitrary number of incoherent layers is depicted in Scheme 2-9.
Scheme 2-9. Forward and backward traveling beams in a system with m’ layers. The incidence medium is medium 0’ and the exit medium is medium m’+1.
Due to the correspondence of the formalisms we can formulate the relations between an incoming beam with intensity U 0 R and the intensity of the reflected beam U 0 R and the transmitted beam U m1 L as: m U m1 L U 0 R 1 1 . D0 Di Pi Di Dm1 U m1 L i1 U 0 R
(2.83)
M
To indicate that we now deal with intensities rather than with electric fields and to distinguish the matrices from those for the fields, we mark the intensity matrices by horizontal lines above the letters. The propagation matrices Pj are now given by (in perfect analogy to the use of intensities):
exp i 2 j Pj 0
. 2 exp i j 0
(2.84)
As a consequence, only the imaginary part remains. If it is zero, than the beam intensity does not change at all in this medium. Deviations can though be found in the dynamical matrices D j 1 j , which are less symmetric than the corresponding ones in coherent layers (at least if absorption plays a role):
D j 1 j
1 1 2 t j 1 j r j 1 j
r
2
j 1 j
2
2
t j 1 j t j 1 j r j 1 j rj 1 j
. 2
(2.85)
In eqn. (2.85), the reflection and transmission coefficients are those for the coherent layers and can therefore be calculated in the usual way by eqs (2.71) for s-polarization and (2.72) for p-polarization. 46
From the matrix M we can also obtain reflection and transmission coefficients for the whole incoherent layer stack according to: U M r 0 R 21 U 0 R U m1L 0 M 11 U m1 L t U 0 R
1 U m1L 0 M 11
U m1 L r U m1 L
M 12 M 11 U 0 R 0
U t 0 R U m1 L
det M M 11 U 0 R 0
.
(2.86)
These reflection and transmission coefficients, however, are already intensities. Accordingly, the reflection coefficient equals the reflectance, while the transmission coefficients need to be corrected to obtain the transmittance: Rr Ts Tp
ts Re nm 1 cos m1 . n0 cos 0
(2.87)
t p Re nm*1 cos m1 n0 cos 0
This completes the calculation of reflectance and transmittance for an incoherent layer stack. As an observant reader, you may have noticed that the numbering of the media was slightly different compared to the incoherent case. Not that the order of the numbers would have been changed, but the numbers were given an apostrophe. The rationale behind this change is that in the next section we intermix coherent layer and incoherent layer packages and need to distinguish between the different layers. On the first view, the way it is done is highly confusing, at least it was for me. I hope I am able to convey the formalism in a manner, which is less unsettling for you (if not, you were at least warned what to expect in contrast to me).
2.5.8 Mixed coherent and incoherent layers The best way of thinking of such a mixed coherent and incoherent layer stack is to keeping first only the incoherent layers and the corresponding formalism from the preceding section. If we take the examples that we already talked about, e.g. a coherent layer stack on a thick substrate you would first think of incidence medium / substrate / exit medium and use the mathematical framework of the preceding section. Depending on whether the light first hits the coherent layer stack and then the substrate or vice versa, you would then replace either the first or the second interface by the coherent layer stack as this is shown in Scheme 2-10.
47
Scheme 2-10. Forward and backward traveling beams in a mixed coherent/incoherent system with m’ incoherent layers. The incidence medium is medium 0’ and the exit medium is medium m’+1. The enlarged coherent system (lower part), which builds the interface between incoherent layer i’ and (i’+1) consists of m coherent layers whose incidence medium is medium i’ and whose exit medium is medium (i’ + 1).
Let’s assume it is the first interface incidence medium / substrate. Then, to calculate transmittance and reflectance of the whole coherent/incoherent layer stack the dynamical matrix D0’1’ would be given by,
Di i1
M 11 2 2 M 21
2 2 det M M 12 M 21 , 2 M 11 M 12
2
(2.88)
if we set i’ = 0’. The matrix M in (2.88) you would calculate according to section 2.5.5. Your incidence medium for the coherent layer stack would be the incoherent medium at the left side (in our example medium 0’) and the exit medium the incoherent medium on the right side (the substrate). In general, the recipe for the calculation would be the following:
Divide the multilayer packet in incoherent layers and packets of coherent layers. Use for every coherent layer packet the formalism for coherent layers Assume that for a number of m coherent layers the layers 0 and m + 1 correspond to the incoherent layers i’- 1 and i’:
By that every simple interface or packet of coherent layers can be converted into the corresponding incoherent interface and the formalism for the incoherent layers can be used to calculate the transmittance and reflectance. Note that, if we would like to, we certainly could also construct 4x4 matrices in the same way as already explained for a coherent layer stack to perform the calculations for s- and p-polarization in one go. 48
2.5.9 Calculating the electric field strengths of a layered medium – mixed coherentincoherent multilayers The first steps are very similar to the one that we need to perform in order to calculate the electric field strength in a pure coherent layer stack. Therefore we divide in eqn. (2.83), which we repeat here for convenience, U m1 L m U 0 R 1 1 , D0 D j Pj D j Dm1 U m1 L U 0 R j 1
(2.89)
M
the matrix M into the following two parts,
i1 U 0 R U iR 1 1 D D P D D P , 0 j j j i i U 0 R U iR j 1
(2.90)
U m1 L m U iR 1 1 . Di D j Pj D j Dm1 U U j i 1 iR m1 L
(2.91)
and
M i m1
It is, not by accident, an incoherent layer that is adjacent to a coherent layer stack (cf. Scheme 2-10). However, this does not alter the calculation of the field inside the incoherent layer i’, it is just of importance because we will afterwards calculate the fields inside the coherent layer packet. Eqn. (2.91) can be evaluated starting by the intensity in the non-absorbing incidence medium U 0 R in the following way: M111 U iR M i m1 U 0 R . U iR 0
(2.92)
The intensities of the forward traveling beam U i Z and the backward traveling beam U i Z inside layer i' can then be obtained by: M111 U i Z di Z U iR di Z P i P i M U 0 R . di U iR di i m 1 0 U i Z
(2.93)
The overall intensity Ui Z is then the sum of U i Z and U i Z , Ui Z Ui Z Ui Z ,
(2.94)
where Z is measured from the interface (i’ - 1)i'. How do we now evaluate the electric field strength or intensity in the coherent layer stack that is sandwiched by the incoherent layers i' and i' + 1? Partly, we can revert to section 2.5.6. Why just partly? Because we need to identify E0R which is now definitely not equal to unity since the beam/wave has already passed to a number of incoherent layers or mixed layers before arriving at the left side of the interface between i' and i' + 1, therefore it has definitely been weakened by reflection 49
and, potentially, absorption. However, this is not our only problem! In section 2.5.6 we considered the last medium m + 1 a semi-infinite exit medium, which means that Em 1 L equals zero, since there is no backward travelling wave in such a medium. We do not need to inspect Scheme 2-10 more closely (but you are certainly allowed to), to see that medium i' + 1 is not semi-infinite and therefore Em 1 L not zero. But then we cannot use the formalism presented in section 2.5.6?! Yes we can! We simply split the calculation and first drive E0R from the left side through the coherent layer stack assuming Em 1 L to be zero. Afterwards we start a second calculation where we drive Em 1 L backwards through the coherent layer stack where we presume that E0R is zero. Then, we calculate the overall field strengths/intensities simply by the usual superposition. Ok, let’s start by assuming that the wave is coming from the left hand side. To do that we need, as mentioned above, E0R , which is not a problem at all. We obtain it simply from U iR in the following way: 1
E0R U iR 2 . E0 R U 0 R
(2.95)
As usual we can simply set U 0 R , the intensity of the beam that hits the first interface on the left side, unity and obtain from eqn. (2.92): 1
E0R
M1 2 1,0 M i m1 11 E0 R . 0
(2.96)
Based on E0R we can then calculate the electric field of the forward and backward travelling waves in layer j of the coherent layer packet based on eqn. (2.81): d j Z E jR dj Z E j Z M111 M P j P j E0 R . d j E jR d j j m1 0 Ej Z
(2.97)
Finally, we find the intensity in the jth layer of the coherent layer stack due to light coming from the left-hand side at Z by adding the fields due to the forward and backward travelling waves and squaring the sum: U jL Z E j Z E j Z E j Z . 2
2
(2.98)
Next, we determine the intensity due to the beam coming from the right side of the jth layer at position Z in the coherent packet U jR Z . To do this we need the intensity of the light beam that travels in negative direction in the incoherent layer i'+1: Em 1 L E0 R
U i1 L U 0 R
1
2 .
(2.99)
To take into account that Em 1 L 0 and that E0R 0 , we have to alter eqn. (2.81) accordingly:
50
Em1 L M12 E jR M j m1 M11 Em1 L . M j m1 1 E m1 L E jR
(2.100)
Based on eqs. (2.100) and (2.99) we obtain for Em 1 L in terms of E0 R : 1
Em1 L
M1 2 0,1 Pi1 M i1 m1 11 E0 R . 0
(2.101)
Now we are able to calculate the electric field of the forward and backward travelling waves in layer j of the coherent layer packet: MM12 d j Z E jR dj Z E j Z P j M j m1 11 E m1 L . P j E Z E d d j jR j j 1
(2.102)
The calculation of U jR Z is analogous to (2.98): U jR Z E j Z E j Z E j Z . 2
2
(2.103)
Fig. 2-7: Electric field intensity of a coherent layer on top of an incoherent substrate for normal incidence due to the incident light (left panel), the light that is reflected from the backside of the substrate and the sum effect. The coherent layer was non-absorbing with an index of refraction n = 1.5 and the substrate was Si.
Now that we know U jL Z and U jR Z at every point and for every wavenumber of interest (ok, so far I did not tell you, that you, or respectively your program, have to repeat the calculations based on 51
the preceding equations for every wavenumber in the spectral range of interest), we simply sum up both intensities according to:
U j Z U jL Z U jR Z .
(2.104)
For the simple case of a non-absorbing coherent layer on top of an incoherent substrate with light normally inciding and first hitting the layer and then the substrate, the effect due to U jR Z
certainly depends on the index of refraction of the substrate. For highly reflecting materials like Si, the effect can be considerable so that Si can certainly not be seen as a semi-infinite medium as is shown in Fig. 2-7.
2.6 Further reading Again, I highly recommend the book of Pocchi Yeh “Optical waves in layered media”, which I have used for large parts of this chapter as basis and inspiration.7 In addition, Born and Wolf’s “Principles of Optics”, is an excellent read8 and also, Hecht’s “Optics”.6 To understand phenomena related to total internal reflection I recommend “Internal Reflection and ATR Spectroscopy”13 by M. Milosevic, a book which is (necessarily for ATR spectroscopy) fully based on wave optics. Furthermore, it is obvious that this chapter cannot cover the whole range of electromagnetic phenomena, which can be observed in infrared spectra. A very important part that is missing are scattering effects. As long as we restrict ourselves to scalar media, Bohren and Huffman’s book “Absorption and scattering of light by small particles”14 is highly recommended (this, however, does not extend to the treatment of anisotropy, where the book would definitely require a revision and update). In addition to the paper of Centurioni,12 it might be also worthwhile to look at different papers which introduced and updated the (transfer) matrix formalism over the years. The first who introduced the matrix formalism for coherent layer stacks to calculate reflectance and transmittance was Abelès at the end of the 40ies of the last century.15 The first who extended the formalism to mixed coherent-incoherent layers seems to be Harbecke in 1986.16 Worth mentioning are also two papers of Ohta and Ishida, which seem to be the first who occupied themselves with the calculation of electric field strength and intensity at general positions within the layer.17-18 Finally, also partial coherence was introduced.19-20
52
3 Dispersion relations As spectroscopists, our lives are much more complicated than if we were just occupied with wave optics, because then it would often sufficient to solve a problem for a particular dielectric constant or index of refraction. As it stands, it would also be much less interesting, at least in my opinion. First of all we have to understand that the dielectric constant is not really constant, it can considerably change when we vary frequency. Therefore, from here on, we call it dielectric function to reflect this property. At the same time, we should better announce the index of refraction also as index of refraction function. However, the index of refraction function is not in the center of attention in this chapter, because it is, as we will see, only indirectly a material property. Unfortunately, this also means automatically that the index of absorption function as its imaginary part is neither a material property (therefore, it also does not really make sense to try to analyze an absorbance spectrum by a band fit). Schematically, the change of real and imaginary part of the dielectric function is illustrated in Fig. 3-1. In the regime of very high frequencies / wavenumbers where the energies of the waves are too high to excite the electrons of an atom, molecule or solid structure resonantly, the dielectric function has the constant value of unity, which means that there is essentially no difference to vacuum. Once the wavenumber is lowered and interacts with the gas, liquid or solid this changes dramatically since absorption sets in and the imaginary part of the dielectric function begins to have values > 0. At the same time, the real part changes correspondingly, so there seems to be some relation between both (indeed, as we will see later, there is a relation that connects the real with the imaginary part, which requires that changes in one part are reflected in the other). The shape of the bands in the UV-Vis can be very complicated depending on, whether we are looking at an isolated atom (line spectrum), a molecule (might have bands with shapes like in Fig. 3-1 or ionic or metallic solids (comparably complicated band structures). 0
-1
Dielectric function
10
10
-2
10
-3
10
/m -4
10
-5
10
-6
10
-7
10
-8
10
r' r, r''
r,¥V
r,¥Vis
0
Microwaves
-2
10
-1
10
0
10
UV-Vis
IR
1
10
2
10
r,¥
3
10
Wavenumber / cm Fig. 3-1: Schematic dispersion of the dielectric function of a liquid.
53
4
10 -1
5
10
6
10
Fortunately, those are (mostly!) not subject of this book and the only effect those excitations have, is that between the visible spectral range and the onset of excitable vibrations the dielectric function stays real with values greater unity (the deviation from one is the higher the more intense the absorptions in the UV-Vis spectral range become). In this range, the NIR which extends from the visible region down to 4000 cm-1, higher harmonics and combinations of the fundamental vibrations can occur. According to quantum mechanics, these are visible due to anharmonicity, but they usually have low to very low intensities so that we will neglect them in our discussion. Below 4000 cm-1 the fundamental vibrations occur. In gases, which we will also neglect, because deviations from the BeerLambert law are usually not due to wave optic related effects, the vibrations are accompanied by rotational transitions, which cause the bands to have a characteristic fine structure or a characteristic band shape with a local minimum in the center of the band, depending on the spectral resolution. For liquids, this fine structure usually does not occur (very much like in gases under higher pressure due to line broadening) as in solids, but for the latter because rotations are usually not possible (instead they are transformed to lattice vibrations). The number of fundamental vibrations depends on the number of atoms in the molecule or the unit cell and the symmetry of the vibrations (for details we refer to the textbooks of vibrational spectroscopy, see further reading at the end of this chapter). I just want to point out the selection rule, which helps to determine, if a certain vibration is infrared active: M ij . r r0
(3.1)
Accordingly, with a change of the distance between atoms r, the dipole moment must change to make the transition moment M ij non-zero.7 Microscopically, a change in the dipole moment is connected with a change of polarization on the macroscopic level and this will help us to connect the microscopic changes to the macroscopic theory that is represented by Maxwell’s equations. In general, the resonance wavenumber j shifts to lower wavenumber the larger the masses of the vibrating atoms are and the smaller the force constants of the bonds between the atoms are. Therefore, for many salts like e.g. the halides, the resonance wavenumbers (the only one, some or all) shift below 400 cm-1 which is usually considered as the boundary between the far infrared (FIR)8 and the mid infrared (MIR). If we go further down in the wavenumber region, something that we will not undertake, but for completeness, I will bring it up, the frequency is so low that permanent dipoles in liquids can follow the electromagnetic radiation. Accordingly, they can reach very high values of the real part of the dielectric function at low wavenumbers, in contrast to corresponding solids, wherein a free or semi-free rotation is not possible. Basically, we are coming back to the materials equations (eqn. (1.4)), which are necessary to complete Maxwell’s equations and provide a unique solution for the fields in matter: D 0 E P 0 r E .
(3.2)
In general, polarization induced by an electric field will counteract the electric field and reduce it as it is illustrated in Fig. 3-2. The reduction of the field lines is proportional to r . In contrast, the D-field does not change, e.g. in a plate capacitor, if matter is brought into the field. For the static case, if direct current is applied to the plate capacitor the situation will not change after a very short induction phase. 7
The dipole moment can also change from zero to some finite value. In other words, a permanent dipole moment is not needed, just a change of it. 8 Newfangled: Terahertz spectral range
54
When we now switch to alternating current, for low frequencies will not change strongly until the frequency is so high that the permanent dipoles are no longer able to follow the electric field (provided they could as in liquids). This will lower , which will afterwards stay approximately constant until the frequency is so high that dipole changes due to vibrations will no longer be possible. In the following we will convert this semi-quantitative model to a quantitative one which will be able to describe spectra in the infrared spectral region to an astonishing high degree, as you will soon see in this chapter.
Fig. 3-2: An electric field induces and/or orients dipoles in matter, which are antiparallel to the applied field.
55
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2. 3. 4.
5. 6. 7. 8. 9. 10.
11.
12. 13. 14. 15. 16. 17. 18.
19. 20.
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. Chalmers, J. M., Mid-Infrared Spectroscopy: Anomalies, Artifacts and Common Errors. In Handbook of Vibrational Spectroscopy, John Wiley & Sons, Ltd: 2006. Matossi, F., Ergebnisse der Ultrarotforschung. In Ergebnisse der Exakten Naturwissenschaften: Siebzehnter Band, Hund, F., Ed. Springer Berlin Heidelberg: Berlin, Heidelberg, 1938; pp 108-163. 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. Zbinden, R., Infrared spectroscopy of high polymers. Academic Press: 1964. Hecht, E., Optics,4/e. Pearson Education: 2002. Yeh, P., Optical Waves in Layered Media. Wiley: 2005. Born, M.; Wolf, E.; Bhatia, A. B., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press: 1999. Schaefer, C.; Matossi, F., Das Ultrarote Spektrum. Verlag von Julius Springer, Berlin: 1930. Mayerhöfer, T. G.; Dunken, H. H.; Keding, R.; Rüssel, C., Interpretation and modeling of IRreflectance spectra of glasses considering medium range order. Journal of Non-Crystalline Solids 2004, 333 (2), 172-181. Airy, G. B., VI. On the phænomena of Newton's rings when formed between two transparent substances of different refractive powers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1833, 2 (7), 20-30. Centurioni, E., Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers. Appl. Opt. 2005, 44 (35), 7532-7539. Milosevic, M., Internal Reflection and ATR Spectroscopy. Wiley: 2012. Bohren, C. F.; Huffman, D. R., Absorption and scattering of light by small particles. Wiley: 1983. Abelès, F., Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Ann. Phys. 1950, 12 (5), 596-640. Harbecke, B., Coherent and incoherent reflection and transmission of multilayer structures. Appl. Phys. B 1986, 39 (3), 165-170. Ohta, K.; Ishida, H., Matrix formalism for calculation of electric field intensity of light in stratified multilayered films. Appl. Opt. 1990, 29 (13), 1952-1959. Ohta, K.; Ishida, H., Matrix formalism for calculation of the light beam intensity in stratified multilayered films, and its use in the analysis of emission spectra. Appl. Opt. 1990, 29 (16), 24662473. Prentice, J. S. C., Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures. Journal of Physics D: Applied Physics 2000, 33 (24), 3139. Katsidis, C. C.; Siapkas, D. I., General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference. Appl. Opt. 2002, 41 (19), 3978-3987.
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