Optical Properties of Dielectric Films

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Chapter 12

OPTICAL PROPERTIES OF DIELECTRIC AND SEMICONDUCTOR

THIN FILMS

I. ChambouleyronInstituto de Fsica Gleb Wataghin, Universidade Estadual de CampinasUNICAMP,13083-970 Campinas, SP, Brazil

J. M. MartnezInstituto de Matemtica, Universidade Estadual de CampinasUNICAMP,13083-970 Campinas, SP, Brazil

Contents

1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. The General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. LightMatter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4. Basic Formulae for Transmitted and Reflected Waves . . . . . . . . . . . . . . . . . . . . . . . . 71.5. Nonnormal Incidence and Linear-System Computations . . . . . . . . . . . . . . . . . . . . . . . 141.6. The Effect of a Thick Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7. Computational Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.8. Optimization Algorithm for Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2. Applications of Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1. Antireflection Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2. A Reverse Engineering Problem: The Retrieval of the Optical Constants and the

Thickness of Thin Films from Transmission Data . . . . . . . . . . . . . . . . . . . . . . . . . . 223. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1. THEORY

1.1. Historical Note

Interference phenomena of light waves occur frequently in na-ture, as in the colored reflection produced by feathers and wingsof birds and many insects. The interference pattern given byoily substances onto water has been certainly noticed by mansince ancient times. The origin of such effects, however, re-mained unclear until relatively recent times. The first systematicstudy on interference was undertaken by Newton [1], the well-known Newton rings, obtained from multiple reflections of lightin the tiny air layer left between a flat and a little convex pieceof glass. The effect, however, was not properly understood at

that moment due to Newtons belief that light beams were thepropagation of a stream of particles, Newton rejecting the ideaof light being a wavelike perturbation of the ether. In 1690,Huygens [2] published his Treatise on Light, in which he pro-posed that light was a disturbancepropagating through the etheras sound moves through air. There is no reference to wave-lengths, though, or a connection of these and color. In 1768,

Euler [3] advanced the hypothesis that the colors we see de-pends on the wavelength of wave light, in the same way asthe pitch of the sound we hear depends on the wavelength ofsound. In 1801, in the lectureOn the theory of light and col-ors, delivered before the Royal Society, Young sketched thecurrent theory of color vision in terms of three different (pri-

Handbook of Thin Films Materials, edited by H.S. NalwaVolume 3: Volume Title

Copyright 2001 by Academic PressAll rights of reproduction in any form reserved.

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2 CHAMBOULEYRON AND MARTNEZ

mary) colors. Youngs experiments on the interference of twolight beams [4] were made around 1800 and published in 1807.They gave the ultimate proof of the wave nature of light andallowed the determination of the wavelength of visible light. Inthe words of Herschel [5], . . . Dr. Young has established a prin-ciple in optics, which, regarded as a physical law, has hardly its

equal for beauty, simplicity, and extent of applications, in thewhole circle of science. . . . The experimental means by whichDr. Young confirmed this principle, which is known in optics bythe name of interference of the rays of light, were as simple andsatisfactory as the principle itself is beautiful. . . . Newtons col-ors of thin films were the first phenomena to which the authorapplied it with full success. Based on the experiments of Maluson the polarization of light by ordinary reflection at the surfaceof a transparent body, Young proposed a few years later thatlight waves were transverse perturbations of the ether and notlongitudinal, as sound waves. The consequences of the findingkept physicists busy for the next hundred years. It was Fresnelthat put Youngs results on a firm mathematical basis. He estab-

lished the relations between the amplitudes of the reflected andrefracted waves and the incident waves, known as the Fresnelformulae.

Optical coatings remained a subject of laboratory curiosityin the sense that no industrial applications occurred until thetwentieth century, with the notable exception of mirror produc-tion. Until the late Middle Ages, mirrors were simply polishedmetals. At a certain unknown moment, the Venetian glass work-ers introduced glass silvering using an amalgam of tin andmercury [6]. The method lasted until the nineteenth century,when the backing of glass was made with silver. The develop-ment of vacuum apparatus techniques allowed the evaporationof a large class of materials, with the ensuing application tooptical systems, among others. The methods to deposit thinfilms in a controllable way increased at a rapid pace, as didthe applications to different fields of industry. Today, advancedelectronic and optical devices are manufactured involving sin-gle or multilayered structures of materials of very differentnature: electro-optic and optical materials, semiconductors, in-sulators, superconductors, and all kinds of oxide and nitridealloys. This rapid development demands accurate depositionmethods, sometimes with in situ characterization, as well aspowerful algorithms for fast corrections to predicted perfor-mance.

1.2. The General Problem

Figure 1 sketches the general problem. A beam of mono- orpolychromatic light falls onto a p-layered optical structure.Within the structure, or at the interfaces, light may be absorbedand/or scattered. In general, the thickness of the different opti-cal coatings composing the structure varies between 0.05 and afew times the wavelength of the incident light beam. An abruptinterface between materials of different optical properties (ora discontinuity taking place in a distance small compared withthe light wavelength) provokes a partial reflection of the light

Fig. 1. The figure sketches the general optical coating problem. A light beamimpinges onto a stack of dielectric thin layers supported by thick substrate. Theradiation is partly transmitted and partly reflected, both depending on photonenergy and layer thickness. Part of the radiation may be internally absorbed.

beam. Sometimes, a change of phase also occurs. As the dis-tances between layer boundaries are comparable to the photonwavelength, multiplyreflected or transmitted beams are coher-ent with one another. Consequently, the total amount of lightreflected and transmitted by the structure results from the al-gebraic sum of the amplitudes of these partially reflected andpartially transmitted beams. This property is at the base of alarge number of applications. Depending on the specific case,the substrate (S) supporting the structure may be transparent orabsorbing. Self-supported structures may be built also but, inwhat follows, the existence of a thick supporting substrate will

be assumed. Hence, in the present context, a thin film is equiv-alent to an optical coating.The structure depicted in Figure 1 suggests three different

problems:

(a) Optical response. Calculation of the spectral responseof ap-layered structure deposited onto a substrate ofknown properties. The optical constants of the coatingmaterials as a function of photon wavelength aregiven, as well as the thickness of each layer.

(b) Structure design. The problem here is to find theappropriate materials, the number of layers, and theirrespective thicknesses in order to meet a desired

spectral performance.(c) Reverse optical engineering. Retrieval of the opticalparameters and the thicknesses of the layerscomposing the structure from measured spectralresponses.

In mathematical terms, the optical responseproblem is adirectproblem. The phenomenon is governed by a partial differ-ential wave equation (coming from Maxwell equations) wherewe assume that the parameters and boundary conditions aregiven. The direct problem consists of computing the state of thewave within each layer, using the information mentionedabove.


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OPTICAL PROPERTIES OF DIELECTRIC AND SEMICONDUCTOR THIN FILMS 3

Analytic solutions can be given if we assume that the incidentwave is a purewave and that the boundary layers are regular.In more complicated situations, numerical solutions are nec-essary. However, even in the case where analytic calculationsare possible, the practical computation of the response can bevery costly. This is because pure waves do not exist in most

optical experiments, and the true response corresponds to anaverage among many waves of different wavelength. Althoughit is possible to give the transmitted and reflected energies ofpure waves in a closed analytic form, their analytic integrationcannot be done and the numerical integration process is, com-putationally, very expensive.

Designand reverse engineeringproblems are inverse prob-lems in the sense that, in both cases, the response of the system(or a part of it, such as transmitted or reflected energy for a setof wavelengths) is known, but (some of) the parameters pro-ducing this response must be estimated. In design problems,we have a target response but there is no a priori knowl-edge of whether or not it is possible to obtain that response for

some set of boundary conditions. In reverse opticalproblems,the response of the structure has already been measured butthe specific conditions under which it has been generated arenot known. The key tool for an efficient solution ofdesignandreverse engineeringproblems is an adequate code that solvesthe optical response problem. All computational devices usedto solve those problems obey the following general scheme:(i) try a set of parameters; (ii) solve the optical response prob-lems using them; (iii) accept the parameters if the response issatisfactory and change them in a suitable way if it is not. Thisis also the scheme of most optimization algorithms, which try tominimize (or maximize)some objective function subject to a setof constraints. The choice of the objective (or merit) function

to be used depends on the nature of the problem. Smoothnessconsiderations usually lead to the use of some kind of sum ofsquares of experimental data.

Some features complicating the optimization ofdesignandreverse engineeringproblems are:

(1) Local nonglobal minimizers. In some problems, the so-lution is well defined but, out of this solution, the merit functionpresents a highly oscillatory behavior with similar functionalvalues. In other words, the function has many local (nonglobal)minimizers whichare useless for design or estimation purposes.Most efficient optimization algorithms have guaranteed conver-gence to local minimizers (in fact, something less than that) but

not to global minimizers, so they tend to get stuck in the attrac-tion basin of those undesirable local minimizers.

(2) Ill-posedness. Some mathematical problems can behighly underdetermined, the solution not being unique. Addingto this feature the fact that additional errors due to small inade-quacies of the model can be present, we have situations wherean infinite set of mathematical approximate solutions can befound, but where thetrue physical solutionmay not correspondto the point with smallest merit function value. This feature istypical in the so-called inverse problemsin the estimation lit-erature. The ill-posedness problem can be partially surmounted

by introducing, in the model, some prior information on the be-havior of the parameters to be estimated. When our knowledgeof this behavior is poor, we can use regularization processes [7].Finally, instead of the trueexpensive models, simplified directalgorithms can be used which, sometimes, are sufficient to findsuitable estimates of the parameters, or a good enough approx-

imation to them.(3) Expensive models. Problems (1) and (2) become moreserious if the solution of the associated direct problem iscomputationally expensive. Optimization algorithms need, es-sentially, trial-and-errorevaluations of the optical response and,obviously, if this evaluation demands a great deal of computertime, the possibility of obtaining good practical solutions in areasonable period of time is severely diminished. Partial solu-tions to all the difficulties mentioned above have been found.The multiple-local-minimizers problem can be attacked by theuse ofglobal optimization algorithmsthat, in principle, are ableto jump over undesirable local nonglobal solutions. However,all these procedures are computationally expensive, and their

effectiveness is linked to the possibility of overcoming this lim-itation.

For the sake of completeness, let us note that all these reme-dies may fail. No global optimization method guarantees thefinding of global minimizers of any function in a reasonabletime. The information that must be introduced into the prob-lem to enhance the probability of finding a good estimate ofthe parameters is not always evident. Moreover, it is not alwaysclear in which way the information must be introduced in themodel. Finally, it is not known how much an optical responseproblem can be simplified without destroying its essential char-acteristics. As in many inverse problems exhaustively analyzed

in the literature, each particular situation must be studied fromscratch (although, of course, experience in similar situations isquite useful) and can demand original solutions. In this review,we consider the reverse solution of some simple optical struc-tures.

The structure shown in Figure 1 calls for some additionalconsiderations.

(1) In principle, the angle of incidence of the impingingbeam may be any angle, the simplest case being that of nor-mal incidence. The mathematical formulation of the problemfor light arriving at angles other than normal to the surface ofthe coatings becomes more involved. The general case is treated

in a coming section.(2) The coatings and the substrate are optically homoge-neous and isotropic. The surfaces and the interfaces betweenlayers are perfectly flat and parallel to each other. In otherwords, we will not consider the case of rough surfaces produc-ing some scattering of the light, nor the existence of thicknessgradients or inhomogeneities. The effects produced by these de-viations to the ideal case will not be discussed. See [8, 9].

(3) In this chapter, we review the properties and applica-tions of semiconductor and dielectric thin films; that is, metalcoatings are not included.


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1.3. LightMatter Interaction

Dielectric and semiconductor coatings are films of materialshaving strong ionic or directed covalent bonds. In most cases,they are transparent to visible and/or infrared light. The interac-tion of the electromagnetic radiation with these films is treatedby applying boundary conditions to the solutions of Maxwell

equations at the boundary between different media. In the fieldof optical coatings, the wavelength of the light is always verymuch larger than interatomic dimensions. Thus the interactionof light and matter is averaged over many unit cells. As aconsequence, the optical properties within each layer can bedescribed macroscopically in terms of phenomenological pa-rameters, the so-called optical constants or optical parameters.As shown below, these are the real and the imaginary parts ofa complex index of refraction n. The real part, n(), is the ra-tio of the velocity of light in vacuum to the velocity of lightof wavelength()in the material. The imaginary part, (),is an attenuation coefficient measuring the absorption of lightwith distance. Using Maxwell equations, it is possible to relatethese frequency-dependent constants to other optical param-eters such as the dielectric constant and the conductivity.

The coating materials are composed of charged particles:bound and conduction electrons, ionic cores, impurities, etc.These particles move differently with oscillating electric fields,giving rise to polarization effects. At visible and infrared lightfrequencies, the only contribution to polarization comes fromthe displacement of the electron cloud, which produces an in-duced dipole moment. At frequencies smaller than these, othercontributions may appear, but they are of no interest to thepurpose of the present work. The parameters describing theseoptical effects, that is, the dielectric constant , the dielec-

tric susceptibility , and the conductivity, can be treated asscalars for isotropic materials. In what follows, dielectric andsemiconductor coating materials are considered nonmagneticand have no extra charges other than those bound in atoms.

To find out what kind of electromagnetic waves exist insidedielectric films, we take= Pandj = P/t, whereisan effective charge, P is the polarization induced by the elec-tromagnetic wave, assumed to be proportional to the electricfield, andj is the corresponding current density averaged overa small volume. Under these conditions, average field Maxwellequations in MKS units read:

E= P

0 E= B

t B= 0

c2 B= t

P

0+ E

where the symbols have their usual meaning. Note that the nor-mal component of the electric field E is not conserved at theinterface between materials of different polarizability. Instead,D = 0 + P, called electrical displacement, is conserved acrosssuch interfaces. The solutions to these equations have the form

of harmonic plane waves with wave vectork:

E= E0exp

i(t k r)H= H0exp

i(t k r)

and represent a wave traveling with a phase velocity / k=c/n, where c is the speed of light in vacuum and n is the

index of refraction. When optical absorption is present, thewave vector and the index are complex quantities. From theMaxwell equations, a dispersion relation k 2 = (/c)2 is ob-tained relating the time variation with the spatial variation ofthe perturbation.

In general, then, the wave vectorkand the dielectric constantare complex quantities, that is,k= k1 ik2and= 1 i2.It is useful to define a complex index of refraction:

n k c

= n i

For isotropic materials,k1and k2are parallel and

1= n2

22= 2n

with the converse equations,

n2 = 12

1 +

21 + 22

1/2 2 = 1

2

1 + 21+ 221/2In the photon energy region whereis real,n = 1/2 is also

real and the phase (/ k)andgroup(/k) velocities are equaltoc/ n. In general, the velocity is reduced to v()=1/c()in a medium of a complex dielectric constant c. The real partofndetermines the phase velocity of the light wave, the imag-inary part determining the spatial decay of its amplitude. Theabsorption coefficient measures the intensity loss of the wave.

For a beam traveling in the zdirection, I(x) = I (0) exp(z),which means= 2/c = 4k/. See [8].

Optical Properties of Dielectric Films

The main absorption process in semiconductors and dielectricsoriginates from the interaction of light with electrons [1012].If the photon has a frequency such that its energy matches theenergy needed to excite an electron to a higher allowed state,then the photon may be absorbed. The electron may be anion-core electron or a free electron in the solid. If the energyof the incoming photon does not match the required excita-tion energy, no excitation occurs and the material is transparent

to such radiation. In nonmetal solids, there is a minimum en-ergy separating the highest filled electron states (valence band)and the lowest empty ones (conduction band), known as theenergy band-gap. Electron transitions from band to band con-stitute the strongest source of absorption. In dielectrics, suchas glass, quartz, some salts, diamond, many metal oxides, andmost plastic materials, no excitation resonances exist in thevisible spectrum, because the valence electrons are so tightlybound that photons with energy in the ultraviolet are neces-sary to free them. Ideally, photons having an energy smallerthan the band-gap are not absorbed. On the contrary, metals are


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good reflectors and are opaque to visible and infrared radia-tion. Both effects derive from the existence of a large densityof free electrons which are able to move so freely that noelectric field may propagate within the solid. The reflectingproperties of metals find numerous optical applications, as inmirrors.

The optical properties of semiconductors lie between thoseof metals and those of insulating dielectrics. Semiconductorsare normally transparent in the near infrared and absorbing inthe visible spectrum, whereas the absorption in dielectrics isstrong in the ultraviolet. Thus, the fundamental absorption edgeof semiconductors lies, approximately, between 0.5 eV (2500 nm) and 2.5 eV ( 500 nm). Within a small energyrange around the fundamental absorption edge, semiconductorsgo, ideally, from high transparency to complete opacity. How-ever, the presence of impurities, free conduction electrons orholes, and/or other defect states may affect the transparencyof semiconductor and dielectric materials at photon energiessmaller than the band-gap.

The situation just depicted is ideal in the sense that we haveconsideredperfectdielectric and semiconductor crystalline ma-terials. In most cases, however, the deposited optical coatingsare either microcrystalline or amorphous. Microcrystalline lay-ers are formed by the aggregation of randomly oriented smallcrystals having a typical size of the order of the film thick-ness. The regions between crystallites, or grain boundaries,are highly imperfect regions having a large density of local-ized electron states located at energies between the valence andconduction bands. These surface or interface defect states con-stitute a source of absorption; that is, electrons are excited fromthe valence band to localized states and/or from them to theconduction band. In all cases, however, the absorption due todefects is smaller than that of the fundamental edge, but it mayaffect in a nonnegligible way the optical properties of the coat-ing.

The fundamental characteristic of an amorphous materialis the topological disorder. The atoms of an amorphous net-work do not form a perfect periodic array as in crystallinematerials. The absence of periodicity, or of long-range order,has important consequences on the optical properties of thefilms. Disorder induces the appearance of localized electronstates within the band-gap. These localized states are of twotypes: coordination defects, also called dangling bonds or deepdefects, and valence and conduction band tail states (shallow

states). Both contribute to the absorption of light. Most impor-tant, due to the lack of periodicity, the electron wave vectorkis no longer a good quantum number and the selection rules,valid for optical transitions in crystals, break down. As a con-sequence, more transitions are allowed between electron stateswith an energy separation equal to the photon energy. The ab-sorption coefficient due to-band-to band transitions is muchlarger in amorphous dielectrics and semiconductors than intheir crystalline parents. The density of defect states and thecorresponding optical absorption strength depend strongly ondeposition methods and conditions.

The Absorption Edge of Semiconductors and Dielectrics

In general, the process of electron excitation by photon ab-sorption is very selective. The sharpness of absorption linesis clearly noted in the dark lines of atomic spectra. The re-quirement of energy matching, called resonance, must also bemet in solids, but solids have such an abundant range of possi-

ble excitations that absorption occurs in a broad energy range.Even so, many of the optical properties of materials aroundresonances can be understood in terms of the properties of adamped harmonic oscillator, often called the Lorentz oscillator.In the model, it is assumed that the electrons are bound to theircores by harmonic forces. In the presence of an external fieldE0exp(it), their motion is given by

m2r

t2+ m r

t+ m20r= eE exp(it)

wherem and e are the mass and the charge of the electron, 0is the natural frequency of the oscillator, and is a dampingterm. Solving for the electron displacementr

=r0exp(

it),

one gets

r= eE/m(20 2) i

The response of a system having a density N /Vof such os-cillators can be obtained in terms of the induced polarization:

P= erN/V= (e2N/mV)E

(20 2) i The dielectric constant corresponding to this polarization is

= 1 i2= 1 const 1

( 20

2

i)

with

1= n2r 2 = 1 K(20 2)

(20 2)2 + 222= 2nr = K

(20 2)2 + 22whereKis a constant. Figure 2 shows the shape of the real andimaginary parts of the dielectric constant of a Lorentz oscilla-tor. This model is extremely powerful because, in real systems,it describes quite accurately not only the absorption betweenbands, but also that corresponding to free electrons and phononsystems. The reason for the general applicability of the Lorentz

oscillator is that the corresponding quantum-mechanical equa-tion for , not discussed here, has the same form as the classicalequation for the damped oscillator.

In a solid dielectric, resonances at different frequencies hav-ing different loss values contribute to the dielectric responsewith terms having a shape similar to that shown in Figure 2.The overall dielectric response results from the addition of allthese contributions, as shown in Figure 3 for a hypotheticalsemiconductor. For photon energies smaller than the band-gap( < EG), the material is transparent or partially absorbing.This is the normal dispersionfrequency range in which both


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Fig. 2. Characteristic behavior of the real and the imaginary part of the di-electric constant of a classical damped electron oscillator as a function of therelative excitation frequency (Lorentz oscillator).

Fig. 3. Real and imaginary parts of a hypothetical semiconductor asa functionof photon energy.

Fig. 4. Single-effective-oscillator representation of the index of refraction ofcrystalline silicon and of an optical glass as a function of the square of photonenergy.

the index of refraction and the absorption increase with pho-ton energy. Moreover, the slope of n/() also increaseswith increasing photon energy. It is important to remark atthis point that the refractive index dispersion data below theinterband absorption edge of a large quantity of solids canbe plotted using a single-effective-oscillator fit of the formn2

1

=FdF0/(F

20

22), whereF0is the single-oscillator

energy, and Fdis the dispersion energy [13]. These parame-ters obey simple empirical rules in large groups of covalent andionic materials. Figure 4 shows the single-effective-oscillatorrepresentation of the refractive index of an optical glass and ofcrystalline silicon. The fit is very good for a wide photonenergyrange.

Let us end the section with a few words on the shape ofthe absorption edge in crystalline and amorphous layers. Theabsorption coefficient has an energy dependence near the ab-sorption edge expressed as ( EG)s , where s is aconstant. In the one-electron approximation, the values takenbysdepend on the selection rules for optical transitions and on

the materials band structure. For allowed transitions, s= 1/2for direct transitions (i.e., GaAs) and s= 2 for indirect tran-sitions (i.e., Si or Ge). In amorphous materials, the absorptionedge always displays an exponential dependence on photon en-ergy. The characteristic energy of the exponential edge, calledUrbach energy, depends on the material and on deposition con-ditions [14, 15].

For reasons that will become clear in a forthcoming section,knowledge of the physics behind the dependence of the opti-cal constants on photon energy may be of great help in solvingoptical reverse engineering problems.


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1.4. Basic Formulae for Transmitted and Reflected Waves

1.4.1. Single Interface

Consider the complex wave function

uI(z, t) =

EIexp

i

t

k0z

(1)

where is real and nonnegative, whereasEI andk0are com-plex. By (1), the wavelength is 2/|(k0)|. ((z) will denotealways the real part of the complex number zand (z)denotesits imaginary part.) So, k0 = 2

(2)

Let the complex numbern0= n0 i0denote the generalized index of refraction in the x domainofu, and

k0= n0c (3)If(k) >0, the wave travels in the forward direction, whereasif(k) L.

Continuity inz = Limposes:EIexpitk0L+ERexpiRtkRL=ETexp iTtkTL

for all realt. So,EIexpik0L exp[it] +ERexpikRL exp[iRt]=ETexpikTL exp[iTt]

for allt R. Since this identity is valid for allt, we must haveR= T=

Now, the direction of the reflected wave must be opposite tothat of the incident wave, and the direction of the transmittedwave must be the same. So, by (3),

kR= k0 kT=n1n0k0Therefore,

EIexpik0L+ERexpik0L =ETexpin1n k0L

(6)

Fig. 5. Incident, reflected, and transmitted waves at an abrupt interface be-tween two different media.

Now, using continuity of the derivative with respect to z inz = L, and the above expressions ofR ,T,kR ,kT, we haveik0EIexpitk0z+ ik0ERexpit+k0z

= ikn1n0 ETexp

i

tn1n0k0z

forz = Land for allt R. Therefore,

EIexp

i

t

k0L

ERexp

i

t+

k0L

=n1n0 ETexpit n1n0k0L

Thus, EIexpik0L exp[it] ERexpik0L exp[it]=n1n0 ETexp

in1n0k0L

exp[it]

which implies

EIexpik0LERexpik0L =n1

n0ETexpin1

n0k0L

(7)Adding (6) and (7), we get

2EIexpik0L = 1 + n1n0ETexpin1n0k0L

Therefore,

1 + n1n0ET= 2EIexpik0L1 n1n0

so,

ET= 2n

n +

n1EIexpik0Ln1

n0

1

(8)


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Subtracting (7) from (6), we obtain

2ERexpik0L= 1 n1n0ETexpin1n0k0L

=

1 n1n0

2n0n0 +

n1

EIexp

ik0L

So, ER=n0 n1n0 +n1 EIexp2ik0L (9)

Summing up:If the incident wave acting onz < LisuI(z,t) =EIexpitk0z

then the reflected wave (z < L) is

uR(z,t) =n0 n1n0 +n1 EIexp2ik0L expit+k0z (10)and the transmitted wave (z > L) is

uT (z,t)= 2n0n0 +n1 EIexpik0Ln1n0 1 exp

i

tn1n0k0z

(11)

1.4.2. Recursive Formulae formLayers

Assume now, as shown in Figure 6, a system of m layerswith generalized index of refractionn0,n1, . . . ,nm1, respec-tively, separated by abrupt interfaces at z= L1, z= L2, . . . ,z= Lm1, whereL1 < L2 Lm1), thek -coefficient of the transmitted wave will bek0nm1/n0. The incident wave will also generate a reflectedwave inz < L1whosek-coefficient will be k0.

Mutatis mutandi, if the incident wave is defined in z > Lm1and given by EIexp[i(t+km1z)], the transmitted wave inz < L1 has a k -coefficient equal tokm1n0/nm1 and the re-flected wave in z > Lm1 will have

km1 as k-coefficient.

The-coefficient is always the same.

In this section, we deducehow the E-coefficient of the trans-mitted and reflected waves depends on the parameters of the in-cident wave. We define the coefficients t0,m1, r0,m1, tm1,0,rm1,0in the following way:

(a) t0,m1Eis theE-coefficient of the transmitted wave inthenm1layer generated by the wave (1) in layer n0.

(b) r0,m1Eis theE-coefficient of the reflected wave inthen0layer generated by the wave (1) in layer n0.

(c) tm1,0Eis theE-coefficient of the transmitted wavein the layern0, generated by a wave (1) defined inlayer

nm1.

Fig. 6. Structure of an m-layered optical film system. The figure illustratesmultiple reflections inside each layer characterized by a refractive index and anextinction coefficient.

Fig. 7. Sketch of the multiple transmitted and reflected waves in a multilayersystem.

(d) rm1,0Eis the E-coefficient of the reflected wave inthenm1layer generated by a wave (1) defined inlayer

nm1.

For completeness, we will write, when necessary:t0,m1= t0,m1(k)

= t0,m1k0,n0,n1, . . . ,nm1, L1, L2, . . . , Lm1

r0,m1= r0,m1(k)= r1,m

k0,n0,n1, . . . ,nm1, L1, L2, . . . , Lm1tm1,0= tm1,0(k)

= tm,1km1,n0,n1, . . . ,nm1, L1, L2, . . . , Lm1

rm1,0= rm1,0(k)= rm1,0

km1,n0,

n1, . . . ,

nm1, L1, L2, . . . , Lm1


9/30


From the previous section, we know that

t0,1(k)= 2n0n0 +n1 exp

ik0L1(n1 n0)n0

(12)

r0,1(

k)=

n0

n1

n0 +

n1

exp

2i

k0L1

(13)

t1,0(k)= 2n1n0 +n1 exp ik1L1(n0 n1)n1 (14)r1,0(k)=n1 n0n1 +n0 exp2ik1L1 (15)

Computingt0,m1

Let us compute t0,m1for m 3 assuming thatt0,m2, r0,m2,tm2,0, rm2,0have already been computed.

Consider an incident waveE0exp[i(tk0z)] in layern0,defined forz < L1. The transmitted wave inz > Lm1will bethe sum of an infinite number of small-waves resulting frompartial transmission and partial reflection at the interfaces. Thek-coefficient of all these transmitted small-waves will be givenby

kof the small-waves =k0nm1n0

The E-coefficient of the first small-wave comes from a di-rect transmission from layern0to layernm2(throughL1, L2,Lm2) and, finally transmission throughz = Lm1. Therefore,

Eof the 1st small-wave

= t0,m2

E0

2

nm2

nm2 +

nm1

expik0Lm1nm2n0 nm1nm2 1The E-coefficient of the second small-wave comes from cal-culating the transmission from layern0to layernm2(throughL1, L2, Lm2) [reflection in z= Lm1, reflection in[Lm2,. . . , L1]] and, finally, transmission throughz = Lm1.

Therefore,

Eof the 2nd small-wave

= t0,m2E0nm2 nm1nm2 +

nm1

exp

2i

k0nm2n0

Lm1

r

m2,0k0nm2n0 2nm2nm2 +nm1 exp

ik0Lm1(nm1 nm2)n0

Proceeding in the same way for all >2, we have

Eof the th small-wave

= t0,m2E0nm2 nm1nm2 +nm1 exp2i

k0nm2n0 Lm1

rm2,0k0nm2n0

1

2nm2nm2 +nm1 exp

ik0Lm1(nm1 nm2)n0

Adding all the above small-waves, we obtain

t0,m2E02

nm2

nm2 +nm1exp

i

k0Lm1(

nm1

nm2)

n0

1 nm2 nm1nm2 +nm1 exp2ik0nm2n0 Lm1 rm1,0

k0nm2n0

1Therefore,

t0,m1= t0,m1(k)= t0,m2 2nm2nm2 +nm1 exp

ik0Lm1(nm1 nm2)n0

1 nm2

nm1

nm2 +nm1 exp2ik0

nm2

n0 Lm1 rm2,0

k0nm2n0

1(16)

A brief explanation on notation becomes necessary. Whenwe write, for example, rm2,0((k0nm2/n0)), this means thefunction rm2,0 applied on(k0nm2/n0). When we writet0,m2, this meanst0,m2(k). When, say,tijis used without ar-gument, it means tij(k). Otherwise the argument is explicitlystated, as in the case ofrm2,0in the formula above.

Computingr0,m1

Assume, again, that the incident waveE0exp[i(tk0z)]isdefined forz < L1. The reflection in [L1, L2, . . . , Lm1] gen-erates a reflected wave, also defined inz < L1. This wave is thesum of an infinite number of small-waves and an additionalwave called here the zeroth-wave. Thek-coefficient of all thesewaves is k0.

The zeroth-wave is the first reflection of the incident wave at[L1, . . . , Lm1], so

Eof the 0th wave = r0,m2k0E0

TheE-coefficient of the first small-wave comes from trans-mission through L1, . . . , Lm

2, followed by reflection at z

=Lm1and, finally, transmission throughLm2, . . . , L1. So,Eof the 1st small-wave

= t0,m2E0nm2 nm1nm2 +nm1 exp2i

k0nm2n0 Lm1

tm2,0k0nm2n0

The E-coefficient of the second small-wave comes from

transmission through L1, . . . , Lm2, followed by reflection atz= Lm1 [then reflection at[Lm2, . . . , L1], then reflection


10/30


atz = Lm1] and, finally, transmission through Lm2, . . . , L1.So,


= t0,m2

E0

nm2

nm1

nm2 +

nm1

exp

2i

k0nm2

n0

Lm1

rm2,0k0nm2n0 nm2 nm1nm2 +nm1 exp

2i

k0nm2n0 Lm1

tm2,0k0nm2n0

So far, theE -coefficient of the th small-wave comes from

transmission through L1, . . . , Lm2, then reflection at z =Lm1 [then reflection at[Lm2, . . . , L1], then reflection atz = Lm1] 1 times and, finally, transmission throughLm2, . . . , L1. So,

Eof the th small-wave

= t0,m2E0nm2 nm1nm2 +nm1 exp2ik0nm2n0 Lm1

rm2,0k0nm2n0

nm2 nm1nm2 +nm1 exp

2i

k0nm2n0 Lm11

tm2,0k0nm2n0

Therefore, theE-coefficient of the reflected wave defined in

z < L1is

r0,m2(k)E0 + t0,m2E0nm2 nm1

nm2 +

nm1

exp2ik0nm2n0 Lm1tm2,0k0nm2n0

1 rm2,0k0nm2n0

nm2 nm1nm2 +nm1 exp

2i

k0nm2n0 Lm11

So,

r0,m1(k)= r0,m2(k) + t0,m2nm2 nm1nm2 +nm1 exp

2i

k0

nm2

n0Lm1

tm2,0

k

nm2

n0

1 rm2,0k0nm2n0 nm2 nm1nm2 +nm1 exp

2i

k0nm2n0 Lm11

(17)

Computingtm1,0

Assume that the waveEm1exp[i(t+km1z)] is defined forz > Lm1, where the refraction index isnm1(see Fig. 6). Thek-coefficient of the transmitted wave defined for z < L1, as

well as the k-coefficient of all the small-waves whose additionis the transmitted wave, will bekm1n0/nm1.

As in the previous cases, the transmitted wave is the sum ofinfinitely many small-waves.

The E-coefficient of the first small-wave is obtained bytransmission through z= Lm1 (from right to left) fol-lowed by transmission through [Lm2, . . . , L1]. So,

Eof the 1st small-wave

=Em1 2nm1nm1 +nm2 exp

ikm1Lm1(nm2 nm1)nm1

tm2,0km1nm2nm1

The E-coefficient of the second small-wave is obtained bytransmission through Lm1(from right to left), followed by [re-flection at [Lm2, . . . , L1], reflection at z = Lm1] and, finally,transmission through [Lm2, . . . , L1]. So,



ikm1Lm1(nm2 nm1)nm1

rm2,0nm2km1nm1

nm2 nm1nm2 +nm1 exp

2ikm1nm2Lm1nm1

tm2,0

km1nm2nm1

The E-coefficient of the th small-wave is obtained bytransmission through Lm1(from right to left), followed by [re-flection at [Lm2, . . . , L1], reflection at z = Lm1] 1 times,and finally, transmission through [Lm2, . . . , L1]. So,Eof theth small-wave


ikm1Lm1(nm2 nm1)nm1

rm2,0nm2km1nm1

nm2 nm1nm2 +nm1 exp

2ikm1nm2Lm1nm1

1tm1,1

km1nm2nm1

Adding all the small-waves, we obtain

Eof the transmitted wave inz < L1


ikm1Lm1(nm2 nm1)nm1

tm2,0km1nm2nm1

1 rm2,0nm2km1nm1

nm2 nm1nm2 +nm1 exp

2ikm1nm2Lm1

nm1

1


11/30


Therefore,

tm1,0(k)= 2nm1nm1 +nm2 exp

ikm1Lm1(nm2 nm1)nm1

tm2,0km1

nm2

nm1

1 rm2,0nm2km1nm1 nm2 nm1nm2 +nm1

exp

2ikm1nm2Lm1nm11

(18)

Computingrm1,0(k)Assume, again, that the incident waveEm1exp[i(t+km1z)]is defined for z > Lm1. We wish to compute the reflected wavedefined, also, in the layernm1. The k-coefficient, as always,will be

km1. The reflected wave we wish to compute is the

sum of infinitely many small-waves plus a zeroth-wave.

The zeroth-wave is the reflection of the incident wave atz = Lm1. So,

Eof the 0th wave

=Em1nm1 nm2nm1 +nm2 exp2ikm1Lm1The first small-wave comes from transmission from right

to left through z= Lm1, followed by reflection at[Lm2,. . . , L1], followed by transmission from left to right throughz = Lm1. So,Eof the 1st small-wave

=Em1 2nm1nm1 +nm2 exp ikm1Lm1(nm2 nm1)nm1 rm2,0

km1nm2nm1

2nm2nm2 +nm1 expikm1Lm1nm2nm1 nm1 nm2nm2

The second small-wave comes from transmission from right

to left through z = Lm1, then reflection at[Lm2, . . . , L1],then [reflection at z = Lm1, reflectionat [Lm2, . . . , L1]] and,finally, transmission (leftright) throughz = Lm1. So,



km1nm2nm1

nm2 nm1nm2 +nm1 exp

2ikm1nm2Lm1nm1

rm1,1km1nm2nm1

2nm2nm2 +

nm1

exp

ikm1Lm1nm1 nm2

nm1

The th small-wave comes from transmission (rightleft)through z = Lm1, then reflection at [Lm2, . . . , L1], then [re-flection at z = Lm1, reflection at [Lm2, . . . , L1]] 1 times,and, finally, transmission leftright throughz = Lm1. So,

Eof theth small-wave


km1nm2nm1

nm2 nm1nm2 +nm1 exp

2ikm1Lm1nm2nm1

rm2,0

km1nm2nm11

2

nm2

nm2 +

nm1

exp

i

km1Lm1

nm1

nm2

nm1

Adding all these waves, we obtain

Eof the reflected wave in z > Lm1

=Em1nm1 nm2nm1 +nm2 exp2ikm1Lm1+Em1 2nm1nm1 +nm2 exp

ikm1Lm1(nm2 nm1)nm1

1 nm2 nm1nm2 +nm1 exp

2ikm1Lm1nm2nm1

rm2,0

km1

nm2

nm1 1

rm2,0km1nm2nm1

2nm2nm2 +nm1

expikm1Lm1nm1 nm2nm1

So,

rm1,0(km1)=nm1 nm2nm1 +

nm2

exp2ikm1Lm1

+ 2nm1nm1 +nm2 exp ikm1Lm1(nm2 nm1)nm1


2ikm1Lm1nm2nm1

rm2,0km1nm2nm1

1 rm2,0

km1nm2nm1

2nm2nm2 +nm1 exp

ikm1Lm1nm1 nm2

nm1


12/30


Thus,

rm1,0(km1)=nm1 nm2nm1 +

nm2

exp2ikm1Lm1

+ 4nm1nm2(nm1 +nm2)2exp2ikm1Lm1(nm2 nm1)nm1

rm2,0km1nm2nm1


2ikm1Lm1nm2nm1

rm2,0km1nm2nm1

1(19)

1.4.3. Organizing the Computations

The analysis of Egs. (16)(19) reveals that for computing, say,t0,m1(k), we need to compute t0, (k) for < m1 andr,0(kn1/n0)for < m 1.

Analogously, for computingr0,m1(k), we need to computet0, (k) for < m 1, t,0(kn1/n0) for < m1,r,0(kn1/n0)for < m 2, andr0, (k)for= m 2.

Let us see this in the following tables, for m= 4. Underthe (j+ 1)th column of the table, we write the quantities thatare needed to compute the quantities that appear under col-umn j. For example, we read that for computing t0,4(k), weneedt0,3(k)andr3,0(kn3/n0), and so on.

Computing Tree oft0,4(k)t0,4(k) t0,3(k) t0,2(k) t0,1(k)

r2,0(kn2/n0) r1,0(kn1/n0)r2,0(kn2/n0) r1,0(kn1/n0)

r3,0(kn3/n0) r2,0(kn2/n0) r1,0(kn1/n0)Computing Tree ofr0,4(k)

r0,4(k) t0,3(k) t0,2(k) t0,1(k)r1,0(kn1/n0)

r2,0(

k

n2/

n0) r1,0(

k

n1/

n0)

t3,0(kn3/n0) t2,0(kn2/n0) t1,0(kn1/n0)r1,0(kn1/n0)

r2,0(kn2/n0) r1,0(kn1/n0)r0,3(k) t0,2(k) t0,1(k)

r1,0(kn1/n0)t2,0(kn2/n0) t1,0(kn1/n0)

r1,0(kn1/n0)r0,2(k) t0,1(k)

t1,0(kn1/n0)r0,1(k)

Therefore, all the computation can be performed using thefollowing conceptual algorithm.

Algorithm

Step 1. Computet0,1(

k),r0,1(

k),t1,0(

k

n1/

n0),

r0,1(

kn1/n0).Step 2. For= 3, . . . , m 1,Computet0, (k),r0,(k),t,0(kn1/n0),r0, (kn1/n0).

1.4.4. Computation by Matricial Methods

The previous sections provide a practical way of computingtransmissions and reflections and understanding how complexwaves are generated as summations of simpler ones. In this sec-tion, we compute the same parameters using a more compactcalculation procedure, although some insight is lost [16]. Theassumptions are exactly the ones of the previous sections. In

layern , for= 0, 1, . . . , m 1, we have two waves given byETexp

i

wtk zand

ERexp

i

wt+k zThe first is a summation of transmitted small-waves and thesecond is a summation of reflected small-waves. As before,k= 2/,

k0=

k and

k=

knn0

for all = 0, 1, . . . , m1. We can intepret that E0Texp[i(wtkz)] is the incident wave. Since there are no reflected wavesin the last semi-infinite layer, we have that

Em1R = 0

Using the continuity of the waves and their derivatives withrespect to x at the interfaces L1, . . . , Lm1, we get, for =1, 2, . . . , m 1,

E1T exp

ik1L+ E1R expik1L= ETexp

ik L

+ ERexpik L

and

k1E1T expik1L+k1E1R expik1L= k ETexpikL+k ERexpik L

In matricial form, and usingk=kn /n0, we obtain 1 1

n1n1

exp(ik1L ) 00 exp(ik1L )

E

1T

E1R

=

1 1

n

n

exp(ikL ) 0

0 exp(ikL )

ET

ER


13/30


Therefore,ET

ER

=

exp(ikL ) 0

0 exp(ikL )

12

n

n+

n1

n

n1

nn1n+n1

exp(ik1L ) 0

0 exp(ik1L )

E1T

E1R

Let us write, for= 1, . . . , m 1,

A= 12n

n+n1nn1nn1n+n1

Then,E

+1T

E+1R

= exp(ik+1L+1) 00 exp(ik+1L+1) A+1

exp(ik[L+1 L ]) 0

0 exp(ik[L+1 L ])

A

exp(ik1L ) 00 exp(ik1L )

E1TE1R

Let d L+1 L (= 1, . . . , m 2) be the thickness oflayer . We define, for= 1, . . . , m 2,

M= A+1

exp(i

kd ) 0

0 exp(ik d )

Then, setting for simplicity and without loss of generality,L1= 0,

Em1T

Em1R

=

exp(ikm1Lm1) 0

0 exp(ikm1Lm1)

Mm2

M1A1

E0T

E0R

Define

M= Mm2 M1A1=

M11 M12

M21 M22

and

M=

exp(ikm1Lm1) 00 exp(ikm1Lm1)

M

=

M11 M12

M21 M22

UsingEm1R = 0, we obtain that

E0R= M21M22

E0T= M21

M22E0T

and

Em1T =

M11

M12M21

M22

E0T

= expikm1Lm1

M11 M12M21

M22

E0T

So,r0,m1= M21

M22

and

t0,m1= exp

ikm1Lm1M11 M12M21M22

(20)

In this deduction, we assumed that the transmitted wave in thefinal layernm1is E m1T exp[i(wtkm1z)]. For this reason,the factor exp(ikm1Lm1)appeared in the final computationof t1,m1. In other words, according to (20), the transmittedwave in the final layer is

M11 M12M21M22

E0Texpiwtkm1z Lm1For energy computations, since| exp(ikm1Lm1)| = 1, thepresence of this factor in the computation oft0,m1is irrelevant.

1.4.5. Transmitted and Reflected Energy

Assume that the layers z < L1and z > Lm1are transparent.In this case, the coefficientsn0and nm1are real, so we have

incident energy= n0|E|2transmitted energy= nm1|E-coefficient of the

transmitted wave

|2

and

reflected energy = n0|E-coefficient of the reflected wave|2

Accordingly, we define

transmittance = trasmitted energyincident energy

and

reflectance = reflected energyincident energy

In other words,

transmittance = nm

1

n0 |t0,m1|2

(21)

and

reflectance = |r0,m1|2 (22)Ifallthe layers were transparent, we would necessarily have

that

incident energy = transmitted energy + reflected energyand, in that case:

transmittance+ reflectance = 1


14/30


Assume now that the layer Lm2 < z < Lm1 is alsotransparent, so thatnm2 is also real. Consider the transmis-sion coefficientt0,m1as a function ofLm1, keeping fixed allthe other arguments. By (16), we see that |t0,m1|2 is periodicand that its period is n0/(2nm2).

Assume now that all the layers are transparent and, without

loss of generality,L1=0. We consider |t0,m1|2

as a functionof the thicknessesd1L2 L1, . . . , d m2 Lm1 Lm2.As above, it can be seen that |t0,m1|2 is periodic with respectto each of the above variables, and that its period with respecttodi(i= 1, 2, . . . , m 2) isn0/(2ni ).

1.5. Nonnormal Incidence and Linear-System

Computations

In the previous section, we deduced explicit formulae for trans-mitted and reflected energies when light impinges normally tothe surface of the layers. Explicit formulae are important be-cause they help to understand how waves effectively behave.

However, it is perhaps simpler to perform computations usingan implicit representation of transmitted and reflected waves in-side each layer. As we are going to see, in this way the electricand magnetic vectors arise as solutions of a single linear systemof equations. This approach allows us to deal in a rather simpleway with a more involved situation: the case in which the in-cidence is not normal, so that the plane of propagation is notparallel to the interfaces. The case considered in the previoussection is a particular case of this.

Assume that, in the three-dimensional space xy z, we havem layers divided by the interface planes z = L1, . . . , z =Lm1 and characterized by the complex refraction indices

n0, . . . ,nm1. Therefore, we have, for= 0, 1, . . . , m 1,n= n iwhere, as always,nrepresents the real refraction index andis the attenuation coefficient. We assume that 0= m1= 0,so the first and last layer are transparent.

Suppose that light arrives at the first surfacez = L1with anangle0with respect to the normal to the surface. This meansthat the vector of propagation of the incident light in the firstlayer is

s0=sin(0), 0, cos(0)

Accordingly (for example, invokingSnells law), the angle with

the normal of the transmitted wave in layer is

= nn0

sin(0)

and, consequently, the vector of propagation of the transmittedwave in the layer will be

s=sin( ), 0, cos( )

Reflected waves are generated in the layers 0, 1, . . . , m2.By Snells laws, their vectors of propagation are (sin( ), 0, cos( ))for= 0, 1, . . . , m 2.

Incident light in layer n0 is represented by the electric vec-tor E and the magnetic vector H. Electromagnetic theory [8]tells that bothE and H are orthogonal to s0. Moreover, the re-lation between these vectors is

H = n0s0 E (23)where

denotes the vectorial product. We consider that the

electromagnetic vectors are linearly polarized, so the consider-ations above lead to the following expression forE:

E= Epcos(0), Ey , Epsin(0) expit kxsin(0) + z cos(0) (24)

wherek= 2/andis the wavelength.By (23), we also have

H= n0Eycos(0), Ep, Eysin(0)

expit kxsin(0) + z cos(0)The incidence of light on the plane z= L1produces trans-

mitted vectors E1T, . . . , Em1T in the layers 1, . . . , m

1 and

the corresponding magnetic vectors H1T, . . . , Hm1T . Simultane-ously, reflected waves are produced, represented by the electricvectors E0R, . . . , E

m2R in the layers 0, 1, . . . , m2 and the cor-

responding magnetic vectorsH0R, . . . , Hm2R .

As in (24), for = 1, . . . , m 1, taking into account thevelocity of light in layer and the attenuation factor, we canwrite

ET=ET ,pcos(), ET ,y , ET ,psin( ) exp

i

t k n

n0

xsin( ) + z cos()

exp

k z

n0cos( ) (25)So, by a relation similar to (23) in layer ,

HT= nET ,ycos( ), ET ,p, ET ,ysin( )

exp

i

t k n

n0

xsin( ) + z cos( )

exp

k z

n0cos()

(26)

Similarly, for the reflected fields, with= 0, 1, . . . , m 2, wehave

ER=

ER,pcos( ), E

R,y , E

R,psin( )

expit k nn0 xsin( ) z cos() exp

k z

n0cos()

(27)

and

HR= n

ER,ycos( ), ER,p , ER,ysin( )

exp

i

t k n

n0

xsin( ) z cos( )

exp

k z

n0cos( )

(28)


15/30


From electromagnetic theory, we know that the tangentialcomponent of the electric and the magnetic fields must be con-tinuous at the interfaces. This means that

(i) Thex and y components ofE + E0Rmust coincidewith thex and ycomponents ofE1T+ E1Rfor z = L1.

(ii) Thex and y components ofH+

H0Rmust coincide

with thex and y components ofH1T+ H1Rforz = L1.

(iii) Thex and y components ofET+ ERmust coincidewith thex and y components ofE+1T + E+1R forz = L for = 1, . . . , m 3.

(iv) Thex and y components ofHT+ HRmust coincidewith thex and y components ofH+1T + H+1R forz = L for = 1, . . . , m 3.

(v) Thex and y components ofEm2T + Em2R mustcoincide with thex and y components ofEm1T forz = Lm1.

(vi) Thex and y components ofHm2T +

Hm2R must

coincide with thex and y components ofHm1T forz = Lm1.

Let us define, for= 0, 1, 2, . . . , m 2,

= exp

k L

n0cos( )+ ik n Lcos( )

n0

(29)

= exp

k1Ln0cos(1)

+ ik n1Lcos(1)nr0

(30)

Then, by (24)(30), the condition (iii) takes the form

E1T ,p

cos(1)

E1R,p cos(1)

ET ,p cos( ) + ER,pcos( )= 0 (31)

E1T ,y

1

+ E1R,y ET ,y

1

ER,y = 0 (32)

for = 1, . . . , m3. Analogously, condition (iv) takes theform

E1T ,y

n1cos(1)

E1R,y n1cos(1)

ET ,yncos( )

+ ER,y ncos( )= 0 (33)

E1T ,p

n1

+ E1R,p n1 ET ,pn

ER,p n = 0 (34)

Moreover, adopting the definition E0T = E and writingE

m1R = 0 (since there is no reflection in the last layer), we

see that (31)(34) also represent the continuity conditions (i),(ii), (v), and (vi). So, far, (31) and (34) form a system of2(m 1)linear equations with 2(m 1)complex unknowns(ET ,p, = 1, . . . , m 1 and E R,p , = 0, . . . , m 2). Onthe other hand, (32)(33) form a similar system whose equa-tions areET ,y ,= 1, . . . , m 1 andER,y ,= 0, . . . , m 2.Both systems have a very special band-structure (each row ofthe matrix of coefficients has only four nonnull entries) and can

be efficiently solved by Gaussian elimination. As a result, wecan compute the transmitted energy and the reflected energy inthe usual way:

Transmitted energy = nm1Em1T 2

and

Reflected energy = n0E0R21.6. The Effect of a Thick Substrate

Assume that we have a stack ofp films deposited on a thicksubstrate. By thick substrate we mean that the substrate thick-ness is much larger than the wavelength of light, whereas thefilm thicknesses are of the same order of magnitude as thatwavelength. Consider first the case of an infinitely thick sub-strate. The transmittance is then defined in the substrate andreflections at the back substrate surface are neglected. Suppose,for example, that the incident medium is air (n0

=1) and one

has an incident radiation of wavelength 995 nm. n1= 2.1 and1= 0.1 are the index of refraction and the extinction coeffi-cient, respectively, of a uniqued=127-nm-thick film (p=1)deposited onto a semi-infinite transparent substrate of refractiveindexnS= 1.57. In this case, the transmittance is

Tsemi-inf= 0.673819 (35)The above optical configuration is not realistic because, in gen-eral, the substrate supporting the films has a finite thickness.Reflections at the back surface of the transparent substrate willoccur and the transmittance is now measured in an additionallayer (usually air) which, in turn, is considered to be semi-infinite. Assuming a substrate thickness of exactly 106 nm= 1 mm, and performing the corresponding four-layer calcu-lation ((p + 3)-layer in the general case), we obtain

Tthick=106 nm= 0.590441 (36)The difference between Tsemi-inf and Tthick=106 is significant.Performing the same calculations with a substrate of thickness106 + 150 nm, we get

Tthick=(106+150)nm= 0.664837 (37)Again, an important difference appears between Tthick=106 nmand Tthick=(106+150)nm . Note that, in practice, the two substratesare indistinguishable, their respective thicknesses differing by

only 0.15 m. This leads to the question of what we actuallymeasure when we perform such experiments in the laboratory.Remember that the transmittance is a periodic function of thesubstrate thickness with a period of one-half wavelength. So,assuming random substrate thickness variations (dS dSdS), with dS s /2, we are led to conjecture that themeasured transmission is better approximated by an average oftransmissions over the period. This involves the integration ofthe transmittance as a function of the substrate thickness. In theabove-mentioned example, this operation gives

Taverage= 0.646609 (38)


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We arrive at the same value when we consider other indetermi-nations or perturbations to exact situations which normallyhappen in experimental setups, for example, slight deviationsfrom normal incidence, or the fact that the measured wave ()is not a pure wave but contains wavelengths between and + , where depends on the physical configuration

of the measuring apparatus. It is worth mentioning that thisslit effect alone does not explain the behavior of thetransmission in real experiments. Slit widths commonly usedin spectrophotometers are not large enough to eliminate com-pletely interference effects that appear when one consider purewaves.

Fortunately, the integral leading to Taverage in the generalcase can be solved analytically. The analytical integral has beendone in [17]. The final formula is simple and admits a nice in-terpretation, which we give below. (See [17].)

Assume that the refractive index of the (transparent) sub-strate is nS and the refractive index of the (transparent) finallayer isnF. By the arguments of previous sections on the two-

layers case, when a wave defined in the substrate arrives at theback surface, it is refracted to the last layer with its energy mul-tiplied by

T= nFnS

2nS

nS+ nF

2(39)

and is reflected with its energy multiplied by

R =

nS nFnS+ nF

2(40)

Consider, for a moment, that the substrate is semi-infinite anddefine

T= transmittance from layern0to the semi-infinite layernSof a wave with length;

T= transmittance from layernSto layern0of a wave withlength(n0/nS);

R= reflectance in the same situation considered forT(T+ R 1).

In the above example, we have T = 0.673819 and R =0.186631.

Assume a unit energy incident wave. Reasoning in energyterms (that is, not taking into account phase changes), we detectthe total transmitted energyas the sum of infinitely many smallenergies, as follows.

The first small energy comes from transmission throughthe films (T) followed by transmission through the substrate(T), so it is equal toT T.

The second small energy comes from transmissionthrough the films (T), followed by reflection at the back-substrate (R), followed by reflection at the back surface of thestack of films (R), followed by transmission through the sub-strate (T). Therefore, this small energy is equal toR RR T.

The th small energy comes from transmission throughthe films (T), followed by [reflection at the back-substrate (R)and reflection at the back surface of the stack of films (R)] 1

times, followed by transmission through the substrate (T).Therefore, this small energy is equal to T[RR]1 T.

Summing all the small energies, we obtain

Average transmittance = TT1 RR

Analogously, it can be obtained that

Average reflectance = T TR1 RR

The proof that these formulae correspond to the average in-tegration of the transmittance and reflectance along the periodcan be found in [17].

1.6.1. Substrate on Top or Between Thin Films

Suppose now that a stack of thin films is covered by a thicktransparent material. All the considerations above can be re-peated, and the formulae for average transmittance and average

reflectance turn out to be

Average transmittance = TT1 RR

and

Average reflectance = R + T2R

1 RRwhere now

T= 4n0nS(n0 + nS)2 and R=

n0 nSn0 + nS

2Tis the transmittance from the layer nS(first layer now) to

the last semi-infinite layer of a wave of length n0/n, andRisthe corresponding reflectance.

In some applications, a transparent thick substrate is coveredwith stacks of films on both sides with the purpose of gettingsome desired optical properties. As before, since the thicknessof the substrate is much larger than the wavelength and, thus,than the period of transmittance and reflectance as functions ofthis thickness, a reasonable model for the measured transmit-tance and reflectance considers integrating these functions withrespect to the period of the substrate. As shown in previous sec-tions, ifnSanddSare the refractive index and the thickness ofthe substrate, respectively, the period of the transmittance and

of the reflectance as a function ofd

S isn

0/(

2n

S), for an in-cident wave of length . At first sight, it is hard to integrate

analytically T and R over the period. However, reasoning inenergy terms, as done in a previous subsection, leads to the de-sired result. Let us define:

T1= transmittance atthrough the upper stack of films, fromlayern0to layernS;

R1= reflectance of the previous situation;T1= transmittance, for wavelengthn0/nS, through the

upper stack of films from layer nSto layern0;R1= reflectance corresponding to theT1situation;


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T2= transmittance, for wavelengthn0/nS, through theupper stack of films, from layern0to layernS;

R2=reflectance of theT2situation.Then, we get

Transmittance from top stack to the substrate

=

T1T2

1 R2R1and

Reflectance from top stack to the substrate = R1 + T1T1R21 R2R1

This leads us to the previous case, a top substrate, with an effec-tive transmittance and reflectance given by the above formulae.

1.7. Computational Filter Design

In this section, we consider that all the layers are transparentand, moreover,n0= nm1=1. Therefore, the transmitted en-ergy is

|t0,m

1

|2 and the reflected energy is

|r0,m

1

|2. For given

wavelength and refraction coefficients n1, . . . , nm2, theseenergies are periodic functions ofd1, . . . , d m2, the thicknessesof the intermediate layers. For some applications (lens design),it is necessary to determine the thicknesses of these layers insuch a way that some objectives in terms of transmission andreflection are fulfilled [18]. For example, suppose that we wantto design a system by the alternated superposition of four filmsof two different materials with refraction indicesnL= 1.8 andnH= 3.4, in such a way that the transmission for a wavelengthof 1000 nm is maximized, whereas the transmission for wave-lengthssuch that | 1000| 10 is minimal. We can do thissolving the following optimization problem:

Maximizef (d1, . . . , d m2) (41)where

f (d1, . . . , d m2)

= t5,0 = 1000, (d1, . . . , d m2)2 Average|1000|10

t6,1,(d1, . . . , d m2)2Note that, although the function |t5,0|2 is periodic with respectto each thickness, this is not the case off. As a result,f hasmany local maximizers that are not true solutions of (41).

The gradient of fcan be computed recursively using thescheme presented in the previous section, so that a first-order

algorithmfor optimization can be used to solve the problem. Weused the algorithm [19] (see also [20]) for solving the problem,with 500 nm as an initial estimate of all the thicknesses. Forthese thicknesses, the transmission for= 1000 is 0.181 andthe average of the side-transmissions is 0.515. The algorithmhas no difficulties in finding a (perhaps local) minimizer usingeight iterations and less than five seconds of computer time ina Sun workstation. The resulting transmission for=1000 is0.9999, whereas the average of the side-transmissions is 0.358.The thicknesses found are 556.3, 508.7, 565.6, and 509.9 nm,respectively.

Fig. 8. Reflectance in the visible and infrared spectral regions of the optimizeddual-band antireflection coating.

Dual-band antireflection filter. Consider, as a final exam-ple, the following design problem. Given air as the incidentmedium, a nonabsorbing sibstrate (sub), and two nonabsorb-ing coating materials (Land H), produce an antireflection thatreduces simultaneously the reflectance to the lowest possiblevalues in two spectral regions: (1) 420 nm to 680 nm; and(2) 10500 nm to 10700 nm.

The foregoing requirements are to be for dielectric and sub-strate layers having the following index of refraction:

nsub(1)= 2.60 nsub(2) = 2.40nL(1)= 1.52 nL(2) = 1.43nH(1)= 2.32 nH(2) = 2.22

There is restriction neither to the thickness of the individuallayers nor to the number of layers. The dual-band problem hasbeen solved as an optimization problem. The objective functionwas the sum of the reflectances in the two bands. This functionwas minimized with respect to the thicknesses using the algo-rithm described in [19]. This algorithm can be freely obtainedin [20]. The minimization was performed in several steps, in-creasing the number of films that are present in the stack. Giventhe solution for a number of films, an additional film was addedwith zero thickness, in an optimal position determined by theneedle technique [21]. According to this technique, the positionof the new zero-thickness film is the best possible one in thesense that the partial derivative that corresponds to its position

is negative and minimal. In other words, the potentiality of de-creasing locally the objective function for the needle-positionof the zero-film is maximized.

Figure 8 shows the results of the optimization process for atotal of 20 layers. The total thickness of the stack is 1560 nm.Note that the maximum reflection in the two bands is approxi-mately 0.1%, the reflectance being smaller in the infrared thanin the visible spectral region. It is clear that this is not the onlysolution to the problem [22]. In principle, an increasing num-ber of layers should improve the performance of the dual-bandantireflection filter. The manufacturing of filters always entails


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small randomvariations of the individual layer thickness. Whena random thickness variation of, say, 1% is applied to the stack,a tenfold increase of the mean reflectance is normally found.The effect becomes more severe as the number of layers in-creases.

1.8. Optimization Algorithm for Thin Films

The method used for solving the optimization problems de-rived from thin-film calculations is a trust-region box-constraintoptimization software developed at the Applied Mathemat-ics Department of the State University of Campinas. Duringthe last five years, it has been used for solving many prac-tical and academic problems with box constraints, and ithas been incorporated as a subalgorithm of Augmented La-grangian methods for minimization with equality constraintsand bounds. This software is called BOX-QUACAN here.Its current version is publicly available as the code EASY!in www.ime.unicamp.br/

martinez. The problems

solved in the present context are of the typeMinimizef (x)

subject to x Rn. is a set defined by simple con-straints. In our case, the thicknesses must be nonnegative orrestricted to some lower bounds. The art of minimization withsimple constraints is known as box-constrained optimization.Actually, this is a well-developed area of numerical analysis.

BOX-QUACAN is a box-constraint solver whose basic prin-ciples are described in [19]. It is an iterative method which,at each iteration, approximates the objective function by aquadratic and minimizes this quadratic model in the box de-termined by the natural constraints and an auxiliary box thatrepresents the region where the quadratic approximation is re-liable (trust region). If the objective function is sufficientlyreduced at the (approximate) minimizer of the quadratic, thecorresponding trial point is accepted as the new iterate. Other-wise, the trust region is reduced. The subroutine that minimizesthe quadratic is called QUACAN and the main algorithm, whichhandles trust-region modifications, is BOX.

Assume thatis ann-dimensional box, given by

= x Rn x uSo, the problem consists of finding an approximate solution of

Minimizex

f(x) subject to

x

u (42)

BOX-QUACAN uses sequential (approximate) minimiza-tion of second-order quadratic approximations with simplebounds. The bounds for the quadratic model come from theintersection of the original box with a trust region defined bythe -norm. No factorization of matrices are used at any stage.The quadratic solver used to deal with the subproblems of thebox-constrained algorithm visits the different faces of its do-main using conjugate gradients on the interior of each face andchopped gradients as search directions to leave the faces. See[19, 23, 24] for a description of the 1998 implementation of

QUACAN. At each iteration of this quadratic solver, a matrix-vector product of the Hessian approximation and a vector isneeded. Since Hessian approximationsare cumbersome to com-pute, we use the Truncated Newton approach, so that eachHessian vectorproduct is replaced by an incremental quo-tient offalong the direction given by the vector.

The box-constraint solver BOX is a trust-region methodwhose convergence results have been given in [19]. Roughlyspeaking, if the objective function has continuous partial deriva-tives and the Hessian approximations are bounded, every limitpoint of a sequence generated by BOX is stationary. Whenone uses true Hessians or secant approximations (see [25]) andthe quadratic subproblems are solved with increasing accuracy,quadratic or superlinear convergence can be expected. In ourimplementation, we used a fixed tolerance as stopping crite-rion for the quadratic solver QUACAN, because the benefitsof high-order convergence of BOX would not compensate in-creasing the work of the quadratic solver. It is interesting tonote that, although one usually talks about Hessian approxi-

mations, second derivatives of the objective function are notassumed to exist at all. Moreover, global convergence resultshold even without the Hessian-existence assumption.

2. APPLICATIONS OF THIN FILMS

In this section, we discuss a few situations where the theoreti-cal considerations of the previous section are put to work. Thesubject of thin-film applications is very vast and impossible tobe detailed in a book chapter. The selection to follow consid-ers, as usual, the most classical structures and those with whichthe authors have been particularly involved. For a discussion ofremaining subjects we refer the readers to the specialized liter-ature [8, 9, 16, 18, 2730].

2.1. Antireflection Coatings

The reflectance between two different (thick) dielectric mate-rials for a light beam falling normally to the interface is givenapproximately by R= (n1 n2)2/(n1 + n2)2 and it may bequite important in dense materials. In an air/glass interface,around4% of the incident light is reflected. Hence, for a normallens, the loss amounts to8%. For an optical system includ-ing a large number of lenses, the optical performance may be

severely affected. For a silicon solar cell (nSi4), the opticallosses, of the order of 35%, reduce considerably the conversionefficiency.

The amount of reflected light may be diminished signif-icantly by depositing thin intermediate antireflection (AR)layers. The subject is addressed in most textbooks on el-ementary optics as well as in the specialized literature onthe physics of thin films. An exact theoretical derivation ofreflection-reducing optical coatings from Maxwell equationsand boundary conditions between layer and substrate has beengivenby Mooney [31]. The analysis considers normal incidence


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Fig. 9. Single antireflection layer onto a thick substrate. The figure shows theconditions of zero reflectance at wavelength0 .

and isotropic flat parallel dielectricmaterials. The analytical ex-pressions for the reflectivity, defined as the ratio of the intensityof the reflected beam to that of the incident beam, is

R=

(1)1/2 + (S)1/2

21 (1)1/2

2+ (1)1/2 (S)1/221 + (1)1/22+ 2(1 S)(1 1) cos(2k1t)

(1)1/2 + (S)1/221 + (1)1/22

+ (1)1/2 (S)1/221 (1)1/22+ 2(1 S)(1 1) cos(2k1t)

1where medium 0 is air, medium 1 is the optical coating, andS is the substrate. The minimum R occurs for t= /2k1=0/4

1 and for all odd integral multiples of this thickness.

Under these circumstances, the beams reflected at the air/layerand at the layer/substrate interfaces interfere destructively, asshown in Figure 9. The minimum reflectance reads

Rmin(1)

= (S)

1/2 1(S)1/2 + 1

2

The condition for zero reflection of a single coating can be ob-tained at a wavelength0if1(0) =

s (0), or n1= nsif

the material is transparent at 0. The relation guarantees that theintensity reflected at each of the two interfaces is the same. If airis not the medium where the light beam comes from ( 0 >1),the condition for zero reflection for odd multiples of a quarter-wavelength thickness becomes1= (0s )1/2. For the case ofa typical glass nG 1.5/air interface, the antireflection layershould possess an index n1= 1.225 for a perfect match. Sta-ble materials having this low index value do not exist. Instead,

Table I. Refractive Indices of Some Materials

Material Index Material Index

MgF2 1.31.4 Si3N4 1.91.95

SiO2 1.41.5 TiO2 2.32.4

Al2O3 1.71.8 Ta2O5 2.12.3

SiO 1.81.9 ZnS 2.5SnO2 1.81.9 Diamond 2.4

MgF2 (n 1.38)can be used, allowing a larger than 2% re-duction of the reflectance. Table I lists the refractive indices ofsome materials.

Note that the values given in Table I correspond to refractiveindices measured in the transparent regions of the spectrum, inother words, far from the absorption edge, where the index ofrefraction increases with photon energy. Variations of the re-fractive index with photon energy can be included in the design

of antireflection coatings. The calculations, however, becomemore involved and numerical methods are required.A double-layer coating using two different dielectrics,

1 and 2, allows zero reflectance at two different wavelengths.The condition for an equal intensity of reflection at each of thethree interfaces is now expressed as n0/n1= n1/n2= n2/ns .Quarter-wavelength-thick layers of such materials,0/4n1and0/4n2, produce destructive interference at wavelengths 3/40and 3/20, at which the phase differencebetween the beamsreflected at the three interfaces equals 120 and 240, respec-tively, as shown in Figure 10. The theory can be extendedto a larger number of layers giving an achromatic antireflec-tion coating. Three appropriate AR layers offers the possibility

of zero reflectance at three different wavelengths, etc. As thenumber of layers increases, however, the benefits may not com-pensate the additional technical difficulties associated with thedeposition of extra films.

The refractive index match of a large number of layersneeded to obtain zero reflection may become difficult to meet.However, as the resulting reflectance is approximately givenby the vector sum of the Fresnel coefficients at the interfaces,a zero reflectance at a given wavelength can be met using thefilm thickness as a design variable. The required condition isthat the Fresnel coefficients r1, . . . , rnform a closed polygon atthe selected wavelength [17].

The same kind of analysis may be applied to the cir-

cumstance where a high reflectance is desired at a particularwavelength. Besides metals, dielectric and semiconductor ma-terials can be used for this purpose. Quarter-wavelength layersare still used, but the refractive index of the coating should begreater than that of the substrate; in fact, the difference shouldbe as large as possible. ZnS, TiO2, and Ta2O5 are used ontoglass for applications in the visible spectrum. For infrared ra-diation, a collection of very dense materials exists, such as Si(n 3.5), Ge (n 4), and the lead salts (PbS, PbSe, andPbTe), to cite a few. With these materials, it is possible to havereflected intensities larger than 80%.


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Fig. 10. Quarter-wavelength double antireflection coating. The figure illus-trates the conditions for index match to produce zero reflectance at wavelengths3/40and 3/20 .

Multiple-layer reflector structures employ an odd largenum-ber of stacked dielectric films of quarter-wavelength thickness.The stacking alternates layers of a large and a small index of re-

fraction. The reflected rays of all the interfaces of the structureemerge in phase for the selected wavelength, at which a re-flectance of nearly 100% can be obtained. The high reflectance,however, exists for a rather narrow frequency range because thephase change at the interfaces depends on wavelength. The ex-pressions giving the reflectance and the transmittance of suchstructures as a function of wavelength are rather lengthy butcan be optimized without difficulty using the formulation givenin the previous section.

2.1.1. Antireflection Coatings for Solar Cells

Solar cells are devices that directly convert sunlight into elec-

tricity. We will not review here the basic mechanisms ofsolar electricity generation, but simply note that antireflectioncoatings are of utmost importance to improve the conversionefficiency of the device. As a consequence, the subject hasreceived much attention in the last decades, when solar elec-tricity generation became an interesting alternative for manyterrestrial applications. Silicon, either single-crystal or poly-crystalline, is the base semiconductor material dominating theindustrial manufacturing of solar cells. Other semiconductor al-loys, such as hydrogenated amorphous silicon (a-Si:H), CdTe,and CuInSe2, are emerging materials for photovoltaic applica-

Fig. 11. Reflectance (%) of a polished crystalline silicon wafer as a functionof wavelength. The open circles indicate the reflectance measured after sprayinga 77-nm-thick SnO2layer.

tions. Let us consider the reflectance of a single-crystal siliconsolar cell. Figure 11 illustrates the situation. The continuousline is the measured reflectance of a polished, 400-m-thick,

Si wafer (nSi 3.4). The vertical dashed line roughly indicatesthe boundary between complete absorption and transparency. Inthe transparent region, the reflectance is high due to the largeindex of silicon and to the contribution of the two faces ofthe wafer to the overall reflectance. For wavelengths smallerthan 1130 nm, the reflectance decreases because of light ab-sorption in the wafer, which diminishes the contribution of thereflectionat the back Siair interface. At 1050 nm, the backinterface does not contribute any more and the measured re-flectance corresponds to the front interface only. The increasedreflectance at 1050 nm derives from an increasing nSi. Theopen symbols indicate the reflectance measured of the same Siwafer after spraying an SnO

2AR coating. The situation de-

picted in Figure 11 deserves some additional comments.

(a) First, in order to work as an electric generator, photonsmust be absorbed in the silicon substrate; that is, the photons ofinterest possess an energy larger than the Si band-gap [ >Eg (Si) 1.1 eV, or < gap 1130 nm].

(b) Second, the solar spectrum is very broad [32]. Thecontinuous line in Figure 12 shows the irradiance of sunlight,or its spectral intensity distribution, in the wavelength rangeof interest for the present discussion. Note the radiance unitson the left ordinate. The area below the complete curve gives


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Fig. 12. Air Mass 1.5 solar spectrum. Full line: irradiance (power density)per unit wavelength interval. Open circles: photon flux density per unit energyinterval. Note that for solar cells (quantum detectors), it is the reflected photonflux that has to be minimized, not the irradiance.

834.6Wm2

, and this value corresponds to a clear day with thesun inclined 48(AM 1.5) with respect to the normal [32]. Thespectrum shows several absorption bands originating mainlyfrom water vapor and carbon oxides.

(c) Third, absorbed photons means working in a photonenergy region where the optical constants of the substrate sili-con strongly depend on wavelength.

(d) Fourth, solar cells are quantum detectors; that is, elec-trons are excited through photon absorption. In other words, inorder to maximize the photocurrent of the solar cell, the totalamount of reflected photons has to be minimized in the energyrange of interest, and not the reflectance (intensity). Remem-ber that photon energy is inversely proportional to wavelength.

The open circles in Figure 12 show the photon density per unitwavelength interval versus wavelength for the same radiancespectrum given by the continuous line. The area below thiscurve for energies smaller than the energy gap gives the totalnumber of absorbed photons. This is the spectrum to be consid-ered for photovoltaic AR coatings.

(e) Fifth, the optimization of the photocurrent in a solarcell must also consider: (1) the internal collection efficiency,which measures the amount of collected electrons at the elec-trical contacts per absorbed photon, and (2) the variations ofthe refractive index with wavelength. In a good device, the

Fig. 13. Spectral photon flux density under AM 1.5 irradiation conditions.The dotted-dashed curve indicates the photon flux density reflected by a pol-ished silicon surface (solar cell). The open circles at the bottom of the figureshow the photon reflection in a system consisting of a cover glass, an optimizedAR coating, and a crystalline silicon solar cell. The optimization finds the in-dex of refraction and the thickness of the AR coating that minimizes the sum ofreflected useful photons of the AM 1.5 solar spectrum. See text.

collection efficiency

1 for most absorbed photons. In gen-eral, it decreases at short wavelengths due to surface effects.At long wavelengths, photon absorption decreases because ofa decrease of the absorption coefficient, and the photocurrentdrops.

Summarizing, the optimization of an antireflection coatingon a solar cell should include the variations of the refractive in-dex of the substrate material in the range of high absorptionandthe quantum nature of the conversion process (photon spectraldistribution of the solar spectrum).

As an illustrative case, we present below the results of min-imizing the reflected photon flux in a solar converter systemconsisting of a 4-mm protecting glass (n

=1.52), an AR coat-

ing (to be optimized), and a crystalline silicon solar cell. Theinput data are the index of refraction of the glass cover, the opti-cal properties of crystalline silicon, and the photon flux densityof the AM 1.5 solar spectrum (see Fig. 12). We consider normalincidence for the photon flux and the 400- to 1130-nm photonwavelength range. The minimization algorithm finds n=2.41and d= 66 nm for the AR coating that will maximize the en-ergy output of the system. The results on reflectance are shownin Figure 13.

The above optimization will just consider the photon distri-bution of a particular sunlight spectrum. In fact, the important


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parameter to be maximized is the electric energy delivered bythe device. In other words, a complete optimization should con-sider the position of the sun in the sky along the day and alongthe year (angle of incidence), as well as the variations of thesolar spectrum with prevailing weather conditions.

2.2. A Reverse Engineering Problem: The Retrieval of the

Optical Constants and the Thickness of Thin Films

from Transmission Data

Thin-film electronics includes thin-film transistors and amor-phous semiconductor solar cells. In many circumstances, thethickness dand the optical properties n() i() must beknown with some degree of accuracy. As shown in a previ-ous section, measurements of the complex amplitudes of thelight transmitted and reflected at normal or oblique incidenceat the film and substrate sides, or different combinations ofthem, enable the explicit evaluation of the thickness and theoptical constants in a broad spectral range. For electronic ap-

plications, however, the interesting photon energy range coversthe neighborhood of the fundamental absorption edge in whichthe material goes from complete opacity to some degree oftransparency. As optical transmittance provides accurate andquick information in this spectral range, efforts have beenmade to develop methods allowing the retrieval of the thick-ness and the optical constants of thin films from transmittancedata only.

Figure 14 shows three transmission spectra typical of thinisotropic and homogeneous dielectric films, bounded by plane-parallel surfaces and deposited onto transparent thick sub-strates. They correspond to: (a) a film, thick enough to display

interference fringes coming from multiple coherent reflectionsat the interfaces and being highly transparent below the funda-mental absorption edge; (b) a thick film, as above, but stronglyabsorbing in all the measured spectral range; and (c) a very thinfilm not showing any fringe pattern in the spectral region ofinterest. The interference fringes of films having a transparentspectral region, such as the ones shown by film 14a, can be usedto estimate the film thickness and the real part of the index ofrefraction in this region [8, 3335]. The approach is known asthe envel

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Optical Properties of Dielectric Films