# Substrate Effect on the Optical Reflectance and Transmittance of Thin-Film Structures

Authors: Anatoly Barybin; Victor Shapovalov
Journal: International Journal of Optics
 (2010)

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## Abstract

A rigorous and consistent approach is demonstrated to develop a model of the 4M structure (the four-media structure of a film on a substrate of finite thickness). The general equations obtained for the reflectance and transmittance spectra of the 4M structure are simplified by employing a procedure of the so-called device averaging to reduce them to a succinct form convenient for processing of experimental spectra for the structures with a thick substrate. The newly derived equations are applied to two special cases: (i) an arbitrary film on highly absorbing substrates and (ii) a slightly absorbing film on transparent substrates. The reflectance and transmittance spectra represented in the simplified (with the device averaging) form have a practical application for determining the film thickness and optical constants from experimental spectra by using the known techniques.


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## Body

## 1. Introduction

Nowadays, there are diverse modern optical measurement methods, which are based on electromagnetic theory applied to interference and absorption phenomena in thin-film layered structures. Advance in developing new physical principles of optical measurements has been achieved by extensive works and intensive efforts of a great number of scientists and researchers. Theoretical and practical knowledge in thin-film optics and optical measurements has been accumulated in numerous scientific publications; well-known books and reviews may be quoted as an example [1–12]. They describe optical properties of various films, techniques of measuring the reflectance and transmittance spectra, and calculations of the film thickness and optical constants from experimental spectra.The present interest in these subjects is caused by the wide application of thin-film structures in various optical devices and the fact that the optical methods of measurement give necessary information about the structural and physical properties of films that are used in optics and microelectronics [13]. Optical properties of thin films present a vital issue due to the increase of thin films quality standards in traditional areas of application [14] and the possibility to solve some recently discovered technical tasks only in thin-film performance; first, we refer to biomedical and ecological applications [15–19].The problem of determining the film thickness and dispersion properties of optical constants is usually solved by analyzing experimental spectra of the reflection and/or transmission for film-on-substrate structures of interest. Results of the analysis depend on a choice of physical model connecting experimentally measured spectra with geometrical and optical parameters of the structure under study. In order to obtain some reliable relations for practical calculations, as applied to processing of the results of measurements, it is necessary to develop a physically correct model of the thin-film structure, allowing interference and absorption in both the film and substrate materials.Various physical approaches and mathematical models describing the optical spectra for thin-film structures can be found in the literature. They usually present identical results in complicated forms difficult for comparison [1–12, 20–27]. A wide variety of different formulae available in the literature, which have been also derived by using dissimilar designations, essentially complicate the choice of the application to a given practical problem. In search of reliable technique for calculations needed for processing the results of spectral measurements an experimenter has to waste time on mathematical transformations to compare the theoretical expressions obtained by different authors.Although most theoretical approaches generally begin with studying a system of multiple layers [1, 5, 6, 8, 9], practical application is found only for the simplest three-media structure (3M structure) appropriate to a single film on a semiinfinite substrate. In particular, Heavens [10] analyzes some special cases of the 3M structure, as applied to the measurement of optical constants of thin films. The same author has developed the more general model for the four-media structure (4M structure) appropriate to a single film on a substrate of finite thickness in the earlier published book [5]. This model looks more practical in the application to optical measurements, as the expressions for the reflectance and transmittance of the 4M structure in addition to the finite substrate thickness contain also optical parameters of both the input and output media adjacent to the film and substrate. However, the general formulae obtained by Heavens [5] describing the 4M structure turn out to be extremely complicated, which restricts their practical applicability to the processing of experimental spectra.Section2 is devoted to a critical analysis of existing models for the reflectance and transmittance spectra and formulates the theoretical problem for further investigation. Section 3 describes the matrix approach to evaluation of the reflectance and transmittance of multilayer structures. The general matrix relations derived by involving the Fresnel reflection coefficients of multiple interfaces are employed to the special cases of the 3M and 4M structures. In Section 4 formulae for the reflectance and transmittance originally expressed in terms of the Fresnel coefficients are adduced in a physical form containing the optical constants of media (refractive and absorptive indices) together with the geometrical parameters of the considered structures. Mathematical details transforming the relevant formulae to the physical form are set forth in Appendix A. The general formulae are simplified by using a procedure of the so-called device averaging (see Appendix B) to reduce them to a succinct form convenient for the processing of experimental spectra for structures with a thick substrate. In Section 5, the simplified equations are applied to two special cases: (i) an arbitrary film on highly absorbing substrates and (ii) a slightly absorbing film on transparent substrates.


## 2. Formulation of the Problem

We shall restrict our consideration to normal incidence of light from a transparent input medium of the real refractive indexn0. In the general case, the film under examination (numbered by 1) has optical losses so that its refractive index is complex and dispersive: ñ1(λ)=n1(λ)-ik1(λ) with the real refractive and absorptive indices n1(λ) and k1(λ) being functions of the light wavelength λ. Depending on application of the thin-film structure, its substrate (numbered by 2) is produced from different materials including transparent dielectrics as well as absorbing metals or semiconductors. Thus, in general, the refractive index of substrate also has the complex form ñ2(λ)=n2(λ)-ik2(λ).Examination of various experimental situations allows one to select the two above-mentioned thin-film structures, which are available for thorough mathematical study. They are depicted in Figure1: (a) the three-media structure (3M structure) appropriate to a film of thickness d1 on a semiinfinite substrate and (b) the four-media structure (4M structure) appropriate to a single film of thickness d1 on a substrate of finite thickness d2. Media 0 and 3 may be considered respectively an input medium (non absorbing one with real index n0) and an output medium (absorbing one with complex index ñ3(λ)=n3(λ)-ik3(λ)). The reflected flows of light are shown in Figure 1 at some angle only for pictorial rendition; the case under consideration has the normal incidence.Schematic geometry for: (a) the 3M structure with a semiinfinite substrate and (b) the 4M structure with a single film (medium 1) on a substrate of finite thicknessd2 (medium 2); media 0 and 3 correspond to the input and output ones.
(a)(b)The main task of theoretical analysis is a derivation of mathematical expressions for the spectra of reflectanceR(λ) and transmittance T(λ), which can be applied to models of the 3M and 4M structures. In our opinion, the final expressions for R(λ) and T(λ) within the framework of any mathematical model should, first, correctly describe physical phenomena occurring under action of light (wave interference and energy absorption) and, second, have a sufficiently succinct form to be convenient for both the physical interpretation and numerical computation. From succinctness standpoint, an expression for the transmittance spectrum given by Swanepoel [24, 25] is more successful, unlike many others of cumbersome forms [5, 9, 10, 12]. However, Swanepoel’s approach casts some doubts upon its mathematical justification, as shown below.In Swanepoel’s notation [24], the transmittance spectrum of a slightly absorbing film (k12≪n12) on a transparent substrate (k2=0) has the following form:(1)T(λ)=AxB-Cxcos⁡φ+Dx2,
where x=exp(-αd1),  α=4πk1/λ,φ=4πn1d1/λ, while A, B, C, and D are the coefficients depending on optical constants of media taken into account by the model under consideration. In publications, there are different expressions for these coefficients; in particular, authors in [21, 23] apply them in the following form (with notation of nm, m=0,1,2, according to Figure 1(a))(2)A=16n0n12n2,C=-2(n02-n12)(n12-n22),B=(n0+n1)2(n1+n2)2,D=(n0-n1)2(n1-n2)2.When using (1)-(2), one implicitly means that a substrate is semiinfinite, which conforms to the 3M structure (Figure 1(a)), and then a receiver of transmitted light should be placed inside the substrate. In fact, such is not the case and the light receiver must be outside a substrate. So, the model of semiinfinite substrate does not work in real situations. Consequently, (1)-(2) yield some higher values of the transmittance because they disregard an additional reflection from another interface of a real substrate (for a glass one the error is around 4% [4]). For this reason, in order to take into account the above-stated fact, Swanepoel [24, 25], followed by other authors [26–31], has applied (1) for the transmittance of a single film on a transparent substrate and suggested that the coefficients A and C should be left in the unchanged form (2), while B and D are written as follows:(3)A=16n0n12n2,C=-2(n02-n12)(n12-n22),B=(n0+n1)3(n1+n22),D=-(n0-n1)3(n1-n22).A distinguishing feature of expressions (3) as compared with (2) consists of appearance of the third power of the first parentheses and asymmetry in powers of items inside the second parentheses for the coefficients B and D. Both facts are very doubtful but have no mathematical substantiation by Swanepoel [24]. In effect, (1) and (3), as applied to the 3M structure in [24], implicitly correspond to the 4M structure shown in Figure 1(b). However, these formulae do not take into account the optical constants of an output medium (numbered by 3), quite apart from lack of the finite substrate thickness d2 inherent in the 4M structure. The above-stated reasons cause some doubts on the correctness of Swanepoel’s approach [24, 25].Consequently, the problem of reflection and transmission of light, as applied to the film-on-substrate structures especially with regard for optical absorption, can correctly be solved only within the scope of the 4M structure model. The 3M structure model is of practical interest solely in the special case that the substrate has high optical losses, which is typical for semiconductor and metal substrates. In this case, the sole reflectance spectrumR(λ) is practically accessible from experimental measurements, unlike the transmittance spectrum T(λ). As proved later, both the spectra R(λ) and T(λ) for the 3M structures can be theoretically obtained as a special case of the general physical situation examined within the framework of the 4M structure model.Certain expressions for the spectraR(λ) and T(λ) of the 4M structure were also derived by Heavens and represented in the following form [5]:(4)R(λ)=t132+u132p132+q132,T(λ)=l132+m132p132+q132.
Here, the items l132, m132, p132, q132, t132, and u132 depend on the optical constants ñm=nm-ikm of media (m=1,2,3), their thicknesses dm, and the light wavelength λ.Expressions (4) only look succinct because the items l132, m132, and so forth are very involved and physically tangled, being composed of a number of other terms by successive substitution of them one after the other. Such a “step-by-step” construction of Heavens’ items, unlike the simple one of Swanepoel’s coefficients (3), practically eliminates the processing ability of formulae (4), as applied to the experimental spectra R(λ) and T(λ). Also, this makes it difficult, if not impossible, to provide insight into influence of physical parameters of the structure on its optical spectra. We do not adduce here expressions for l132, m132, and so forth, because of their complexity and refer an interested reader to the original book [5] or paper [32], which have also given them.Formulae (4) describe the physical situation corresponding to interference of waves transmitted and reflected by three interfaces, as shown in Figure 1(b). These formulae are applicable to any film-on-substrate structure with arbitrary properties including a double absorbing film (media 1 and 2) on semiinfinite absorbing substrates (medium 3).Figure2 shows spectra R(λ) and T(λ) of film Ta2O5 on a quartz-glass substrate (n0=n3=1,  n1=2.21,n2=1.472, k1=k2=k3=0, d1=1μm, and d2=500μm) calculated on the basis of Heavens’ formulae (4). In both spectra, there are oscillations of two types caused by wave interference in the thin film Ta2O5 (slow oscillations) and in the thick substrate (fast oscillations in the form of a “beard”). Space periods of these oscillations differ from each other by the factor η≡n2d2/n1d1, which exceeds two orders in magnitude for the given instance. However, when measuring the spectra an experimental result takes the form of a dashed curve in Figure 2 caused by interference only in the thin film Ta2O5. This stems from the fact that the operating slit of a spectrophotometer is not infinitely narrow but has finite width (of a few nanometers). Because of this, the input irradiation is not monochromatic, and the fast oscillations caused by interference in the thick substrate with n2d2≫n1d1 prove to be averaged down to zero, as is mathematically justified in Appendix B. In other words, any spectrophotometer with an operating slit of finite width performs a procedure that will be called the device averaging. As a result of this procedure, the “beard” produced by interference oscillations in a thick substrate is liquidated, and only the thin-film interference curve similar to dashed ones in Figure 2 is left in experimental spectra.Reflectance (R) and transmittance (T) spectra of the Ta2O5-quartz-glass structure calculated from (4) and shown by gray lines in the form of a “beard”. Dashed curves on the spectra result from measurements by using a spectrophotometer with operating slit of finite width.
(a)(b)Our subsequent task consists of deriving the analytical expressions forR(λ) and T(λ) in such a form that allows us to carry out the device averaging.


## 3. General Matrix Theory for Multilayer Structures

### 3.1. Desired Form of the Fresnel Reflection Coefficients

Consider any two adjacent layers of the multilayer structure under consideration, say, the(m-1)th and mth absorbing layers specified by the complex refractive indices(5)ñm-1=nm-1-ikm-1,ñm=nm-ikm,
where m=1,2,… and ñ0=n0 since k0≡0 for a non-absorbing input medium. Each mth interface is characterized by the Fresnel reflection and transmission coefficients rm and tm=1+rm defined as [1–7](6)rm=ñm-1-ñmñm-1+ñm,tm=2ñm-1ñm-1+ñm.Heavens [5] represents the Fresnel reflection coefficient rm as a sum of its real and imaginary parts(7)rm=gm+ihm,
where in accordance with (5) and (6), we have(8)gm=|ñm-1|2-|ñm|2(nm-1+nm)2+(km-1+km)2,hm=2(nm-1km-nmkm-1)(nm-1+nm)2+(km-1+km)2,
with the squared moduli of the refractive indices (5) being equal to(9)|ñm-1|2=nm-12+km-12,|ñm|2=nm2+km2.Unlike Heavens’ approach, for our analysis, it is more convenient to apply forrm instead of (7) the exponential form(10)rm=ρmexp⁡(iθm).
The squared modulus ρm2 and phase θm of the Fresnel reflection coefficient (7) or (10) are defined by the following relations(11)ρm2=gm2+hm2=|ñm-1-ñm|2|ñm-1+ñm|2≡N-mN+m,(12)tan⁡θm=hmgm=2(nm-1km-nmkm-1)|ñm-1|2-|ñm|2,
where we have introduced the following quantities(13)N±m≡|ñm-1±ñm|2=(nm-1±nm)2+(km-1±km)2=(|ñm-1|2+|ñm|2)±2(nm-1nm+km-1km).In the absence of optical absorption (km-1=km=0), expression (6) for rm gives purely real values (negative or positive) so that rm<0 or rm>0 when, respectively, nm-1<nm or nm-1>nm. Yet, the exponential form (10) does not display explicitly this property. Indeed, from (12), it follows that in the lossless situation tanθm=0 and by definition always ρm>0. In order for the above-stated property to be explicitly displayed from relation (10), it is necessary to require the phase θm to be positively defined. This requirement is readily realized by considering the phase angle θm to be a function of the quantity(14)xm=smhmgm≡hm|gm|=2(nm-1km-nmkm-1)||ñm-1|2-|ñm|2|=2(nm-1km-nmkm-1)|(nm-12-nm2)+(km-12-km2)|,
where the sign of gm is inserted as(15)sm=gm|gm|=sgn⁡{|ñm-1|2-|ñm|2}={-1if|ñm-1|<|ñm|,+1if|ñm-1|>|ñm|.From formulae (12) and (14) explained by Figure 3 it follows that(16)θm=arctan(smxm)={arctan(+xm)=0+ϕmforsm=+1arctan(-xm)=π-ϕmforsm=-1,(17)exp⁡(iθm)=smexp⁡(ismϕm),
where the new phase angle ϕm is defined by the following relation:(18)tan⁡ϕm=xm≡hm|gm|=2(nm-1km-nmkm-1)||ñm-1|2-|ñm|2|.Figure 3
Illustration of the phase anglesθm and ϕm for representing the Fresnel reflection coefficient in the exponential form rm=ρmexp(iθm)=smρmexp(ismϕm).Substitution of relation (17) into (10) yields the desired exponential form of the Fresnel reflection coefficient(19)rm=ρ̅mexp⁡(iϕ̅m)withρ̅m=smρm,ϕ̅m=smϕm.From (18) and (19), it evidently follows that in the lossless situation, when ϕm=0, the Fresnel coefficient rm=smρm<0 or >0 depending on whether nm-1<nm or nm-1>nm, which is what we set out to obtain.


### 3.2. Matrix Approach to Evaluating the Reflectance and Transmittance

Let us consider the(m-1)th and mth layers which are incorporated into the N-media (including input and output ones) structure (NM structure) and depicted in Figure 4. Electromagnetic fields in each of them are formed by superposition of two plane waves—positive going (of a complex amplitude marked by superscript +) and negative going (of a complex amplitude marked by superscript −) with respect to the axis z directed transversely to interfaces. As seen from Figure 4, light travels from an input medium to output one. So, for the mth layer the positive-going wave of amplitude Em+ corresponds to light transmitted across its left boundary at zm-1, while the negative-going wave of amplitude Em-—to light reflected from its right boundary at zm.Figure 4
Schematic geometry of the multilayer structure consisting ofN media (including the input/output ones) with two highlighted (m-1)th and mth layers having an interface at zm-1 specified by the Fresnel reflection coefficient rm=smρmexp(ismϕm).We shall describe light propagation in themth absorbing medium of the complex refractive index ñm=nm-ikm by using the wave factor exp(iωt∓γmz), where the propagation constant (20)γm=i2πλñm≡αm+iβm
consists of the amplitude and phase constants defined as(21)αm=2πλkm,βm=2πλnm.
Let us write down the electromagnetic fields inside the two layers under examination(i)
for the(m-1)th layer located at zm-2<z<zm-1(22)Em-1(z)=Em-1+e-γm-1(z-zm-2)+Em-1-eγm-1(z-zm-2),Hm-1(z)=ñm-1Em-1+e-γm-1(z-zm-2)-ñm-1Em-1-eγm-1(z-zm-2),(ii)
for themth layer located at zm-1<z<zm(23)Em(z)=Em+e-γm(z-zm-1)+Em-eγm(z-zm-1),Hm(z)=ñmEm+e-γm(z-zm-1)-ñmEm-eγm(z-zm-1).Imposing on the electromagnetic fields (22) and (23) at interface point z=zm-1 the following matching conditions:(24)Em-1(zm-1)=Em(zm-1),Hm-1(zm-1)=Hm(zm-1),
we couple the complex amplitudes in the (m-1)th and mth layers by the matrix relation(25)(Em-1+Em-1-)=1tmCm(Em+Em-),
where tm is the Fresnel transmission coefficient (6) for the mth interface at z=zm-1. The coupling matrix appearing in (25)(26)Cm=(eΔm-1rmeΔm-1rme-Δm-1e-Δm-1)withΔm-1=γm-1dm-1
is identical in form to those obtained previously [5, 9].To express the input electric fieldsE0± in terms of the output field EN-1+, it is necessary to successively multiply the coupling matrices (26) for all layers entering into the NM-structure depicted in Figure 4. As a result, we obtain from (25) the following relation:(27)(E0+E0-)=1t1t2⋯tN-1(aNbNcNdN)(EN-1+0),
where EN-1-=0 for a semiinfinite output medium and the resultant coupling matrix is(28)(aNbNcNdN)=C1C2⋯CN-1.The reflectance and transmittance (defined as ratios of the reflected and transmitted power to the incident power) immediately follow from (27) as(29)R≡|E0-|2|E0+|2=|cN|2|aN|2,T≡nN-1n0|EN-1+|2|E0+|2=nN-1n0|t1|2|t2|2⋯|tN-1|2|aN|2.Therefore, in order to obtainR and T, it is necessary to find only two matrix elements for calculating |aN|2 and |cN|2 and also to know the Fresnel transmission coefficients tm of all interfaces (m=1,2,…,N-1) for which from (19) it follows that(30)|tm|2=|1+rm|2=1+ρ¯m2+2ρ¯mcos⁡ϕ¯m.Let us apply (26)–(29) to the special cases appropriate to the 3M structure (Figure 1(a)) and the 4M structure (Figure 1(b)).


### 3.3. Reflectance and Transmittance of the 3M Structure

For the 3M structure shown in Figure1(a) the coupling matrix product equals(31)C1C2=(1r1r11)(eΔ1r2eΔ1r2e-Δ1e-Δ1)=(eΔ1+r1r2e-Δ1r1e-Δ1+r2eΔ1r1eΔ1+r2e-Δ1e-Δ1+r1r2eΔ1)≡(a3b3c3d3),
where Δ1≡γ1d1=α1d1+iβ1d1 (see formulae (21) for m=1) and the Fresnel coefficients r1 and r2 have the exponential form (19).The general formulae (29) yield the required expressions for the reflectance and transmittance of the 3M structure(32)R=|c3|2|a3|2,T=n2n0|t1|2|t2|2|a3|2.
From elements of the coupling matrix (31) and (30), we have obtained(33)|a3|2=(e2α1d1+ρ¯12ρ¯22e-2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1-ϕ¯1)-ϕ¯2],(34)|c3|2=(ρ¯12e2α1d1+ρ¯22e-2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1+ϕ¯1)-ϕ¯2],(35)|t1|2|t2|2=(1+ρ¯12+2ρ¯1cos⁡ϕ¯1)(1+ρ¯22+2ρ¯2cos⁡ϕ¯2),
where ρ¯1,2≡s1,2ρ1,2 and ϕ¯1,2≡s1,2ϕ1,2.


### 3.4. Reflectance and Transmittance of the 4M Structure

For the 4M structure shown in Figure1(b) the coupling matrix product has the form(36)C1C2C3=(a3b3c3d3)(eΔ2r3eΔ2r3e-Δ2e-Δ2)≡(a4b4c4d4),
with the elements a4 and c4 being equal to(37)a4=e(Δ1+Δ2)+r1r2e-(Δ1-Δ2)+r1r3e-(Δ1+Δ2)+r2r3e(Δ1-Δ2),c4=r1e(Δ1+Δ2)+r2e-(Δ1-Δ2)+r3e-(Δ1+Δ2)+r1r2r3e(Δ1-Δ2).
Here, Δ1,2≡γ1,2d1,2=α1,2d1,2+iβ1,2d1,2 (see formulae (21) for m=1,2) and the Fresnel coefficients rm are of exponential form (19), namely, rm=ρ¯mexp(iϕ¯m) with ρ¯m=smρm and ϕ¯m=smϕm (see (11), (15), and (18) for m=1,2,3).The general formulae (29) yield the required expressions for the reflectance and transmittance of the 4M-structure (cf. expressions (32))(38)R=|c4|2|a4|2,T=n3n0|t1|2|t2|2|t3|2|a4|2,
where the product |t1|2|t2|2|t3|2 is written similar to expression (35).From (37), after some laborious transformations, we obtain the desired expressions(39)|a4|2=(|a3|2e2α2d2+ρ¯32|a3′|2e-2α2d2)+ρ¯3fa(β2d2),|c4|2=(|c3|2e2α2d2+ρ¯32|c3′|2e-2α2d2)+ρ¯3fc(β2d2),
where the terms |a3|2 and |c3|2 have (33) and (34) inherent in the 3M structure. Besides, (39) include certain additional terms(i)
the terms due to the interference and absorption in layer 1 of optical thicknessβ1d1(40)|a3′|2=(ρ¯12e-2α1d1+ρ¯22e2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1-ϕ¯1)+ϕ¯2],|c3′|2=(e-2α1d1+ρ¯12ρ¯22e2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1+ϕ¯1)+ϕ¯2],(ii)
the terms due to the interference and absorption in layer 2 of optical thicknessβ2d2(41)ρ¯3fa(β2d2)=2ρ¯1ρ¯3{cos⁡[(2β1d1-ϕ¯1)+(2β2d2-ϕ¯3)]+ρ¯22cos⁡[(2β1d1-ϕ¯1)-(2β2d2-ϕ¯3)]}+2ρ¯2ρ¯3{e2α1d1cos⁡[(2β2d2-ϕ¯3)-ϕ¯2]+ρ¯12e-2α1d1cos⁡[(2β2d2-ϕ¯3)+ϕ¯2]},ρ¯3fc(β2d2)=2ρ¯1ρ¯3{cos⁡[(2β1d1+ϕ¯1)+(2β2d2-ϕ¯3)]+ρ¯22cos⁡[(2β1d1+ϕ¯1)-(2β2d2-ϕ¯3)]}+2ρ¯2ρ¯3{e-2α1d1cos⁡[(2β2d2-ϕ¯3)+ϕ¯2]+ρ¯12e2α1d1cos⁡[(2β2d2-ϕ¯3)-ϕ¯2]}.From a comparison of (33)-(34) and (40), it follows that both terms (|a3|2,|c3|2) and (|a3′|2,|c3′|2) are produced by the interference and absorption phenomena only inside layer 1. Both of them, being inserted in the 4M composition (|a4|2,|c4|2), as seen from (39), prove to be multiplied by the factors exp(±2α2d2) due to optical losses in layer 2. The interference oscillations in this layer (together with an additional contribution from interference and absorption in layer 1) are solely taken into account by the functions ρ¯3fa(β2d2) and ρ¯3fc(β2d2) given in the form of (41).It is very important to note that both the additional terms (|a3′|2,|c3′|2) and (fa,fc) contribute to expressions (39) for the 4M structure being multiplied by ρ3≡|r3|∝|ñ2-ñ3|. Therefore, their contributions to (38) for R and T disappear and, as a result of this, they are converted into (32) for the 3M structure when(42)ñ3=ñ2,d2=0.Both requirements (42) are needed for the conversion 4M → 3M, as applied to the transmittance T, while the reflectance conversion is immediately fulfilled if ñ3=ñ2, that is, if only medium 3 does not differ from medium 2.Hence, our further transformations will deal with the 4M structure, as a general one, to reduce the reflectanceR and transmittance T given by (38)–(41) to the so-called physical form. Such a form has to contain instead of the Fresnel coefficients the physical constants of media (refractive and absorptive indices) as well as the geometrical parameters of a structure under consideration.


## 3.1. Desired Form of the Fresnel Reflection Coefficients

Consider any two adjacent layers of the multilayer structure under consideration, say, the(m-1)th and mth absorbing layers specified by the complex refractive indices(5)ñm-1=nm-1-ikm-1,ñm=nm-ikm,
where m=1,2,… and ñ0=n0 since k0≡0 for a non-absorbing input medium. Each mth interface is characterized by the Fresnel reflection and transmission coefficients rm and tm=1+rm defined as [1–7](6)rm=ñm-1-ñmñm-1+ñm,tm=2ñm-1ñm-1+ñm.Heavens [5] represents the Fresnel reflection coefficient rm as a sum of its real and imaginary parts(7)rm=gm+ihm,
where in accordance with (5) and (6), we have(8)gm=|ñm-1|2-|ñm|2(nm-1+nm)2+(km-1+km)2,hm=2(nm-1km-nmkm-1)(nm-1+nm)2+(km-1+km)2,
with the squared moduli of the refractive indices (5) being equal to(9)|ñm-1|2=nm-12+km-12,|ñm|2=nm2+km2.Unlike Heavens’ approach, for our analysis, it is more convenient to apply forrm instead of (7) the exponential form(10)rm=ρmexp⁡(iθm).
The squared modulus ρm2 and phase θm of the Fresnel reflection coefficient (7) or (10) are defined by the following relations(11)ρm2=gm2+hm2=|ñm-1-ñm|2|ñm-1+ñm|2≡N-mN+m,(12)tan⁡θm=hmgm=2(nm-1km-nmkm-1)|ñm-1|2-|ñm|2,
where we have introduced the following quantities(13)N±m≡|ñm-1±ñm|2=(nm-1±nm)2+(km-1±km)2=(|ñm-1|2+|ñm|2)±2(nm-1nm+km-1km).In the absence of optical absorption (km-1=km=0), expression (6) for rm gives purely real values (negative or positive) so that rm<0 or rm>0 when, respectively, nm-1<nm or nm-1>nm. Yet, the exponential form (10) does not display explicitly this property. Indeed, from (12), it follows that in the lossless situation tanθm=0 and by definition always ρm>0. In order for the above-stated property to be explicitly displayed from relation (10), it is necessary to require the phase θm to be positively defined. This requirement is readily realized by considering the phase angle θm to be a function of the quantity(14)xm=smhmgm≡hm|gm|=2(nm-1km-nmkm-1)||ñm-1|2-|ñm|2|=2(nm-1km-nmkm-1)|(nm-12-nm2)+(km-12-km2)|,
where the sign of gm is inserted as(15)sm=gm|gm|=sgn⁡{|ñm-1|2-|ñm|2}={-1if|ñm-1|<|ñm|,+1if|ñm-1|>|ñm|.From formulae (12) and (14) explained by Figure 3 it follows that(16)θm=arctan(smxm)={arctan(+xm)=0+ϕmforsm=+1arctan(-xm)=π-ϕmforsm=-1,(17)exp⁡(iθm)=smexp⁡(ismϕm),
where the new phase angle ϕm is defined by the following relation:(18)tan⁡ϕm=xm≡hm|gm|=2(nm-1km-nmkm-1)||ñm-1|2-|ñm|2|.Figure 3
Illustration of the phase anglesθm and ϕm for representing the Fresnel reflection coefficient in the exponential form rm=ρmexp(iθm)=smρmexp(ismϕm).Substitution of relation (17) into (10) yields the desired exponential form of the Fresnel reflection coefficient(19)rm=ρ̅mexp⁡(iϕ̅m)withρ̅m=smρm,ϕ̅m=smϕm.From (18) and (19), it evidently follows that in the lossless situation, when ϕm=0, the Fresnel coefficient rm=smρm<0 or >0 depending on whether nm-1<nm or nm-1>nm, which is what we set out to obtain.


## 3.2. Matrix Approach to Evaluating the Reflectance and Transmittance

Let us consider the(m-1)th and mth layers which are incorporated into the N-media (including input and output ones) structure (NM structure) and depicted in Figure 4. Electromagnetic fields in each of them are formed by superposition of two plane waves—positive going (of a complex amplitude marked by superscript +) and negative going (of a complex amplitude marked by superscript −) with respect to the axis z directed transversely to interfaces. As seen from Figure 4, light travels from an input medium to output one. So, for the mth layer the positive-going wave of amplitude Em+ corresponds to light transmitted across its left boundary at zm-1, while the negative-going wave of amplitude Em-—to light reflected from its right boundary at zm.Figure 4
Schematic geometry of the multilayer structure consisting ofN media (including the input/output ones) with two highlighted (m-1)th and mth layers having an interface at zm-1 specified by the Fresnel reflection coefficient rm=smρmexp(ismϕm).We shall describe light propagation in themth absorbing medium of the complex refractive index ñm=nm-ikm by using the wave factor exp(iωt∓γmz), where the propagation constant (20)γm=i2πλñm≡αm+iβm
consists of the amplitude and phase constants defined as(21)αm=2πλkm,βm=2πλnm.
Let us write down the electromagnetic fields inside the two layers under examination(i)
for the(m-1)th layer located at zm-2<z<zm-1(22)Em-1(z)=Em-1+e-γm-1(z-zm-2)+Em-1-eγm-1(z-zm-2),Hm-1(z)=ñm-1Em-1+e-γm-1(z-zm-2)-ñm-1Em-1-eγm-1(z-zm-2),(ii)
for themth layer located at zm-1<z<zm(23)Em(z)=Em+e-γm(z-zm-1)+Em-eγm(z-zm-1),Hm(z)=ñmEm+e-γm(z-zm-1)-ñmEm-eγm(z-zm-1).Imposing on the electromagnetic fields (22) and (23) at interface point z=zm-1 the following matching conditions:(24)Em-1(zm-1)=Em(zm-1),Hm-1(zm-1)=Hm(zm-1),
we couple the complex amplitudes in the (m-1)th and mth layers by the matrix relation(25)(Em-1+Em-1-)=1tmCm(Em+Em-),
where tm is the Fresnel transmission coefficient (6) for the mth interface at z=zm-1. The coupling matrix appearing in (25)(26)Cm=(eΔm-1rmeΔm-1rme-Δm-1e-Δm-1)withΔm-1=γm-1dm-1
is identical in form to those obtained previously [5, 9].To express the input electric fieldsE0± in terms of the output field EN-1+, it is necessary to successively multiply the coupling matrices (26) for all layers entering into the NM-structure depicted in Figure 4. As a result, we obtain from (25) the following relation:(27)(E0+E0-)=1t1t2⋯tN-1(aNbNcNdN)(EN-1+0),
where EN-1-=0 for a semiinfinite output medium and the resultant coupling matrix is(28)(aNbNcNdN)=C1C2⋯CN-1.The reflectance and transmittance (defined as ratios of the reflected and transmitted power to the incident power) immediately follow from (27) as(29)R≡|E0-|2|E0+|2=|cN|2|aN|2,T≡nN-1n0|EN-1+|2|E0+|2=nN-1n0|t1|2|t2|2⋯|tN-1|2|aN|2.Therefore, in order to obtainR and T, it is necessary to find only two matrix elements for calculating |aN|2 and |cN|2 and also to know the Fresnel transmission coefficients tm of all interfaces (m=1,2,…,N-1) for which from (19) it follows that(30)|tm|2=|1+rm|2=1+ρ¯m2+2ρ¯mcos⁡ϕ¯m.Let us apply (26)–(29) to the special cases appropriate to the 3M structure (Figure 1(a)) and the 4M structure (Figure 1(b)).


## 3.3. Reflectance and Transmittance of the 3M Structure

For the 3M structure shown in Figure1(a) the coupling matrix product equals(31)C1C2=(1r1r11)(eΔ1r2eΔ1r2e-Δ1e-Δ1)=(eΔ1+r1r2e-Δ1r1e-Δ1+r2eΔ1r1eΔ1+r2e-Δ1e-Δ1+r1r2eΔ1)≡(a3b3c3d3),
where Δ1≡γ1d1=α1d1+iβ1d1 (see formulae (21) for m=1) and the Fresnel coefficients r1 and r2 have the exponential form (19).The general formulae (29) yield the required expressions for the reflectance and transmittance of the 3M structure(32)R=|c3|2|a3|2,T=n2n0|t1|2|t2|2|a3|2.
From elements of the coupling matrix (31) and (30), we have obtained(33)|a3|2=(e2α1d1+ρ¯12ρ¯22e-2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1-ϕ¯1)-ϕ¯2],(34)|c3|2=(ρ¯12e2α1d1+ρ¯22e-2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1+ϕ¯1)-ϕ¯2],(35)|t1|2|t2|2=(1+ρ¯12+2ρ¯1cos⁡ϕ¯1)(1+ρ¯22+2ρ¯2cos⁡ϕ¯2),
where ρ¯1,2≡s1,2ρ1,2 and ϕ¯1,2≡s1,2ϕ1,2.


## 3.4. Reflectance and Transmittance of the 4M Structure

For the 4M structure shown in Figure1(b) the coupling matrix product has the form(36)C1C2C3=(a3b3c3d3)(eΔ2r3eΔ2r3e-Δ2e-Δ2)≡(a4b4c4d4),
with the elements a4 and c4 being equal to(37)a4=e(Δ1+Δ2)+r1r2e-(Δ1-Δ2)+r1r3e-(Δ1+Δ2)+r2r3e(Δ1-Δ2),c4=r1e(Δ1+Δ2)+r2e-(Δ1-Δ2)+r3e-(Δ1+Δ2)+r1r2r3e(Δ1-Δ2).
Here, Δ1,2≡γ1,2d1,2=α1,2d1,2+iβ1,2d1,2 (see formulae (21) for m=1,2) and the Fresnel coefficients rm are of exponential form (19), namely, rm=ρ¯mexp(iϕ¯m) with ρ¯m=smρm and ϕ¯m=smϕm (see (11), (15), and (18) for m=1,2,3).The general formulae (29) yield the required expressions for the reflectance and transmittance of the 4M-structure (cf. expressions (32))(38)R=|c4|2|a4|2,T=n3n0|t1|2|t2|2|t3|2|a4|2,
where the product |t1|2|t2|2|t3|2 is written similar to expression (35).From (37), after some laborious transformations, we obtain the desired expressions(39)|a4|2=(|a3|2e2α2d2+ρ¯32|a3′|2e-2α2d2)+ρ¯3fa(β2d2),|c4|2=(|c3|2e2α2d2+ρ¯32|c3′|2e-2α2d2)+ρ¯3fc(β2d2),
where the terms |a3|2 and |c3|2 have (33) and (34) inherent in the 3M structure. Besides, (39) include certain additional terms(i)
the terms due to the interference and absorption in layer 1 of optical thicknessβ1d1(40)|a3′|2=(ρ¯12e-2α1d1+ρ¯22e2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1-ϕ¯1)+ϕ¯2],|c3′|2=(e-2α1d1+ρ¯12ρ¯22e2α1d1)+2ρ¯1ρ¯2cos⁡[(2β1d1+ϕ¯1)+ϕ¯2],(ii)
the terms due to the interference and absorption in layer 2 of optical thicknessβ2d2(41)ρ¯3fa(β2d2)=2ρ¯1ρ¯3{cos⁡[(2β1d1-ϕ¯1)+(2β2d2-ϕ¯3)]+ρ¯22cos⁡[(2β1d1-ϕ¯1)-(2β2d2-ϕ¯3)]}+2ρ¯2ρ¯3{e2α1d1cos⁡[(2β2d2-ϕ¯3)-ϕ¯2]+ρ¯12e-2α1d1cos⁡[(2β2d2-ϕ¯3)+ϕ¯2]},ρ¯3fc(β2d2)=2ρ¯1ρ¯3{cos⁡[(2β1d1+ϕ¯1)+(2β2d2-ϕ¯3)]+ρ¯22cos⁡[(2β1d1+ϕ¯1)-(2β2d2-ϕ¯3)]}+2ρ¯2ρ¯3{e-2α1d1cos⁡[(2β2d2-ϕ¯3)+ϕ¯2]+ρ¯12e2α1d1cos⁡[(2β2d2-ϕ¯3)-ϕ¯2]}.From a comparison of (33)-(34) and (40), it follows that both terms (|a3|2,|c3|2) and (|a3′|2,|c3′|2) are produced by the interference and absorption phenomena only inside layer 1. Both of them, being inserted in the 4M composition (|a4|2,|c4|2), as seen from (39), prove to be multiplied by the factors exp(±2α2d2) due to optical losses in layer 2. The interference oscillations in this layer (together with an additional contribution from interference and absorption in layer 1) are solely taken into account by the functions ρ¯3fa(β2d2) and ρ¯3fc(β2d2) given in the form of (41).It is very important to note that both the additional terms (|a3′|2,|c3′|2) and (fa,fc) contribute to expressions (39) for the 4M structure being multiplied by ρ3≡|r3|∝|ñ2-ñ3|. Therefore, their contributions to (38) for R and T disappear and, as a result of this, they are converted into (32) for the 3M structure when(42)ñ3=ñ2,d2=0.Both requirements (42) are needed for the conversion 4M → 3M, as applied to the transmittance T, while the reflectance conversion is immediately fulfilled if ñ3=ñ2, that is, if only medium 3 does not differ from medium 2.Hence, our further transformations will deal with the 4M structure, as a general one, to reduce the reflectanceR and transmittance T given by (38)–(41) to the so-called physical form. Such a form has to contain instead of the Fresnel coefficients the physical constants of media (refractive and absorptive indices) as well as the geometrical parameters of a structure under consideration.


## 4. Physical Form of Expressions for the Reflectance and Transmittance

### 4.1. General Expressions

AppendixA is devoted to expressing the quantities |a4|2 and |c4|2 obtained above in the form of (39)–(41) in terms of the physical and geometrical parameters of the 4M structure. The final result is given by relations (A.39)-(A.40) and so their substitution into (38) yields the desired expressions for the reflectance and transmittance(43)R=L-+Mcos⁡(2β1d1-φ-)+F-(β2d2)L++Mcos⁡(2β1d1-φ+)+F+(β2d2),T=16n0|ñ1|2|ñ2|2n3L++Mcos⁡(2β1d1-φ+)+F+(β2d2),
where all the subscripts + and − are in line with double signs appearing in the proper coefficients given below.In (43), we have introduced the following new quantities:(i)the amplitudeMand phase anglesφ± of the interference oscillations in layer 1 of the optical thickness β1d1 (see (A.14)–(A.16) and (A.18)-(A.19))(44)M=s1s2S12(Acos⁡ϕ¯2)2+(Bsinϕ¯2)2,(45)φ±=ψ±ϕ¯1withtanψ=(BA)tanϕ¯2,(46)A=(|ñ2|2+|ñ3|2)cosh⁡2α2d2+2(n2n3+k2k3)sinh⁡2α2d2,(47)B=(|ñ2|2+|ñ3|2)sinh⁡2α2d2+2(n2n3+k2k3)cosh⁡2α2d2,(48)S12={[(n0-n1)2+k12][(n0+n1)2+k12]×[(n1-n2)2+(k1-k2)2]×[(n1+n2)2+(k1+k2)2]}1/2,(ii) the losses parametersL± due to optical absorption (see (A.26)-(A.27))(49)L±=a±cosh⁡2α1d1+b±sinh⁡2α1d1,(50)a±=(n02+|ñ1|2)(|ñ1|2+|ñ2|2)A±4n0n1(n1n2+k1k2)B,(51)b±=2[(n02+|ñ1|2)(n1n2+k1k2)B±n0n1(|ñ1|2+|ñ2|2)A],(iii) the functionsF±(β2d2) taking into account the interference oscillations in layer 2 of the optical thickness β2d2 (together with an additional contribution from the interference and absorption in layer 1) (see (A.32)-(A.33) and (A.36)–(A.38))(52)F±(β2d2)=N±cos⁡[(2β2d2-ϕ¯3)-ψ±]+Pcos⁡(2β1d1∓ϕ¯1)cos⁡(2β2d2-ϕ¯3)-Qsin(2β1d1∓ϕ¯1)sin(2β2d2-ϕ¯3),(53)N±=s2s3S23(A±cos⁡ϕ¯2)2+(B±sinϕ¯2)2,(54)tanψ±=(B±A±)tanϕ¯2,(55)A±=(n02+|ñ1|2)cosh⁡2α1d1±2n0n1sinh⁡2α1d1,(56)B±=(n02+|ñ1|2)sinh⁡2α1d1±2n0n1cosh⁡2α1d1,(57)P=(|ñ1|2+|ñ2|2)s1s3S13,Q=2(n1n2+k1k2)s1s3S13,(58)S13={[(n0-n1)2+k02][(n0+n1)2+k02]×[(n2-n3)2+(k2-k3)2]×[(n2+n3)2+(k2+k3)2]}1/2,(59)S23={[(n1-n2)2+(k1-k2)2][(n1+n2)2+(k1+k2)2]×[(n2-n3)2+(k2-k3)2]×[(n2+n3)2+(k2+k3)2]}1/2.The first term in (52) for F±(β2d2) is a contribution caused by single interference oscillations in layer 2 having the amplitudes N± and phase angles ψ±, which is similar to the interference term of layer 1 with the amplitude M and phase angles φ±. The last two terms in (52) involving products of trigonometric functions with the amplitudes P and Q reflect a contribution from the double (intermodulation) interference due to mutual oscillations in both layers 1 and 2.


### 4.2. Particular Expressions Taking Account of the Device Averaging

It is the functionsF±(β2d2) that take into account the fast oscillations generated by interference in a thick substrate (when n2d2≫n1d1) and result in appearing a “beard” on the optical spectra similar to that shown in Figure 2. As discussed above, any spectrophotometer with a operating slit of finite width cannot register these fast oscillations. In other words, such a spectrophotometer performs the so-called device averaging whose mathematical justification is given in Appendix B. As follows from (B.11) and (B.12), the device averaging allows us to assume F±(β2d2)~0 in (43) for the spectra R(λ) and T(λ).Thus, when taking into account the device averaging, (43) for the 4M structure take the simplest and succinct form(60)R=L-+Mcos⁡(2β1d1-φ-)L++Mcos⁡(2β1d1-φ+),(61)T=16n0|ñ1|2|ñ2|2n3L++Mcos⁡(2β1d1-φ+),
with all the quantities (44)–(51) being left unchanged.Accordingly, by means of these quantities the spectraR(λ) and T(λ) now take into account the optical constants (n2, k2) of a thick substrate and its finite thickness (d2≫d1) but fully disregard interference effects in the substrate.The analogous simplified form ofR and T, as applied to the special case of the 3M structure, follows immediately from (43). Indeed, if (42) of the conversion 4M→3M are fulfilled, (58) and (59) provide S13=S23=0 (because of n3=n2 and k3=k2) so that F±(β2d2)=0 because of N±=P=Q=0. Besides, in this case from (46) and (47) it follows that(62)A=B=2|ñ2|2,
which ensures two consequences: (i) the phase angles φ±=ϕ¯2±ϕ¯1 since tanψ=tanϕ¯2 when A=B (see (45)), (ii) the quantities M and L± get a common multiplier equal to (62) (see (44) and (49)–(51)), which is cancelled after substituting into the numerator and denominator of (43)  for R and T.In such a case, (43) assume the forms specific to the 3M structure of arbitrary physical and geometrical properties(63)R=L-+Mcos⁡(2β1d1-φ-)L++Mcos⁡(2β1d1-φ+),(64)T=8n0|ñ1|2n2L++Mcos⁡(2β1d1-φ+),
where (cf. (44), (45), (45) and (49)–(51))(65)M=s1s2S12,φ±=ϕ¯2±ϕ¯1,(66)S12={[(n0-n1)2+k12][(n0+n1)2+k12]×[(n1-n2)2+(k1-k2)2]×[(n1+n2)2+(k1+k2)2]}1/2,(67)L±=a±cosh⁡2α1d1+b±sinh⁡2α1d1,(68)a±=(n02+|ñ1|2)(|ñ1|2+|ñ2|2)±4n0n1(n1n2+k1k2),(69)b±=2[(n02+|ñ1|2)(n1n2+k1k2)±n0n1(|ñ1|2+|ñ2|2)].Expressions (63)–(69) are valid for any one of the 3M structures and give appropriate equations for the different special cases encountered in the literature [1–12].The anglesϕ¯1 and ϕ¯2 appearing in (45) and (65) display a phase of the Fresnel reflection coefficients (19) for interfaces 1 and 2 (shown in Figure 1(b)) and, in accordance with formulae (A.2) and (A.3), they are defined in the form(70)tanϕ¯1≡tan(s1ϕ1)=s1tanϕ1=2n0k1n02-|ñ1|2,tanϕ¯2≡tan(s2ϕ2)=s2tanϕ2=2(n1k2-n2k1)|ñ1|2-|ñ2|2.
As follows from (70), these angles may be called the losses angles because in a lossless situation they vanish tanϕ¯1=tanϕ¯2=0 when k1=k2=0.It is extremely remarkable that (60)-(61) for the 4M structures with the device averaging and expressions (63)-(64) for the arbitrary 3M structures completely coincide in form and differ only by the appropriate parameters (44)–(51) for the former and (65)–(69) for the latter, which are also of the same configuration. Such a coincidence enable the reflectance and/or transmission spectra to be analyzed below in the same way, as applied to both types of the film-substrate structures.There is a variety of the 3M structures and the 4M structures but only two of them find the wide application in optical measurement techniques, namely(i)
an arbitrary film on a highly absorbing (metal) substrate(ii)
a slightly absorbing film on a transparent (dielectric) substrate.Let us consider modifications of the above expressions forR and T, as applied to the two particular cases.


#### 4.2.1. Arbitrary Films on Highly Absorbing Substrates

are characterized by the only requirement(71)α2d2issomuchlargethatcosh⁡2α2d2≃sinh⁡2α2d2≃12e2α2d2.
Then, from (46) and (47), it follows that(72)A=B=12[(n2+n3)2+(k2+k3)2]e2α2d2.
The condition (72) works as well as (62) did before; that is, it ensures the same consequences: (i) the phase angles φ±=ϕ¯2±ϕ¯1 because tanψ=tanϕ¯2 when A=B (see (45)) and (ii) the quantities M and L± acquire a common multiplier equal to (72) (see (44) and (49)–(51)).Hence, the reflectanceR of the 4M structure with a highly absorbing substrate, provided (71) holds true, is described by (63) inherent in the 3M structure with (65)–(70) left unchanged. As for the transmittance T, its expression (61) proves to be divided by (72) so that values of T practically vanish because of the factor exp(-2α2d2)→0. Therefore, for thin films on a highly absorbing substrates, there is no point in using formulae for the 4M structure.


#### 4.2.2. Slightly Absorbing Films on Transparent Substrates

are characterized by the following requirements:(73)k12≪n12,|ñ1|2≃n12,k2=k3=α2d2=0,|ñ2|2=n22|ñ3|2=n32.
Then, from (46)-(47) and (48) or (66), it, respectively, follows that(74)A=n22+n32,B=2n2n3,(75)S12=|n02-n12||n12-n22|,s1s2S12=(n02-n12)(n12-n22).Requirements (73) modify the losses angles ϕ¯1 and ϕ¯2 so that (70) take the following small-losses form(76)tanϕ¯1≃2n0k1n02-n12,tanϕ¯2≃-2n2k1n12-n22.
In terms of (74) and (76), (45) assumes the form(77)tanψ=BAtanϕ¯2≃-4n22n3k1(n12-n22)(n22+n32).Formulae (74)–(77) are used below to obtain some simplified expressions for the basic quantities M,φ± and L± defining the reflectance and transmission spectra. In so doing, we allow for the small-losses consequences (tanϕ¯2)2≪1 and (tanψ)2≪1.The amplitudeM of interference oscillations follows from formulae (65) and (44) in the following form:(i)
for the 3M structure(78)M=s1s2S12=(n02-n12)(n12-n22),(ii)
for the 4M structure(79)M≃s1s2S12A=(n02-n12)(n12-n22)(n22+n32).The phase anglesφ± of interference oscillations for the 3M and 4M structures given by (65) and (45) can be represented in a common form(80)tanφ±=m±Mk1n1,
where(i)
for the 3M structure(81)m±=2n1[-(n02-n12)n2±n0(n12-n22)],(ii)
for the 4M structure(82)m±=2n1[-(n02-n12)n2B±n0(n12-n22)A]=2n1[-2(n02-n12)n22n3±n0(n12-n22)(n22+n32)].The losses parametersL± for the 3M and 4M structures retain the common form (67) or (49) with the factors a± and b± of the following form:(i)
for the 3M structure(83)a±=(n02+n12)(n12+n22)±4n0n12n2,b±=2n1[(n02+n12)n2±n0(n12+n22)],(ii)
for the 4M structure(84)a±=(n02+n12)(n12+n22)A±4n0n12n2B=(n02+n12)(n12+n22)(n22+n32)±8n0n12n22n3,b±=2n1[(n02+n12)n2B±n0(n12+n22)A]=2n1[2(n02+n12)n22n3±n0(n12+n22)(n22+n32)].Let us analyze the reflectance and transmittance spectra of the 4M structure with the device averaging and the 3M structure by taking advantage of coinciding with (60)-(61) for the former and with (63)-(64) for the latter.


## 4.1. General Expressions

AppendixA is devoted to expressing the quantities |a4|2 and |c4|2 obtained above in the form of (39)–(41) in terms of the physical and geometrical parameters of the 4M structure. The final result is given by relations (A.39)-(A.40) and so their substitution into (38) yields the desired expressions for the reflectance and transmittance(43)R=L-+Mcos⁡(2β1d1-φ-)+F-(β2d2)L++Mcos⁡(2β1d1-φ+)+F+(β2d2),T=16n0|ñ1|2|ñ2|2n3L++Mcos⁡(2β1d1-φ+)+F+(β2d2),
where all the subscripts + and − are in line with double signs appearing in the proper coefficients given below.In (43), we have introduced the following new quantities:(i)the amplitudeMand phase anglesφ± of the interference oscillations in layer 1 of the optical thickness β1d1 (see (A.14)–(A.16) and (A.18)-(A.19))(44)M=s1s2S12(Acos⁡ϕ¯2)2+(Bsinϕ¯2)2,(45)φ±=ψ±ϕ¯1withtanψ=(BA)tanϕ¯2,(46)A=(|ñ2|2+|ñ3|2)cosh⁡2α2d2+2(n2n3+k2k3)sinh⁡2α2d2,(47)B=(|ñ2|2+|ñ3|2)sinh⁡2α2d2+2(n2n3+k2k3)cosh⁡2α2d2,(48)S12={[(n0-n1)2+k12][(n0+n1)2+k12]×[(n1-n2)2+(k1-k2)2]×[(n1+n2)2+(k1+k2)2]}1/2,(ii) the losses parametersL± due to optical absorption (see (A.26)-(A.27))(49)L±=a±cosh⁡2α1d1+b±sinh⁡2α1d1,(50)a±=(n02+|ñ1|2)(|ñ1|2+|ñ2|2)A±4n0n1(n1n2+k1k2)B,(51)b±=2[(n02+|ñ1|2)(n1n2+k1k2)B±n0n1(|ñ1|2+|ñ2|2)A],(iii) the functionsF±(β2d2) taking into account the interference oscillations in layer 2 of the optical thickness β2d2 (together with an additional contribution from the interference and absorption in layer 1) (see (A.32)-(A.33) and (A.36)–(A.38))(52)F±(β2d2)=N±cos⁡[(2β2d2-ϕ¯3)-ψ±]+Pcos⁡(2β1d1∓ϕ¯1)cos⁡(2β2d2-ϕ¯3)-Qsin(2β1d1∓ϕ¯1)sin(2β2d2-ϕ¯3),(53)N±=s2s3S23(A±cos⁡ϕ¯2)2+(B±sinϕ¯2)2,(54)tanψ±=(B±A±)tanϕ¯2,(55)A±=(n02+|ñ1|2)cosh⁡2α1d1±2n0n1sinh⁡2α1d1,(56)B±=(n02+|ñ1|2)sinh⁡2α1d1±2n0n1cosh⁡2α1d1,(57)P=(|ñ1|2+|ñ2|2)s1s3S13,Q=2(n1n2+k1k2)s1s3S13,(58)S13={[(n0-n1)2+k02][(n0+n1)2+k02]×[(n2-n3)2+(k2-k3)2]×[(n2+n3)2+(k2+k3)2]}1/2,(59)S23={[(n1-n2)2+(k1-k2)2][(n1+n2)2+(k1+k2)2]×[(n2-n3)2+(k2-k3)2]×[(n2+n3)2+(k2+k3)2]}1/2.The first term in (52) for F±(β2d2) is a contribution caused by single interference oscillations in layer 2 having the amplitudes N± and phase angles ψ±, which is similar to the interference term of layer 1 with the amplitude M and phase angles φ±. The last two terms in (52) involving products of trigonometric functions with the amplitudes P and Q reflect a contribution from the double (intermodulation) interference due to mutual oscillations in both layers 1 and 2.


## 4.2. Particular Expressions Taking Account of the Device Averaging

It is the functionsF±(β2d2) that take into account the fast oscillations generated by interference in a thick substrate (when n2d2≫n1d1) and result in appearing a “beard” on the optical spectra similar to that shown in Figure 2. As discussed above, any spectrophotometer with a operating slit of finite width cannot register these fast oscillations. In other words, such a spectrophotometer performs the so-called device averaging whose mathematical justification is given in Appendix B. As follows from (B.11) and (B.12), the device averaging allows us to assume F±(β2d2)~0 in (43) for the spectra R(λ) and T(λ).Thus, when taking into account the device averaging, (43) for the 4M structure take the simplest and succinct form(60)R=L-+Mcos⁡(2β1d1-φ-)L++Mcos⁡(2β1d1-φ+),(61)T=16n0|ñ1|2|ñ2|2n3L++Mcos⁡(2β1d1-φ+),
with all the quantities (44)–(51) being left unchanged.Accordingly, by means of these quantities the spectraR(λ) and T(λ) now take into account the optical constants (n2, k2) of a thick substrate and its finite thickness (d2≫d1) but fully disregard interference effects in the substrate.The analogous simplified form ofR and T, as applied to the special case of the 3M structure, follows immediately from (43). Indeed, if (42) of the conversion 4M→3M are fulfilled, (58) and (59) provide S13=S23=0 (because of n3=n2 and k3=k2) so that F±(β2d2)=0 because of N±=P=Q=0. Besides, in this case from (46) and (47) it follows that(62)A=B=2|ñ2|2,
which ensures two consequences: (i) the phase angles φ±=ϕ¯2±ϕ¯1 since tanψ=tanϕ¯2 when A=B (see (45)), (ii) the quantities M and L± get a common multiplier equal to (62) (see (44) and (49)–(51)), which is cancelled after substituting into the numerator and denominator of (43)  for R and T.In such a case, (43) assume the forms specific to the 3M structure of arbitrary physical and geometrical properties(63)R=L-+Mcos⁡(2β1d1-φ-)L++Mcos⁡(2β1d1-φ+),(64)T=8n0|ñ1|2n2L++Mcos⁡(2β1d1-φ+),
where (cf. (44), (45), (45) and (49)–(51))(65)M=s1s2S12,φ±=ϕ¯2±ϕ¯1,(66)S12={[(n0-n1)2+k12][(n0+n1)2+k12]×[(n1-n2)2+(k1-k2)2]×[(n1+n2)2+(k1+k2)2]}1/2,(67)L±=a±cosh⁡2α1d1+b±sinh⁡2α1d1,(68)a±=(n02+|ñ1|2)(|ñ1|2+|ñ2|2)±4n0n1(n1n2+k1k2),(69)b±=2[(n02+|ñ1|2)(n1n2+k1k2)±n0n1(|ñ1|2+|ñ2|2)].Expressions (63)–(69) are valid for any one of the 3M structures and give appropriate equations for the different special cases encountered in the literature [1–12].The anglesϕ¯1 and ϕ¯2 appearing in (45) and (65) display a phase of the Fresnel reflection coefficients (19) for interfaces 1 and 2 (shown in Figure 1(b)) and, in accordance with formulae (A.2) and (A.3), they are defined in the form(70)tanϕ¯1≡tan(s1ϕ1)=s1tanϕ1=2n0k1n02-|ñ1|2,tanϕ¯2≡tan(s2ϕ2)=s2tanϕ2=2(n1k2-n2k1)|ñ1|2-|ñ2|2.
As follows from (70), these angles may be called the losses angles because in a lossless situation they vanish tanϕ¯1=tanϕ¯2=0 when k1=k2=0.It is extremely remarkable that (60)-(61) for the 4M structures with the device averaging and expressions (63)-(64) for the arbitrary 3M structures completely coincide in form and differ only by the appropriate parameters (44)–(51) for the former and (65)–(69) for the latter, which are also of the same configuration. Such a coincidence enable the reflectance and/or transmission spectra to be analyzed below in the same way, as applied to both types of the film-substrate structures.There is a variety of the 3M structures and the 4M structures but only two of them find the wide application in optical measurement techniques, namely(i)
an arbitrary film on a highly absorbing (metal) substrate(ii)
a slightly absorbing film on a transparent (dielectric) substrate.Let us consider modifications of the above expressions forR and T, as applied to the two particular cases.


### 4.2.1. Arbitrary Films on Highly Absorbing Substrates

are characterized by the only requirement(71)α2d2issomuchlargethatcosh⁡2α2d2≃sinh⁡2α2d2≃12e2α2d2.
Then, from (46) and (47), it follows that(72)A=B=12[(n2+n3)2+(k2+k3)2]e2α2d2.
The condition (72) works as well as (62) did before; that is, it ensures the same consequences: (i) the phase angles φ±=ϕ¯2±ϕ¯1 because tanψ=tanϕ¯2 when A=B (see (45)) and (ii) the quantities M and L± acquire a common multiplier equal to (72) (see (44) and (49)–(51)).Hence, the reflectanceR of the 4M structure with a highly absorbing substrate, provided (71) holds true, is described by (63) inherent in the 3M structure with (65)–(70) left unchanged. As for the transmittance T, its expression (61) proves to be divided by (72) so that values of T practically vanish because of the factor exp(-2α2d2)→0. Therefore, for thin films on a highly absorbing substrates, there is no point in using formulae for the 4M structure.


### 4.2.2. Slightly Absorbing Films on Transparent Substrates

are characterized by the following requirements:(73)k12≪n12,|ñ1|2≃n12,k2=k3=α2d2=0,|ñ2|2=n22|ñ3|2=n32.
Then, from (46)-(47) and (48) or (66), it, respectively, follows that(74)A=n22+n32,B=2n2n3,(75)S12=|n02-n12||n12-n22|,s1s2S12=(n02-n12)(n12-n22).Requirements (73) modify the losses angles ϕ¯1 and ϕ¯2 so that (70) take the following small-losses form(76)tanϕ¯1≃2n0k1n02-n12,tanϕ¯2≃-2n2k1n12-n22.
In terms of (74) and (76), (45) assumes the form(77)tanψ=BAtanϕ¯2≃-4n22n3k1(n12-n22)(n22+n32).Formulae (74)–(77) are used below to obtain some simplified expressions for the basic quantities M,φ± and L± defining the reflectance and transmission spectra. In so doing, we allow for the small-losses consequences (tanϕ¯2)2≪1 and (tanψ)2≪1.The amplitudeM of interference oscillations follows from formulae (65) and (44) in the following form:(i)
for the 3M structure(78)M=s1s2S12=(n02-n12)(n12-n22),(ii)
for the 4M structure(79)M≃s1s2S12A=(n02-n12)(n12-n22)(n22+n32).The phase anglesφ± of interference oscillations for the 3M and 4M structures given by (65) and (45) can be represented in a common form(80)tanφ±=m±Mk1n1,
where(i)
for the 3M structure(81)m±=2n1[-(n02-n12)n2±n0(n12-n22)],(ii)
for the 4M structure(82)m±=2n1[-(n02-n12)n2B±n0(n12-n22)A]=2n1[-2(n02-n12)n22n3±n0(n12-n22)(n22+n32)].The losses parametersL± for the 3M and 4M structures retain the common form (67) or (49) with the factors a± and b± of the following form:(i)
for the 3M structure(83)a±=(n02+n12)(n12+n22)±4n0n12n2,b±=2n1[(n02+n12)n2±n0(n12+n22)],(ii)
for the 4M structure(84)a±=(n02+n12)(n12+n22)A±4n0n12n2B=(n02+n12)(n12+n22)(n22+n32)±8n0n12n22n3,b±=2n1[(n02+n12)n2B±n0(n12+n22)A]=2n1[2(n02+n12)n22n3±n0(n12+n22)(n22+n32)].Let us analyze the reflectance and transmittance spectra of the 4M structure with the device averaging and the 3M structure by taking advantage of coinciding with (60)-(61) for the former and with (63)-(64) for the latter.


## 4.2.1. Arbitrary Films on Highly Absorbing Substrates

are characterized by the only requirement(71)α2d2issomuchlargethatcosh⁡2α2d2≃sinh⁡2α2d2≃12e2α2d2.
Then, from (46) and (47), it follows that(72)A=B=12[(n2+n3)2+(k2+k3)2]e2α2d2.
The condition (72) works as well as (62) did before; that is, it ensures the same consequences: (i) the phase angles φ±=ϕ¯2±ϕ¯1 because tanψ=tanϕ¯2 when A=B (see (45)) and (ii) the quantities M and L± acquire a common multiplier equal to (72) (see (44) and (49)–(51)).Hence, the reflectanceR of the 4M structure with a highly absorbing substrate, provided (71) holds true, is described by (63) inherent in the 3M structure with (65)–(70) left unchanged. As for the transmittance T, its expression (61) proves to be divided by (72) so that values of T practically vanish because of the factor exp(-2α2d2)→0. Therefore, for thin films on a highly absorbing substrates, there is no point in using formulae for the 4M structure.


## 4.2.2. Slightly Absorbing Films on Transparent Substrates

are characterized by the following requirements:(73)k12≪n12,|ñ1|2≃n12,k2=k3=α2d2=0,|ñ2|2=n22|ñ3|2=n32.
Then, from (46)-(47) and (48) or (66), it, respectively, follows that(74)A=n22+n32,B=2n2n3,(75)S12=|n02-n12||n12-n22|,s1s2S12=(n02-n12)(n12-n22).Requirements (73) modify the losses angles ϕ¯1 and ϕ¯2 so that (70) take the following small-losses form(76)tanϕ¯1≃2n0k1n02-n12,tanϕ¯2≃-2n2k1n12-n22.
In terms of (74) and (76), (45) assumes the form(77)tanψ=BAtanϕ¯2≃-4n22n3k1(n12-n22)(n22+n32).Formulae (74)–(77) are used below to obtain some simplified expressions for the basic quantities M,φ± and L± defining the reflectance and transmission spectra. In so doing, we allow for the small-losses consequences (tanϕ¯2)2≪1 and (tanψ)2≪1.The amplitudeM of interference oscillations follows from formulae (65) and (44) in the following form:(i)
for the 3M structure(78)M=s1s2S12=(n02-n12)(n12-n22),(ii)
for the 4M structure(79)M≃s1s2S12A=(n02-n12)(n12-n22)(n22+n32).The phase anglesφ± of interference oscillations for the 3M and 4M structures given by (65) and (45) can be represented in a common form(80)tanφ±=m±Mk1n1,
where(i)
for the 3M structure(81)m±=2n1[-(n02-n12)n2±n0(n12-n22)],(ii)
for the 4M structure(82)m±=2n1[-(n02-n12)n2B±n0(n12-n22)A]=2n1[-2(n02-n12)n22n3±n0(n12-n22)(n22+n32)].The losses parametersL± for the 3M and 4M structures retain the common form (67) or (49) with the factors a± and b± of the following form:(i)
for the 3M structure(83)a±=(n02+n12)(n12+n22)±4n0n12n2,b±=2n1[(n02+n12)n2±n0(n12+n22)],(ii)
for the 4M structure(84)a±=(n02+n12)(n12+n22)A±4n0n12n2B=(n02+n12)(n12+n22)(n22+n32)±8n0n12n22n3,b±=2n1[(n02+n12)n2B±n0(n12+n22)A]=2n1[2(n02+n12)n22n3±n0(n12+n22)(n22+n32)].Let us analyze the reflectance and transmittance spectra of the 4M structure with the device averaging and the 3M structure by taking advantage of coinciding with (60)-(61) for the former and with (63)-(64) for the latter.


## 5. Analysis of the Reflectance and Transmittance Spectra

All the relations obtained above are valid for the general case when eachmth medium incorporated into the film-substrate structure is absorbing and dispersive, that is, ñm(λ)=nm(λ)-ikm(λ),m=1,2,3. Such a dependence of the refractive indices on light wavelength λ naturally manifests itself in the optical spectra R(λ) and T(λ). However, a leading contribution into interference oscillations of the spectra is made by the periodic functions cos(2β1d1-φ±) appearing in (60)-(61) or (63)-(64) because usually the functions ñm(λ) are slowly varying as compared with the periodic functions.For this reason, below we take into account only the main contribution to the dependenciesR(λ) and T(λ) from the cosine functions.


### 5.1. Reflectance Spectrum

Let us write down the reflectance spectrum (60) or (63) in the following form:(85)R(x)=L-+Mcos⁡(2x-φ-)L++Mcos⁡(2x-φ+).
We consider instead of R(λ) the frequency spectrum R(x) where x≡β1d1=(ω/c)n1d1 is the dimensionless frequency.Positions of maxima and minima in the spectrum (85) are found from the condition ∂R(x)/∂x=0 which gives the following transcendental equation to find them:(86)L-sin(2β1d1-φ+)-L+sin(2β1d1-φ-)=Msin(φ+-φ-).
Here, the quantities M,φ± and L± have different expressions for either the 4M structure (with the device averaging) or the 3M structure, namely, they are (44)–(48) and (49)–(51) for the former or (65)-(66) and (67)–(69) for the latter.An essential simplification of (86) takes place in the particular case of practical interest when a transparent film (k1=0, α1d1=(2π/λ)k1d1=0) is situated on a highly absorbing substrate being subject to the requirement (71). As proved above, for such a physical situation the 3M structure model is workable even if the substrate is of finite thickness d2 because values of α2d2=(2π/λ)k2d2 are sufficiently large.In this simplified case, from (70), it follows that the losses angles of interfaces 1 and 2 are equal to(87)ϕ¯1=0,ϕ¯2=arctan2n1k2n12-|ñ2|2.
Hence, the phase angles φ+ and φ- given by (65) as φ±=ϕ¯2±ϕ¯1 become equal to each other owing to (87), namely,(88)φ≡φ+=φ-=ϕ¯2,tanφ=2n1k2n12-|ñ2|2,
where |tanφ| is not small because of noticeable losses in a highly absorbing substrate.Then, (85) for the reflectance assumes the simplified form(89)R(λ)=L-+Mcos⁡(2β1d1-φ)L++Mcos⁡(2β1d1-φ),
where the quantities M and L± generally given by (65)–(69) are also reduced with taking into account (88) to the following simplified form:(90)M=s1s2S12=(n02-n12)(n12-|ñ2|2)1+tan2φ,L±=a±=(n02+n12)(n12+|ñ2|2)±4n0n12n2.Formulae (88)–(90) have been used to compute the reflectance spectrum R(λ) for a non-absorbing film (n0=1,n1=2.3,k1=0) of thickness d1=1μm deposited on a titanium substrate (n2=1.89,k2=2.58). Result of the computation is shown in Figure 5 by a solid line. For comparison, here, there is also a dashed line corresponding to the same film on a transparent substrate (n2=1.89,k2=0).Figure 5
Reflectance spectrum of a transparent film on a metal substrate (solid line) and on a nonabsorbing substrate (dashed line).As seen from Figure5, the high absorption of a metal substrate not only influences the amplitude of interference oscillations but also changes the positions of maxima and minima in the spectrum R(λ). To find them, it is necessary to apply an equation resulting from (86) as a particular case for which φ+=φ- and having the utterly simplified form(91)sin(2β1d1-φ)=0.As follows from (91), the absorption of a substrate indeed shifts the spectrum R(x) along the frequency axis x≡β1d1=(ω/c)n1d1 by the same value for all the maxima and minima equal to the losses angle φ=ϕ¯2.According to our experimental data, for amorphous oxide films TiO2 effect of the dispersion function n1(λ) becomes noticeable for λ<600 nm. Figure 6(a) contains the curve n1(λ) which is drawn on the basis of experimental tabular data taken from [27]. We have approximated this curve by an expression (proposed in [24])(92)n1(λ)=aλb+c,
and found its coefficients a=0.015,b=4.025,c=2.2 by the least-squares method. The spectrum R(λ) for such a dispersive film of thickness d1=1μm on a titanium substrate (n2=1.89,k2=2.58) is computed by using (88)–(90) and shown in Figure 6(b) by solid line. A dashed curve describes the analogous spectrum calculated from the same formulae for a constant value of n1=2.3. From comparison of the two curves, it follows that an increase in n1(λ) at short wavelengths leads to a rise in number of oscillations for both the spectra.(a) Experimental curven1(λ) for an oxide film TiO2 [27] and (b) the reflectance spectrum R(λ) calculated for such a film (solid line) and for a nondispersive film with n1=2.3 (dashed line) both deposited on a titanium substrate.
(a)(b)


### 5.2. Transmittance Spectrum

The transmittance spectra (61) for the 4M structure with the device averaging and (64) for the 3M structure can be represented in a common form(93)T(λ)=KL++Mcos⁡(2β1d1-φ+),
where the quantities M,φ+ and L+ are the same as those in the denominator of R(λ) and the numerator K is equal to(i)
for the 3M structure(94)K=8n0|ñ1|2n2,(ii)
for the 4M structure(95)K=16n0|ñ1|2|ñ2|2n3.Position of maxima and minima in the spectrum (93) is found from the condition ∂T(x)/∂x=0 (with x=β1d1) which gives the following equation to find them:(96)sin(2β1d1-φ+)=0.This equations is identical to (91) for R(x) applicable only to transparent films on a highly absorbing substrate whereas (96) for T(x) holds valid always independently of the physical properties and type (3M or 4M) of a structure. As follows from (96), the position of maxima and minima in the transmission spectrum is solely controlled by the phase angle φ+ given by (45) or (65).As noted above, among various thin-film structures, the most practical interest in the transmittance spectrum measurements has been displayed for the transparent or slightly absorbing films on non-absorbing substrates which satisfy (73). The simplified expressions for M,φ+ and L+ which are applicable to such a kind of the 3M and 4M structures have the form given by (78)–(84). These expressions were used to compute the spectrum T(λ) of a slightly absorbing film on a transparent substrate with the following parameters: n0=n3=1,n1=2.21,k1=0.01, n2=1.472,k2=0 and d1=1μm.The results of calculation are given in Figure7 as two curves corresponding to both the 3M structure (solid thick line) and the 4M structure (dashed line). A comparison of these curves demonstrates that the 3M structure model with a semiinfinite substrate of index n2 (see Figure 1(a)) yields substantially higher values of T(λ) because of neglecting influence of medium 3 with index n3 and so that of light reflection from interface 3 (see Figure 1(b)). Besides, there is a solid thin curve calculated by using the extremely simplified (1) and (3) of Swanepoel's approach [24, 25]. These formulae give also higher values of T(λ) only at points of spectral minimum, as compared with our formulae derived on the basis of a rigorous theoretical approach. Such a difference in minimum points of T(λ) between Swanepoel's and our results amounts to 2.5%–3%. This fact refutes a statement of Swanepoel that his theory provides an accuracy error less than 1%.Figure 7
Transmittance spectrum of a thin dielectric film (d1=1 μm) with small optical losses (n1=2.21,k1=0.01) on a transparent substrate (n2=1.472,k2=0) computed for the 3M structure (solid line) and 4M structure (dashed line); a solid thin curve is calculated from (1) and (3) by using the same numerical values.In order to verify applicability of our formulae to processing of experimental spectra for finding thin-film parameters, a tantalum oxide film was deposited on the quartz-glass substrate (n2=1.472,k2=0,d2=500μm) by using the reactive magnetron sputtering method. The experimental spectrum T(λ) of such a thin-film structure measured by a spectrophotometer is shown in Figure 8 as a thin-line curve. For processing of this spectrum we have used the known iteration procedure [20, 21, 27] which allows one to find values of n1(λi) and k1(λi) at points λi(i=1,2,…) of an experimental spectrum.Figure 8
Experimental transmittance spectrum of an oxide filmTa2O5 on a quartz-glass substrate (thin line) and the spectrum T(λ) calculated from (93) for the 4M structure (thick line) by using d1=0.824μm, n1(λ) and k1(λ) in the form of (97) which are obtained from the experimental spectrum.Unlike the above cited papers, we have applied our newly derived (93) where its coefficients for the 4M structure with the device averaging are given in the form of (49), (79), (80), (82), (84), and (95). The values of n1(λi) and k1(λi) obtained from the experimental spectrum T(λ) (a thin curve in Figure 8) have been approximated in the form of (92) with coefficients found by the least-squares method, namely,(97)n1(λ)=0.003λ4.285+2.045,k1(λ)=0.00027λ2.381+0.00050,
where λ is measured in μm. Thickness of the film d1 has been proved to be equal to 0.824 μm while the phase angle φ+ to be negligibly small because of small optical losses.The spectrumT(λ), calculated from our formulae by using d1=0.824μm and (97), is depicted by a thick curve in Figure 8. A close agreement between the experimental and calculated spectra T(λ) justifies the practical applicability of our theoretical results to the processing of experimental optical spectra.


## 5.1. Reflectance Spectrum

Let us write down the reflectance spectrum (60) or (63) in the following form:(85)R(x)=L-+Mcos⁡(2x-φ-)L++Mcos⁡(2x-φ+).
We consider instead of R(λ) the frequency spectrum R(x) where x≡β1d1=(ω/c)n1d1 is the dimensionless frequency.Positions of maxima and minima in the spectrum (85) are found from the condition ∂R(x)/∂x=0 which gives the following transcendental equation to find them:(86)L-sin(2β1d1-φ+)-L+sin(2β1d1-φ-)=Msin(φ+-φ-).
Here, the quantities M,φ± and L± have different expressions for either the 4M structure (with the device averaging) or the 3M structure, namely, they are (44)–(48) and (49)–(51) for the former or (65)-(66) and (67)–(69) for the latter.An essential simplification of (86) takes place in the particular case of practical interest when a transparent film (k1=0, α1d1=(2π/λ)k1d1=0) is situated on a highly absorbing substrate being subject to the requirement (71). As proved above, for such a physical situation the 3M structure model is workable even if the substrate is of finite thickness d2 because values of α2d2=(2π/λ)k2d2 are sufficiently large.In this simplified case, from (70), it follows that the losses angles of interfaces 1 and 2 are equal to(87)ϕ¯1=0,ϕ¯2=arctan2n1k2n12-|ñ2|2.
Hence, the phase angles φ+ and φ- given by (65) as φ±=ϕ¯2±ϕ¯1 become equal to each other owing to (87), namely,(88)φ≡φ+=φ-=ϕ¯2,tanφ=2n1k2n12-|ñ2|2,
where |tanφ| is not small because of noticeable losses in a highly absorbing substrate.Then, (85) for the reflectance assumes the simplified form(89)R(λ)=L-+Mcos⁡(2β1d1-φ)L++Mcos⁡(2β1d1-φ),
where the quantities M and L± generally given by (65)–(69) are also reduced with taking into account (88) to the following simplified form:(90)M=s1s2S12=(n02-n12)(n12-|ñ2|2)1+tan2φ,L±=a±=(n02+n12)(n12+|ñ2|2)±4n0n12n2.Formulae (88)–(90) have been used to compute the reflectance spectrum R(λ) for a non-absorbing film (n0=1,n1=2.3,k1=0) of thickness d1=1μm deposited on a titanium substrate (n2=1.89,k2=2.58). Result of the computation is shown in Figure 5 by a solid line. For comparison, here, there is also a dashed line corresponding to the same film on a transparent substrate (n2=1.89,k2=0).Figure 5
Reflectance spectrum of a transparent film on a metal substrate (solid line) and on a nonabsorbing substrate (dashed line).As seen from Figure5, the high absorption of a metal substrate not only influences the amplitude of interference oscillations but also changes the positions of maxima and minima in the spectrum R(λ). To find them, it is necessary to apply an equation resulting from (86) as a particular case for which φ+=φ- and having the utterly simplified form(91)sin(2β1d1-φ)=0.As follows from (91), the absorption of a substrate indeed shifts the spectrum R(x) along the frequency axis x≡β1d1=(ω/c)n1d1 by the same value for all the maxima and minima equal to the losses angle φ=ϕ¯2.According to our experimental data, for amorphous oxide films TiO2 effect of the dispersion function n1(λ) becomes noticeable for λ<600 nm. Figure 6(a) contains the curve n1(λ) which is drawn on the basis of experimental tabular data taken from [27]. We have approximated this curve by an expression (proposed in [24])(92)n1(λ)=aλb+c,
and found its coefficients a=0.015,b=4.025,c=2.2 by the least-squares method. The spectrum R(λ) for such a dispersive film of thickness d1=1μm on a titanium substrate (n2=1.89,k2=2.58) is computed by using (88)–(90) and shown in Figure 6(b) by solid line. A dashed curve describes the analogous spectrum calculated from the same formulae for a constant value of n1=2.3. From comparison of the two curves, it follows that an increase in n1(λ) at short wavelengths leads to a rise in number of oscillations for both the spectra.(a) Experimental curven1(λ) for an oxide film TiO2 [27] and (b) the reflectance spectrum R(λ) calculated for such a film (solid line) and for a nondispersive film with n1=2.3 (dashed line) both deposited on a titanium substrate.
(a)(b)


## 5.2. Transmittance Spectrum

The transmittance spectra (61) for the 4M structure with the device averaging and (64) for the 3M structure can be represented in a common form(93)T(λ)=KL++Mcos⁡(2β1d1-φ+),
where the quantities M,φ+ and L+ are the same as those in the denominator of R(λ) and the numerator K is equal to(i)
for the 3M structure(94)K=8n0|ñ1|2n2,(ii)
for the 4M structure(95)K=16n0|ñ1|2|ñ2|2n3.Position of maxima and minima in the spectrum (93) is found from the condition ∂T(x)/∂x=0 (with x=β1d1) which gives the following equation to find them:(96)sin(2β1d1-φ+)=0.This equations is identical to (91) for R(x) applicable only to transparent films on a highly absorbing substrate whereas (96) for T(x) holds valid always independently of the physical properties and type (3M or 4M) of a structure. As follows from (96), the position of maxima and minima in the transmission spectrum is solely controlled by the phase angle φ+ given by (45) or (65).As noted above, among various thin-film structures, the most practical interest in the transmittance spectrum measurements has been displayed for the transparent or slightly absorbing films on non-absorbing substrates which satisfy (73). The simplified expressions for M,φ+ and L+ which are applicable to such a kind of the 3M and 4M structures have the form given by (78)–(84). These expressions were used to compute the spectrum T(λ) of a slightly absorbing film on a transparent substrate with the following parameters: n0=n3=1,n1=2.21,k1=0.01, n2=1.472,k2=0 and d1=1μm.The results of calculation are given in Figure7 as two curves corresponding to both the 3M structure (solid thick line) and the 4M structure (dashed line). A comparison of these curves demonstrates that the 3M structure model with a semiinfinite substrate of index n2 (see Figure 1(a)) yields substantially higher values of T(λ) because of neglecting influence of medium 3 with index n3 and so that of light reflection from interface 3 (see Figure 1(b)). Besides, there is a solid thin curve calculated by using the extremely simplified (1) and (3) of Swanepoel's approach [24, 25]. These formulae give also higher values of T(λ) only at points of spectral minimum, as compared with our formulae derived on the basis of a rigorous theoretical approach. Such a difference in minimum points of T(λ) between Swanepoel's and our results amounts to 2.5%–3%. This fact refutes a statement of Swanepoel that his theory provides an accuracy error less than 1%.Figure 7
Transmittance spectrum of a thin dielectric film (d1=1 μm) with small optical losses (n1=2.21,k1=0.01) on a transparent substrate (n2=1.472,k2=0) computed for the 3M structure (solid line) and 4M structure (dashed line); a solid thin curve is calculated from (1) and (3) by using the same numerical values.In order to verify applicability of our formulae to processing of experimental spectra for finding thin-film parameters, a tantalum oxide film was deposited on the quartz-glass substrate (n2=1.472,k2=0,d2=500μm) by using the reactive magnetron sputtering method. The experimental spectrum T(λ) of such a thin-film structure measured by a spectrophotometer is shown in Figure 8 as a thin-line curve. For processing of this spectrum we have used the known iteration procedure [20, 21, 27] which allows one to find values of n1(λi) and k1(λi) at points λi(i=1,2,…) of an experimental spectrum.Figure 8
Experimental transmittance spectrum of an oxide filmTa2O5 on a quartz-glass substrate (thin line) and the spectrum T(λ) calculated from (93) for the 4M structure (thick line) by using d1=0.824μm, n1(λ) and k1(λ) in the form of (97) which are obtained from the experimental spectrum.Unlike the above cited papers, we have applied our newly derived (93) where its coefficients for the 4M structure with the device averaging are given in the form of (49), (79), (80), (82), (84), and (95). The values of n1(λi) and k1(λi) obtained from the experimental spectrum T(λ) (a thin curve in Figure 8) have been approximated in the form of (92) with coefficients found by the least-squares method, namely,(97)n1(λ)=0.003λ4.285+2.045,k1(λ)=0.00027λ2.381+0.00050,
where λ is measured in μm. Thickness of the film d1 has been proved to be equal to 0.824 μm while the phase angle φ+ to be negligibly small because of small optical losses.The spectrumT(λ), calculated from our formulae by using d1=0.824μm and (97), is depicted by a thick curve in Figure 8. A close agreement between the experimental and calculated spectra T(λ) justifies the practical applicability of our theoretical results to the processing of experimental optical spectra.


## 6. Conclusion

In the literature, there are no correct and consistent formulae relating to the reflectance and/or transmittance optical spectra for a thin film on a substrate of finite thickness, which are appropriate to the 4M structure model. A known model of the 3M structure developed previously in sufficient detail corresponds to the semiinfinite substrates and introduces an accuracy error into processing of measured optical spectra.The paper has demonstrated a rigorous approach to development of a model for the 4M-structure. The general expressions (43)–(59) for the spectra R(λ) and T(λ) are applicable to all kinds of the film-substrate structures including the particular case of semiinfinite substrates (see (63)–(69)). However, the general expressions derived for the 4M structure make no practical sense in the case of a sufficiently thick substrate because of appearing fast interference oscillations in the theoretical spectra. Any spectrophotometer with an operating slit of finite width eliminates such oscillations from the experimental spectra by means of their averaging because of nonmonochromatic irradiation (the so-called device averaging).The crucial point of our theory is an introduction of the device averaging procedure into (43) in order to exclude the thick substrate oscillations nonrequired practically from theoretical spectra. This procedure has produced (60)-(61) for R(λ) and T(λ) in the simplest and succinct form to be convenient for processing of the experimental spectra. The newly derived equations have been analyzed, as applied to two special cases: (i) an arbitrary film on highly absorbing substrates and (ii) a slightly absorbing film on transparent substrates (see (78)–(84)). The results of such an analysis have displayed a close agreement between the obtained theoretical relations and experimental measurements.Thus, the reflectance and transmittance spectra represented in the simplified (with the device averaging) form (60)-(61) are practically useful for determining the film thickness and optical constants from experimental spectra by applying the known techniques [20, 21, 23, 27]. Moreover, the more complex (without the device averaging) general form (43) may be applied to the 4M-structure (see Figure 1(b)) with a double film (of comparable values of n1d1 and n2d2) on semiinfinite substrates.



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*Source: 137572-2010-12-30.xml*