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

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.

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.

Section 2 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 index 𝑛₀. In the general case, the film under examination (numbered by 1) has optical losses so that its refractive index is complex and dispersive: 𝑛₁(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 𝑛₂(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 Figure 1: (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 𝑛₃(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.

(a)

(b)

Figure 1 

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 thickness 𝑑2 (medium 2); media 0 and 3 correspond to the input and output ones.

The main task of theoretical analysis is a derivation of mathematical expressions for the spectra of reflectance 𝑅(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 (𝑘²₁≪𝑛²₁) on a transparent substrate (𝑘₂=0) has the following form:𝑇(𝜆)=𝐴𝑥𝐵−𝐶𝑥cos𝜑+𝐷𝑥², (1) 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 𝑛_𝑚, 𝑚=0,1,2, according to Figure 1(a))𝐴=16𝑛₀𝑛²₁𝑛₂,𝐶=−2𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂,𝐵=𝑛₀+𝑛₁²𝑛₁+𝑛₂²,𝐷=𝑛₀−𝑛₁²𝑛₁−𝑛₂². (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:𝐴=16𝑛₀𝑛²₁𝑛₂,𝐶=−2𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂,𝐵=𝑛₀+𝑛₁³𝑛₁+𝑛²₂,𝐷=−𝑛₀−𝑛₁³𝑛₁−𝑛²₂. (3)

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 spectrum 𝑅(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 spectra 𝑅(𝜆) and 𝑇(𝜆) of the 4M structure were also derived by Heavens and represented in the following form [5]:𝑅(𝜆)=𝑡²₁₃+𝑢²₁₃𝑝²₁₃+𝑞²₁₃(𝜆)=𝑙,𝑇²¹³+𝑚²₁₃𝑝²₁₃+𝑞²₁₃. (4) Here, the items 𝑙²₁₃, 𝑚²₁₃, 𝑝²₁₃, 𝑞²₁₃, 𝑡²₁₃, and 𝑢²₁₃ depend on the optical constants 𝑛_𝑚=𝑛_𝑚−𝑖𝑘_𝑚 of media (𝑚=1,2,3), their thicknesses 𝑑_𝑚, and the light wavelength 𝜆.

Expressions (4) only look succinct because the items 𝑙²₁₃, 𝑚²₁₃, 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 𝑙²₁₃, 𝑚²₁₃, 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).

Figure 2 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.

(a)

(b)

Figure 2 

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.

Our subsequent task consists of deriving the analytical expressions for 𝑅(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 indices𝑛_𝑚−1=𝑛_𝑚−1−𝑖𝑘_𝑚−1,𝑛_𝑚=𝑛_𝑚−𝑖𝑘_𝑚, (5) where 𝑚=1,2,… and 𝑛₀=𝑛₀ since 𝑘₀≡0 for a non-absorbing input medium. Each 𝑚th interface is characterized by the Fresnel reflection and transmission coefficients 𝑟_𝑚 and 𝑡_𝑚=1+𝑟_𝑚 defined as [1–7]𝑟_𝑚=𝑛_𝑚−1−𝑛_𝑚𝑛_𝑚−1+𝑛_𝑚,𝑡_𝑚=2𝑛_𝑚−1𝑛_𝑚−1+𝑛_𝑚. (6)

Heavens [5] represents the Fresnel reflection coefficient 𝑟_𝑚 as a sum of its real and imaginary parts𝑟_𝑚=𝑔_𝑚+𝑖ℎ_𝑚, (7) where in accordance with (5) and (6), we have𝑔_𝑚=𝑛_𝑚−1²−𝑛_𝑚²𝑛_𝑚−1+𝑛_𝑚²+𝑘_𝑚−1+𝑘_𝑚²,ℎ_𝑚=2𝑛_𝑚−1𝑘_𝑚−𝑛_𝑚𝑘_𝑚−1𝑛_𝑚−1+𝑛_𝑚²+𝑘_𝑚−1+𝑘_𝑚², (8) with the squared moduli of the refractive indices (5) being equal to𝑛_𝑚−1²=𝑛²_𝑚−1+𝑘²_𝑚−1,𝑛_𝑚²=𝑛²_𝑚+𝑘²_𝑚. (9)

Unlike Heavens’ approach, for our analysis, it is more convenient to apply for 𝑟_𝑚 instead of (7) the exponential form𝑟_𝑚=𝜌_𝑚exp𝑖𝜃_𝑚. (10) The squared modulus 𝜌²_𝑚 and phase 𝜃_𝑚 of the Fresnel reflection coefficient (7) or (10) are defined by the following relations𝜌²_𝑚=𝑔²_𝑚+ℎ²_𝑚=𝑛_𝑚−1−𝑛_𝑚²𝑛_𝑚−1+𝑛_𝑚²≡𝒩_−𝑚𝒩_+𝑚,tan𝜃_𝑚=ℎ_𝑚𝑔_𝑚=2𝑛_𝑚−1𝑘_𝑚−𝑛_𝑚𝑘_𝑚−1𝑛_𝑚−1²−𝑛_𝑚², (11)(12) where we have introduced the following quantities𝒩_±𝑚≡𝑛_𝑚−1±𝑛_𝑚²=𝑛_𝑚−1±𝑛_𝑚²+𝑘_𝑚−1±𝑘_𝑚²=𝑛_𝑚−1²+𝑛_𝑚²±2𝑛_𝑚−1𝑛_𝑚+𝑘_𝑚−1𝑘_𝑚. (13)

In the absence of optical absorption (𝑘_𝑚−1=𝑘_𝑚=0), expression (6) for 𝑟_𝑚 gives purely real values (negative or positive) so that 𝑟_𝑚<0 or 𝑟_𝑚>0 when, respectively, 𝑛_𝑚−1<𝑛_𝑚 or 𝑛_𝑚−1>𝑛_𝑚. Yet, the exponential form (10) does not display explicitly this property. Indeed, from (12), it follows that in the lossless situation tan𝜃_𝑚=0 and by definition always 𝜌_𝑚>0. In order for the above-stated property to be explicitly displayed from relation (10), it is necessary to require the phase 𝜃_𝑚 to be positively defined. This requirement is readily realized by considering the phase angle 𝜃_𝑚 to be a function of the quantity𝑥_𝑚=𝑠_𝑚ℎ_𝑚𝑔_𝑚≡ℎ_𝑚𝑔_𝑚=2𝑛_𝑚−1𝑘_𝑚−𝑛_𝑚𝑘_𝑚−1𝑛_𝑚−1²−𝑛_𝑚²=2𝑛_𝑚−1𝑘_𝑚−𝑛_𝑚𝑘_𝑚−1𝑛²_𝑚−1−𝑛²_𝑚+𝑘²_𝑚−1−𝑘²_𝑚, (14) where the sign of 𝑔_𝑚 is inserted as𝑠_𝑚=𝑔_𝑚𝑔_𝑚=sgn𝑛_𝑚−1²−𝑛_𝑚²=−1if𝑛_𝑚−1<𝑛_𝑚,+1if𝑛_𝑚−1>𝑛_𝑚. (15)

From formulae (12) and (14) explained by Figure 3 it follows that𝜃_𝑚=arctan𝑠_𝑚𝑥_𝑚=arctan+𝑥_𝑚=0+𝜙_𝑚for𝑠_𝑚=+1arctan−𝑥_𝑚=𝜋−𝜙_𝑚for𝑠_𝑚=−1,exp𝑖𝜃_𝑚=𝑠_𝑚exp𝑖𝑠_𝑚𝜙_𝑚, (16)(17) where the new phase angle 𝜙_𝑚 is defined by the following relation:tan𝜙_𝑚=𝑥_𝑚≡ℎ_𝑚𝑔_𝑚=2𝑛_𝑚−1𝑘_𝑚−𝑛_𝑚𝑘_𝑚−1𝑛_𝑚−1²−𝑛_𝑚². (18)

Figure 3 

Illustration of the phase angles 𝜃𝑚 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 coefficient𝑟_𝑚=𝜌_𝑚exp𝑖𝜙_𝑚with𝜌_𝑚=𝑠_𝑚𝜌_𝑚,𝜙_𝑚=𝑠_𝑚𝜙_𝑚. (19)

From (18) and (19), it evidently follows that in the lossless situation, when 𝜙_𝑚=0, the Fresnel coefficient 𝑟_𝑚=𝑠_𝑚𝜌_𝑚<0 or >0 depending on whether 𝑛_𝑚−1<𝑛_𝑚 or 𝑛_𝑚−1>𝑛_𝑚, 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 𝐸⁺_𝑚 corresponds to light transmitted across its left boundary at 𝑧_𝑚−1, while the negative-going wave of amplitude 𝐸⁻_𝑚—to light reflected from its right boundary at 𝑧_𝑚.

Figure 4 

Schematic geometry of the multilayer structure consisting of 𝑁 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 the 𝑚th absorbing medium of the complex refractive index 𝑛_𝑚=𝑛_𝑚−𝑖𝑘_𝑚 by using the wave factor exp(𝑖𝜔𝑡∓𝛾_𝑚𝑧), where the propagation constant 𝛾_𝑚=𝑖2𝜋𝜆𝑛_𝑚≡𝛼_𝑚+𝑖𝛽_𝑚 (20) consists of the amplitude and phase constants defined as𝛼_𝑚=2𝜋𝜆𝑘_𝑚,𝛽_𝑚=2𝜋𝜆𝑛_𝑚. (21) Let us write down the electromagnetic fields inside the two layers under examination(i)for the (𝑚−1)th layer located at 𝑧_𝑚−2<𝑧<𝑧_𝑚−1𝐸_𝑚−1(𝑧)=𝐸⁺_𝑚−1e^{−𝛾𝑚−1(𝑧−𝑧𝑚−2)}+𝐸⁻_𝑚−1e^{𝛾𝑚−1(𝑧−𝑧𝑚−2)},𝐻_𝑚−1(𝑧)=𝑛_𝑚−1𝐸⁺_𝑚−1e^{−𝛾𝑚−1(𝑧−𝑧𝑚−2)}−𝑛_𝑚−1𝐸⁻_𝑚−1e^{𝛾𝑚−1(𝑧−𝑧𝑚−2)}, (22)(ii)for the 𝑚th layer located at 𝑧_𝑚−1<𝑧<𝑧_𝑚𝐸_𝑚(𝑧)=𝐸⁺_𝑚e^{−𝛾𝑚(𝑧−𝑧𝑚−1)}+𝐸⁻_𝑚e^{𝛾𝑚(𝑧−𝑧𝑚−1)},𝐻_𝑚(𝑧)=𝑛_𝑚𝐸⁺_𝑚e^{−𝛾𝑚(𝑧−𝑧𝑚−1)}−𝑛_𝑚𝐸⁻_𝑚e^{𝛾𝑚(𝑧−𝑧𝑚−1)}. (23)

Imposing on the electromagnetic fields (22) and (23) at interface point 𝑧=𝑧_𝑚−1 the following matching conditions:𝐸_𝑚−1𝑧_𝑚−1=𝐸_𝑚𝑧_𝑚−1,𝐻_𝑚−1𝑧_𝑚−1=𝐻_𝑚𝑧_𝑚−1, (24) we couple the complex amplitudes in the (𝑚−1)th and 𝑚th layers by the matrix relation𝐸⁺_𝑚−1𝐸⁻_𝑚−1=1𝑡_𝑚𝐶_𝑚𝐸⁺_𝑚𝐸⁻_𝑚, (25) where 𝑡_𝑚 is the Fresnel transmission coefficient (6) for the mth interface at 𝑧=𝑧_𝑚−1. The coupling matrix appearing in (25)𝐶_𝑚=e^Δ𝑚−1𝑟_𝑚e^Δ𝑚−1𝑟_𝑚e^{−Δ𝑚−1}e^{−Δ𝑚−1}withΔ_𝑚−1=𝛾_𝑚−1𝑑_𝑚−1 (26) is identical in form to those obtained previously [5, 9].

To express the input electric fields 𝐸^±₀ in terms of the output field 𝐸⁺_𝑁−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:𝐸⁺₀𝐸⁻₀=1𝑡₁𝑡₂⋯𝑡_𝑁−1𝑎_𝑁𝑏_𝑁𝑐_𝑁𝑑_𝑁𝐸⁺_𝑁−10, (27) where 𝐸⁻_𝑁−1=0 for a semiinfinite output medium and the resultant coupling matrix is𝑎_𝑁𝑏_𝑁𝑐_𝑁𝑑_𝑁=𝐶₁𝐶₂⋯𝐶_𝑁−1. (28)

The reflectance and transmittance (defined as ratios of the reflected and transmitted power to the incident power) immediately follow from (27) as𝐸𝑅≡⁻⁰²𝐸⁺₀²=𝑐_𝑁²𝑎_𝑁²,𝑛𝑇≡^𝑁−1𝑛₀𝐸⁺_𝑁−1²𝐸⁺₀²=𝑛_𝑁−1𝑛₀𝑡₁²𝑡₂²⋯𝑡_𝑁−1²𝑎_𝑁². (29)

Therefore, in order to obtain 𝑅 and 𝑇, it is necessary to find only two matrix elements for calculating |𝑎_𝑁|² and |𝑐_𝑁|² and also to know the Fresnel transmission coefficients 𝑡_𝑚 of all interfaces (𝑚=1,2,…,𝑁−1) for which from (19) it follows that𝑡_𝑚²=1+𝑟_𝑚²=1+𝜌²_𝑚+2𝜌_𝑚cos𝜙_𝑚. (30)

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 Figure 1(a) the coupling matrix product equals𝐶₁𝐶₂=1𝑟₁𝑟₁1e^Δ1𝑟₂e^Δ1𝑟₂e^−Δ1e^−Δ1=e^Δ1+𝑟₁𝑟₂e^−Δ1𝑟₁e^−Δ1+𝑟₂e^Δ1𝑟₁e^Δ1+𝑟₂e^−Δ1e^−Δ1+𝑟₁𝑟₂e^Δ1≡𝑎₃𝑏₃𝑐₃𝑑₃, (31) 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 structure𝑐𝑅=³²𝑎₃²𝑛,𝑇=²𝑛₀|𝑡₁|²𝑡₂²𝑎₃². (32) From elements of the coupling matrix (31) and (30), we have obtained𝑎₃²=e^2𝛼1𝑑1+𝜌²₁𝜌²₂e^{−2𝛼1𝑑1}+2𝜌₁𝜌₂cos2𝛽₁𝑑₁−𝜙₁−𝜙₂,𝑐₃²=𝜌²₁e^2𝛼1𝑑1+𝜌²₂e^{−2𝛼1𝑑1}+2𝜌₁𝜌₂cos2𝛽₁𝑑₁+𝜙₁−𝜙₂,𝑡₁²𝑡₂²=1+𝜌²₁+2𝜌₁cos𝜙₁1+𝜌²₂+2𝜌₂cos𝜙₂, (33)(34)(35) where 𝜌_1,2≡𝑠_1,2𝜌_1,2 and 𝜙_1,2≡𝑠_1,2𝜙_1,2.

3.4. Reflectance and Transmittance of the 4M Structure

For the 4M structure shown in Figure 1(b) the coupling matrix product has the form𝐶₁𝐶₂𝐶₃=𝑎₃𝑏₃𝑐₃𝑑₃e^Δ2𝑟₃e^Δ2𝑟₃e^−Δ2e^−Δ2≡𝑎₄𝑏₄𝑐₄𝑑₄, (36) with the elements 𝑎₄ and 𝑐₄ being equal to𝑎₄=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). (37) Here, Δ_1,2≡𝛾_1,2𝑑_1,2=𝛼_1,2𝑑_1,2+𝑖𝛽_1,2𝑑_1,2 (see formulae (21) for 𝑚=1,2) and the Fresnel coefficients 𝑟_𝑚 are of exponential form (19), namely, 𝑟_𝑚=𝜌_𝑚exp(𝑖𝜙_𝑚) with 𝜌_𝑚=𝑠_𝑚𝜌_𝑚 and 𝜙_𝑚=𝑠_𝑚𝜙_𝑚 (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))𝑐𝑅=⁴²𝑎₄²𝑛,𝑇=³𝑛₀𝑡₁²𝑡₂²𝑡₃²𝑎₄², (38) 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 expressions𝑎₄²=𝑎₃²e^2𝛼2𝑑2+𝜌²₃𝑎^′₃²e^{−2𝛼2𝑑2}+𝜌₃𝑓_𝑎𝛽₂𝑑₂,𝑐₄²=𝑐₃²e^2𝛼2𝑑2+𝜌²₃𝑐^′₃²e^{−2𝛼2𝑑2}+𝜌₃𝑓_𝑐𝛽₂𝑑₂, (39) 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 thickness 𝛽₁𝑑₁𝑎^′₃²=𝜌²₁e^{−2𝛼1𝑑1}+𝜌²₂e^2𝛼1𝑑1+2𝜌₁𝜌₂cos2𝛽₁𝑑₁−𝜙₁+𝜙₂,𝑐^′₃²=e^{−2𝛼1𝑑1}+𝜌²₁𝜌²₂e^2𝛼1𝑑1+2𝜌₁𝜌₂cos2𝛽₁𝑑₁+𝜙₁+𝜙₂, (40)(ii)the terms due to the interference and absorption in layer 2 of optical thickness 𝛽₂𝑑₂𝜌₃𝑓_𝑎𝛽₂𝑑₂=2𝜌₁𝜌₃cos2𝛽₁𝑑₁−𝜙₁+2𝛽₂𝑑₂−𝜙₃+𝜌²₂cos2𝛽₁𝑑₁−𝜙₁−2𝛽₂𝑑₂−𝜙₃+2𝜌₂𝜌₃e^2𝛼1𝑑1cos2𝛽₂𝑑₂−𝜙₃−𝜙₂+𝜌²₁e^{−2𝛼1𝑑1}cos2𝛽₂𝑑₂−𝜙₃+𝜙₂,𝜌₃𝑓_𝑐𝛽₂𝑑₂=2𝜌₁𝜌₃cos2𝛽₁𝑑₁+𝜙₁+2𝛽₂𝑑₂−𝜙₃+𝜌²₂cos2𝛽₁𝑑₁+𝜙₁−2𝛽₂𝑑₂−𝜙₃+2𝜌₂𝜌₃e^{−2𝛼1𝑑1}cos2𝛽₂𝑑₂−𝜙₃+𝜙₂+𝜌²₁e^2𝛼1𝑑1cos2𝛽₂𝑑₂−𝜙₃−𝜙₂. (41)

From a comparison of (33)-(34) and (40), it follows that both terms (|𝑎₃|²,|𝑐₃|²) and (|𝑎^′₃|²,|𝑐^′₃|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 𝜌₃𝑓_𝑎(𝛽₂𝑑₂) and 𝜌₃𝑓_𝑐(𝛽₂𝑑₂) given in the form of (41).

It is very important to note that both the additional terms (|𝑎^′₃|²,|𝑐^′₃|²) and (𝑓_𝑎,𝑓_𝑐) contribute to expressions (39) for the 4M structure being multiplied by 𝜌₃≡|𝑟₃|∝|𝑛₂−𝑛₃|. 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 when𝑛₃=𝑛₂,𝑑₂=0. (42)

Both requirements (42) are needed for the conversion 4M → 3M, as applied to the transmittance 𝑇, while the reflectance conversion is immediately fulfilled if 𝑛₃=𝑛₂, 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 reflectance 𝑅 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

Appendix A 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 transmittance𝐿𝑅=⁻+𝑀cos2𝛽₁𝑑₁−𝜑₋+𝐹₋𝛽₂𝑑₂𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊+𝐹₊𝛽₂𝑑₂,𝑇=16𝑛₀𝑛₁²𝑛₂²𝑛₃𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊+𝐹₊𝛽₂𝑑₂, (43) 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 amplitude 𝑀 and phase angles 𝜑_± of the interference oscillations in layer 1 of the optical thickness 𝛽₁𝑑₁ (see (A.14)–(A.16) and (A.18)-(A.19))𝑀=𝑠₁𝑠₂𝑆₁₂𝐴cos𝜙₂²+𝐵sin𝜙₂²,𝜑_±=𝜓±𝜙₁𝐵withtan𝜓=𝐴tan𝜙₂𝑛,𝐴=₂²+𝑛₃²cosh2𝛼₂𝑑₂+2𝑛₂𝑛₃+𝑘₂𝑘₃sinh2𝛼₂𝑑₂,𝐵=𝑛₂²+𝑛₃²sinh2𝛼₂𝑑₂+2𝑛₂𝑛₃+𝑘₂𝑘₃cosh2𝛼₂𝑑₂,𝑆₁₂=𝑛₀−𝑛₁²+𝑘²₁𝑛₀+𝑛₁²+𝑘²₁×𝑛₁−𝑛₂²+𝑘₁−𝑘₂²×𝑛₁+𝑛₂²+𝑘₁+𝑘₂^21/2, (44)(45)(46)(47)(48)(ii) the losses parameters 𝐿_± due to optical absorption (see (A.26)-(A.27))𝐿_±=𝑎_±cosh2𝛼₁𝑑₁+𝑏_±sinh2𝛼₁𝑑₁𝑎_±,=𝑛²₀+𝑛₁²𝑛₁²+𝑛₂²𝐴±4𝑛₀𝑛₁𝑛₁𝑛₂+𝑘₁𝑘₂𝑏𝐵,_±=2𝑛²₀+𝑛₁²𝑛₁𝑛₂+𝑘₁𝑘₂𝐵±𝑛₀𝑛₁𝑛₁²+𝑛₂²𝐴, (49)(50)(51)(iii) the functions 𝐹_±(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))𝐹_±𝛽₂𝑑₂=𝑁_±cos2𝛽₂𝑑₂−𝜙₃−𝜓_±+𝑃cos2𝛽₁𝑑₁∓𝜙₁cos2𝛽₂𝑑₂−𝜙₃−𝑄sin2𝛽₁𝑑₁∓𝜙₁sin2𝛽₂𝑑₂−𝜙₃,𝑁_±=𝑠₂𝑠₃𝑆₂₃𝐴_±cos𝜙₂²+𝐵_±sin𝜙₂²,tan𝜓_±=𝐵_±𝐴_±tan𝜙₂𝐴_±,=𝑛²₀+𝑛₁²cosh2𝛼₁𝑑₁±2𝑛₀𝑛₁sinh2𝛼₁𝑑₁𝐵_±,=𝑛²₀+𝑛₁²sinh2𝛼₁𝑑₁±2𝑛₀𝑛₁cosh2𝛼₁𝑑₁𝑛,𝑃=₁²+𝑛₂²𝑠₁𝑠₃𝑆₁₃,𝑄=2𝑛₁𝑛₂+𝑘₁𝑘₂𝑠₁𝑠₃𝑆₁₃𝑆₁₃,=𝑛₀−𝑛₁²+𝑘²₀𝑛₀+𝑛₁²+𝑘²₀×𝑛₂−𝑛₃²+𝑘₂−𝑘₃²×𝑛₂+𝑛₃²+𝑘₂+𝑘₃^21/2,𝑆₂₃=𝑛₁−𝑛₂²+𝑘₁−𝑘₂²𝑛₁+𝑛₂²+𝑘₁+𝑘₂²×𝑛₂−𝑛₃²+𝑘₂−𝑘₃²×𝑛₂+𝑛₃²+𝑘₂+𝑘₃^21/2. (52)(53)(54)(55)(56)(57)(58)(59)

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 functions 𝐹_±(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 form𝐿𝑅=⁻+𝑀cos2𝛽₁𝑑₁−𝜑₋𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊,𝑇=16𝑛₀𝑛₁²𝑛₂²𝑛₃𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊, (60)(61) with all the quantities (44)–(51) being left unchanged.

Accordingly, by means of these quantities the spectra 𝑅(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 of 𝑅 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 that𝐴=𝐵=2𝑛₂², (62) which ensures two consequences: (i) the phase angles 𝜑_±=𝜙₂±𝜙₁ since tan𝜓=tan𝜙₂ 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 properties𝐿𝑅=⁻+𝑀cos2𝛽₁𝑑₁−𝜑₋𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊,𝑇=8𝑛₀𝑛₁²𝑛₂𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊, (63)(64) where (cf. (44), (45), (45) and (49)–(51))𝑀=𝑠₁𝑠₂𝑆₁₂,𝜑_±=𝜙₂±𝜙₁,𝑆₁₂=𝑛₀−𝑛₁²+𝑘²₁𝑛₀+𝑛₁²+𝑘²₁×𝑛₁−𝑛₂²+𝑘₁−𝑘₂²×𝑛₁+𝑛₂²+𝑘₁+𝑘₂^21/2,𝐿_±=𝑎_±cosh2𝛼₁𝑑₁+𝑏_±sinh2𝛼₁𝑑₁𝑎_±,=𝑛²₀+𝑛₁²𝑛₁²+𝑛₂²±4𝑛₀𝑛₁𝑛₁𝑛₂+𝑘₁𝑘₂𝑏_±,=2𝑛²₀+𝑛₁²𝑛₁𝑛₂+𝑘₁𝑘₂±𝑛₀𝑛₁𝑛₁²+𝑛₂². (65)(66)(67)(68)(69)

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 angles 𝜙₁ and 𝜙₂ 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 formtan𝜙₁≡tan𝑠₁𝜙₁=𝑠₁tan𝜙₁=2𝑛₀𝑘₁𝑛²₀−𝑛₁²,tan𝜙₂≡tan𝑠₂𝜙₂=𝑠₂tan𝜙₂=2𝑛₁𝑘₂−𝑛₂𝑘₁𝑛₁²−𝑛₂². (70) As follows from (70), these angles may be called the losses angles because in a lossless situation they vanish tan𝜙₁=tan𝜙₂=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 for 𝑅 and 𝑇, as applied to the two particular cases.

4.2.1. Arbitrary Films on Highly Absorbing Substrates

are characterized by the only requirement𝛼₂𝑑₂issomuchlargethatcosh2𝛼₂𝑑₂≃sinh2𝛼₂𝑑₂≃12e^2𝛼2𝑑2. (71) Then, from (46) and (47), it follows that1𝐴=𝐵=2𝑛₂+𝑛₃²+𝑘₂+𝑘₃²e^2𝛼2𝑑2. (72) The condition (72) works as well as (62) did before; that is, it ensures the same consequences: (i) the phase angles 𝜑_±=𝜙₂±𝜙₁ because tan𝜓=tan𝜙₂ 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 reflectance 𝑅 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:𝑘²₁≪𝑛²₁,𝑛₁²≃𝑛²₁,𝑘₂=𝑘₃=𝛼₂𝑑₂=0,𝑛₂²=𝑛²₂𝑛₃²=𝑛²₃. (73) Then, from (46)-(47) and (48) or (66), it, respectively, follows that𝐴=𝑛²₂+𝑛²₃,𝐵=2𝑛₂𝑛₃,𝑆₁₂=𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂,𝑠₁𝑠₂𝑆₁₂=𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂. (74)(75)

Requirements (73) modify the losses angles 𝜙₁ and 𝜙₂ so that (70) take the following small-losses formtan𝜙₁≃2𝑛₀𝑘₁𝑛²₀−𝑛²₁,tan𝜙₂≃−2𝑛₂𝑘₁𝑛²₁−𝑛²₂. (76) In terms of (74) and (76), (45) assumes the form𝐵tan𝜓=𝐴tan𝜙₂≃−4𝑛²₂𝑛₃𝑘₁𝑛²₁−𝑛²₂𝑛²₂+𝑛²₃. (77)

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 (tan𝜙₂)²≪1 and (tan𝜓)²≪1.

The amplitude 𝑀 of interference oscillations follows from formulae (65) and (44) in the following form:(i)for the 3M structure 𝑀=𝑠₁𝑠₂𝑆₁₂=𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂, (78)(ii)for the 4M structure 𝑀≃𝑠₁𝑠₂𝑆₁₂𝐴=𝑛²₀−𝑛²₁𝑛²₁−𝑛²₂𝑛²₂+𝑛²₃. (79)

The phase angles 𝜑_± of interference oscillations for the 3M and 4M structures given by (65) and (45) can be represented in a common formtan𝜑_±=𝑚_±𝑀𝑘₁𝑛₁, (80) where(i)for the 3M structure 𝑚_±=2𝑛₁−𝑛²₀−𝑛²₁𝑛₂±𝑛₀𝑛²₁−𝑛²₂, (81)(ii)for the 4M structure𝑚_±=2𝑛₁−𝑛²₀−𝑛²₁𝑛₂𝐵±𝑛₀𝑛²₁−𝑛²₂𝐴=2𝑛₁−2𝑛²₀−𝑛²₁𝑛²₂𝑛₃±𝑛₀𝑛²₁−𝑛²₂𝑛²₂+𝑛²₃. (82)

The losses parameters 𝐿_± 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 𝑎_±=𝑛²₀+𝑛²₁𝑛²₁+𝑛²₂±4𝑛₀𝑛²₁𝑛₂,𝑏_±=2𝑛₁𝑛²₀+𝑛²₁𝑛₂±𝑛₀𝑛²₁+𝑛²₂, (83)(ii)for the 4M structure𝑎_±=𝑛²₀+𝑛²₁𝑛²₁+𝑛²₂𝐴±4𝑛₀𝑛²₁𝑛₂𝐵=𝑛²₀+𝑛²₁𝑛²₁+𝑛²₂𝑛²₂+𝑛²₃±8𝑛₀𝑛²₁𝑛²₂𝑛₃,𝑏_±=2𝑛₁𝑛²₀+𝑛²₁𝑛₂𝐵±𝑛₀𝑛²₁+𝑛²₂𝐴=2𝑛₁2𝑛²₀+𝑛²₁𝑛²₂𝑛₃±𝑛₀𝑛²₁+𝑛²₂𝑛²₂+𝑛²₃. (84)

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 each 𝑚th medium incorporated into the film-substrate structure is absorbing and dispersive, that is, 𝑛_𝑚(𝜆)=𝑛_𝑚(𝜆)−𝑖𝑘_𝑚(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 𝑛_𝑚(𝜆) are slowly varying as compared with the periodic functions.

For this reason, below we take into account only the main contribution to the dependencies 𝑅(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:𝑅(𝑥)=𝐿₋+𝑀cos2𝑥−𝜑₋𝐿₊+𝑀cos2𝑥−𝜑₊. (85) 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:𝐿₋sin2𝛽₁𝑑₁−𝜑₊−𝐿₊sin2𝛽₁𝑑₁−𝜑₋=𝑀sin𝜑₊−𝜑₋. (86) 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 to𝜙₁=0,𝜙₂=arctan2𝑛₁𝑘₂𝑛²₁−𝑛₂². (87) Hence, the phase angles 𝜑₊ and 𝜑₋ given by (65) as 𝜑_±=𝜙₂±𝜙₁ become equal to each other owing to (87), namely,𝜑≡𝜑₊=𝜑₋=𝜙₂,tan𝜑=2𝑛₁𝑘₂𝑛²₁−𝑛₂², (88) where |tan𝜑| is not small because of noticeable losses in a highly absorbing substrate.

Then, (85) for the reflectance assumes the simplified form𝑅(𝜆)=𝐿₋+𝑀cos2𝛽₁𝑑₁−𝜑𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑, (89) where the quantities 𝑀 and 𝐿_± generally given by (65)–(69) are also reduced with taking into account (88) to the following simplified form:𝑀=𝑠₁𝑠₂𝑆₁₂=𝑛²₀−𝑛²₁𝑛²₁−𝑛₂²1+tan²𝐿𝜑,_±=𝑎_±=𝑛²₀+𝑛²₁𝑛²₁+𝑛₂²±4𝑛₀𝑛²₁𝑛₂. (90)

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 Figure 5, 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 formsin2𝛽₁𝑑₁−𝜑=0. (91)

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=𝜙₂.

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])𝑛₁(𝜆)=𝑎𝜆^𝑏+𝑐, (92) 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)

(b)

Figure 6 

(a) Experimental curve 𝑛1(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.

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 form𝑇(𝜆)=𝐾𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊, (93) 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 𝐾=8𝑛₀𝑛₁²𝑛₂, (94)(ii)for the 4M structure𝐾=16𝑛₀𝑛₁²𝑛₂²𝑛₃. (95)

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:sin2𝛽₁𝑑₁−𝜑₊=0. (96)

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 Figure 7 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(𝜆_𝑖) and 𝑘₁(𝜆_𝑖) at points 𝜆_𝑖(𝑖=1,2,…) of an experimental spectrum.

Figure 8 

Experimental transmittance spectrum of an oxide film Ta2O5 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(𝜆_𝑖) and 𝑘₁(𝜆_𝑖) 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,𝑛₁(𝜆)=0.003𝜆^4.285+2.045,𝑘₁(𝜆)=0.00027𝜆^2.381+0.00050, (97) 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 spectrum 𝑇(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.

Appendices

A. Derivation of Expressions for |𝑎₄|² and |𝑐₄|²

A.1. General Relations

Let us begin with employing (11), (13), (15), and (18) to explicitly write down parameters of the Fresnel reflection coefficient (19), that is, 𝑟_𝑚=𝜌_𝑚e^𝑖𝜙𝑚≡𝑠_𝑚𝜌_𝑚e^{𝑖𝑠𝑚𝜙𝑚}, (A.1) as applied to three interfaces (𝑚=1,2,3) which exist in the 4M-structure depicted in Figure 1(b), then (i)for interface 1 with 𝜌₁=𝑠₁𝜌₁ and 𝜙₁=𝑠₁𝜙₁𝜌²₁=𝜌²₁=𝑛₀−𝑛₁²𝑛₀+𝑛₁²≡𝒩₋₁𝒩₊₁,tan𝜙₁=2𝑛₀𝑘₁𝑛²₀−𝑛₁²,𝒩_±1=𝑛²₀+𝑛₁²±2𝑛₀𝑛₁,𝑠₁=sgn𝑛²₀−𝑛₁², (A.2)(ii)for interface 2 with 𝜌₂=𝑠₂𝜌₂ and 𝜙₂=𝑠₂𝜙₂𝜌²₂=𝜌²₂=𝑛₁−𝑛₂²𝑛₁+𝑛₂²≡𝒩₋₂𝒩₊₂,tan𝜙₂=2𝑛₁𝑘₂−𝑛₂𝑘₁𝑛₁²−𝑛₂²,𝒩_±2=𝑛₁²+𝑛₂²±2𝑛₁𝑛₂+𝑘₁𝑘₂,𝑠₂=sgn𝑛₁²−𝑛₂², (A.3)(iii)for interface 3 with 𝜌₃=𝑠₃𝜌₃ and 𝜙₃=𝑠₃𝜙₃𝜌²₃=𝜌²₃=𝑛₂−𝑛₃²𝑛₂+𝑛₃²≡𝒩₋₃𝒩₊₃,tan𝜙₃=2𝑛₂𝑘₃−𝑛₃𝑘₂𝑛₂²−𝑛₃²,𝒩_±3=𝑛₂²+𝑛₃²±2𝑛₂𝑛₃+𝑘₂𝑘₃,𝑠₃=sgn𝑛₂²−𝑛₃². (A.4)

For subsequent transformations it is necessary to employ the following quantities made up on the basis of (A.2)–(A.4), namely,𝜌₁𝜌₂=𝑠₁𝑠₂𝒩₊₁𝒩₊₂𝑆₁₂with𝑆₁₂=𝒩₋₁𝒩₊₁𝒩₋₂𝒩₊₂,𝜌₁𝜌₃=𝑠₁𝑠₃𝒩₊₁𝒩₊₃𝑆₁₃with𝑆₁₃=𝒩₋₁𝒩₊₁𝒩₋₃𝒩₊₃,𝜌₂𝜌₃=𝑠₂𝑠₃𝒩₊₂𝒩₊₃𝑆₂₃with𝑆₂₃=𝒩₋₂𝒩₊₂𝒩₋₃𝒩₊₃. (A.5)(A.6)(A.7)

It is convenient to rewrite (39) for |𝑎₄|² and |𝑐₄|² by using (33)-(34) and (40) in the following form:𝑎₄²=𝒜_abs+𝒜_int+𝜌₃𝑓_𝑎𝛽₂𝑑₂,𝑐₄²=𝒞_abs+𝒞_int+𝜌₃𝑓_𝑐𝛽₂𝑑₂, (A.8) where the terms 𝜌₃𝑓_𝑎(𝛽₂𝑑₂) and 𝜌₃𝑓_𝑐(𝛽₂𝑑₂) have the previous form (41), while the absorption terms are 𝒜_abs=e^2𝛼1𝑑1+𝜌²₁𝜌²₂e^{−2𝛼1𝑑1}e^2𝛼2𝑑2+𝜌²₃𝜌²₁e^{−2𝛼1𝑑1}+𝜌²₂e^2𝛼1𝑑1e^{−2𝛼2𝑑2},𝒞_abs=𝜌²₁e^2𝛼1𝑑1+𝜌²₂e^{−2𝛼1𝑑1}e^2𝛼2𝑑2+𝜌²₃e^{−2𝛼1𝑑1}+𝜌²₁𝜌²₂e^2𝛼1𝑑1e^{−2𝛼2𝑑2}, (A.9) and the interference terms are 𝒜_int=2𝜌₁𝜌₂e^2𝛼2𝑑2cos2𝛽₁𝑑₁−𝜙₁−𝜙₂+𝜌²₃e^{−2𝛼2𝑑2}cos2𝛽₁𝑑₁−𝜙₁+𝜙₂,𝒞_int=2𝜌₁𝜌₂e^2𝛼2𝑑2cos2𝛽₁𝑑₁+𝜙₁−𝜙₂+𝜌²₃e^{−2𝛼2𝑑2}cos2𝛽₁𝑑₁+𝜙₁+𝜙₂. (A.10)

We now pursue further transformations beginning with the interference terms (A.10).

A.2. Transformation of Interference Terms

Using (A.4) and (A.5), we can rewrite (A.10) in the form𝒜_int=4𝒩₊₁𝒩₊₂𝒩₊₃𝑠₁𝑠₂𝑆₁₂𝑓₊𝛽₁𝑑₁,𝒞_int=4𝒩₊₁𝒩₊₂𝒩₊₃𝑠₁𝑠₂𝑆₁₂𝑓₋𝛽₁𝑑₁, (A.11) where we have defined the new quantities as functions of optical thickness 𝛽₁𝑑₁ of layer 1𝑓_±𝛽₁𝑑₁=12𝒩₊₃e^2𝛼2𝑑2cos2𝛽₁𝑑₁∓𝜙₁−𝜙₂+𝒩₋₃e^{−2𝛼2𝑑2}cos2𝛽₁𝑑₁∓𝜙₁+𝜙₂. (A.12) After some trigonometrical transformations, (A.12) take the form𝑓_±𝛽₁𝑑₁=𝐴cos𝜙₂cos2𝛽₁𝑑₁∓𝜙₁+𝐵sin𝜙₂sin2𝛽₁𝑑₁∓𝜙₁≡𝐴cos𝜙₂²+𝐵sin𝜙₂²cos2𝛽₁𝑑₁−𝜑_±, (A.13) where we have denoted1𝐴≡2𝒩₊₃e^2𝛼2𝑑2+𝒩₋₃e^{−2𝛼2𝑑2}=𝑛₂²+𝑛₃²cosh2𝛼₂𝑑₂+2𝑛₂𝑛₃+𝑘₂𝑘₃sinh2𝛼₂𝑑₂,1𝐵≡2𝒩₊₃e^2𝛼2𝑑2−𝒩₋₃e^{−2𝛼2𝑑2}=𝑛₂²+𝑛₃²sinh2𝛼₂𝑑₂+2𝑛₂𝑛₃+𝑘₂𝑘₃cosh2𝛼₂𝑑₂. (A.14)(A.15) The phase angles 𝜑_± appearing in (A.13) are defined as follows:𝜑_±=𝜓±𝜙₁𝐵withtan𝜓=𝐴tan𝜙₂, (A.16) where tan𝜙₂ follows from (A.3).

Substitution of (A.13) into (A.11) finally yields the desired expressions for the interference terms 𝒜_int=4𝒩₊₁𝒩₊₂𝒩₊₃𝑀cos2𝛽₁𝑑₁−𝜑₊,𝒞_int=4𝒩₊₁𝒩₊₂𝒩₊₃𝑀cos2𝛽₁𝑑₁−𝜑₋, (A.17) where𝑀=𝑠₁𝑠₂𝑆₁₂𝐴cos𝜙₂²+𝐵sin𝜙₂² (A.18) and from (A.2), (A.3), and (A.5), it follows that𝑆₁₂≡𝒩₋₁𝒩₊₁𝒩₋₂𝒩₊₂=𝑛₀−𝑛₁²+𝑘²₁𝑛₀+𝑛₁²+𝑘²₁×𝑛₁−𝑛₂²+𝑘₁−𝑘₂²×𝑛₁+𝑛₂²+𝑘₁+𝑘₂^21/2. (A.19)

A.3. Transformation of Absorption Terms

Using relations (A.2)–(A.4) we can rewrite (A.9) in the form 𝒜_abs=4𝒩₊₁𝒩₊₂𝒩₊₃𝐿₊,𝒞_abs=4𝒩₊₁𝒩₊₂𝒩₊₃𝐿₋, (A.20) where we have introduced the new quantities𝐿_±=14𝒩_±1𝒩₊₂e^2𝛼1𝑑1+𝒩_∓1𝒩₋₂e^{−2𝛼1𝑑1}𝒩₊₃e^2𝛼2𝑑2+14𝒩_∓1𝒩₊₂e^{−2𝛼1𝑑1}+𝒩_±1𝒩₋₂e^2𝛼1𝑑1𝒩₋₃e^{−2𝛼2𝑑2}. (A.21) After some transformations, (A.21) take the following form:𝐿_±=12𝑝_±𝒩₊₃e^2𝛼2𝑑2+𝑝_∓𝒩₋₃e^{−2𝛼2𝑑2}cosh2𝛼₁𝑑₁+12𝑞_±𝒩₊₃e^2𝛼2𝑑2−𝑞_∓𝒩₋₃e^{−2𝛼2𝑑2}sinh2𝛼₁𝑑₁, (A.22) where we have denoted𝑝_±≡12𝒩_±1𝒩₊₂+𝒩_∓1𝒩₋₂=𝑛²₀+𝑛₁²𝑛₁²+𝑛₂²±4𝑛₀𝑛₁𝑛₁𝑛₂+𝑘₁𝑘₂,𝑞_±≡12𝒩_±1𝒩₊₂−𝒩_∓1𝒩₋₂=2𝑛²₀+𝑛₁²𝑛₁𝑛₂+𝑘₁𝑘₂±𝑛₀𝑛₁𝑛₁²+𝑛₂². (A.23)

From (A.14) and (A.15), it follows that𝒩₊₃e^2𝛼2𝑑2=𝐴+𝐵,𝒩₋₃e^{−2𝛼2𝑑2}=𝐴−𝐵. (A.24)

Insertion of (A.24) into (A.22) gives𝐿_±=12𝑝_±+𝑝_∓𝐴+𝑝_±−𝑝_∓𝐵cosh2𝛼₁𝑑₁+12𝑞_±−𝑞_∓𝐴+𝑞_±+𝑞_∓𝐵sinh2𝛼₁𝑑₁. (A.25)

The use of (A.23) for 𝑝_± and 𝑞_± reduces (A.25) to the final form𝐿_±=𝑎_±cosh2𝛼₁𝑑₁+𝑏_±sinh2𝛼₁𝑑₁, (A.26) where 𝑎_±=𝑛²₀+𝑛₁²𝑛₁²+𝑛₂²𝐴±4𝑛₀𝑛₁𝑛₁𝑛₂+𝑘₁𝑘₂𝑏𝐵,_±=2𝑛²₀+𝑛₁²𝑛₁𝑛₂+𝑘₁𝑘₂𝐵±𝑛₀𝑛₁𝑛₁²+𝑛₂²𝐴 (A.27) with the factors 𝐴 and 𝐵 introduced by (A.14) and (A.15).

Substitution of (A.26) into (A.20) yields the desired expressions for the absorption terms 𝒜_abs and 𝒞_abs.

A.4. Transformation of the Functions 𝜌₃𝑓_𝑎(𝛽₂𝑑₂) and 𝜌₃𝑓_𝑐(𝛽₂𝑑₂)

These functions are defined in (41). Using (A.2)-(A.3) and (A.6)-(A.7), we can rewrite these equations in the form 𝜌₃𝑓_𝑎𝛽₂𝑑₂=4𝒩₊₁𝒩₊₂𝒩₊₃×𝑠₁𝑠₃𝑆₁₃𝑓₊𝛽₂𝑑₂+𝑠₂𝑠₃𝑆₂₃𝑔₊𝛽₂𝑑₂,𝜌₃𝑓_𝑐𝛽₂𝑑₂=4𝒩₊₁𝒩₊₂𝒩₊₃×s₁𝑠₃𝑆₁₃𝑓₋𝛽₂𝑑₂+𝑠₂𝑠₃𝑆₂₃𝑔₋𝛽₂𝑑₂, (A.28) where we have introduced the following new quantities as functions of optical thickness 𝛽₂𝑑₂ of layer 2 (substrate)𝑓_±𝛽₂𝑑₂=12𝒩₊₂cos2𝛽₁𝑑₁∓𝜙₁+2𝛽₂𝑑₂−𝜙₃+𝒩₋₂cos2𝛽₁𝑑₁∓𝜙₁−2𝛽₂𝑑₂−𝜙₃,𝑔_±𝛽₂𝑑₂=12𝒩₊₁e^{±2𝛼1𝑑1}cos2𝛽₂𝑑₂−𝜙₃∓𝜙₂+𝒩₋₁e^{∓2𝛼1𝑑1}cos2𝛽₂𝑑₂−𝜙₃±𝜙₂. (A.29) After trigonometrical transformations (A.29), take the form𝑓_±𝛽₂𝑑₂=𝑛₁²+𝑛₂²cos2𝛽₁𝑑₁∓𝜙₁×cos2𝛽₂𝑑₂−𝜙₃−2𝑛₁𝑛₂+𝑘₁𝑘₂×sin2𝛽₁𝑑₁∓𝜙₁sin2𝛽₂𝑑₂−𝜙₃,𝑔_±𝛽₂𝑑₂=𝐴_±cos𝜙₂cos2𝛽₂𝑑₂−𝜙₃+𝐵_±sin𝜙₂sin2𝛽₂𝑑₂−𝜙₃≡𝐴_±cos𝜙₂²+𝐵_±sin𝜙₂²×cos2𝛽₂𝑑₂−𝜙₃−𝜓_±, (A.30)(A.31) where we have denoted𝐴_±≡12𝒩₊₁e^{±2𝛼1𝑑1}+𝒩₋₁e^{∓2𝛼1𝑑1}=𝑛²₀+𝑛₁²cosh2𝛼₁𝑑₁±2𝑛₀𝑛₁sinh2𝛼₁𝑑₁,𝐵_±1≡±2𝒩₊₁e^{±2𝛼1𝑑1}−𝒩₋₁e^{∓2𝛼1𝑑1}=𝑛²₀+𝑛₁²sinh2𝛼₁𝑑₁±2𝑛₀𝑛₁cosh2𝛼₁𝑑₁. (A.32) The phase angles 𝜓_± appearing in (A.31) are defined astan𝜓_±=𝐵_±𝐴_±tan𝜙₂, (A.33) where tan𝜙₂ follows from (A.3).

Introducing the following denotation𝐹_±𝛽₂𝑑₂=𝑠₁𝑠₃𝑆₁₃𝑓_±𝛽₂𝑑₂+𝑠₂𝑠₃𝑆₂₃𝑔_±𝛽₂𝑑₂, (A.34) we can represent (A.28) in the form 𝜌₃𝑓_𝑎𝛽₂𝑑₂=4𝒩₊₁𝒩₊₂𝒩₊₃𝐹₊𝛽₂𝑑₂,𝜌₃𝑓_𝑐𝛽₂𝑑₂=4𝒩₊₁𝒩₊₂𝒩₊₃𝐹₋𝛽₂𝑑₂. (A.35) Insertion of (A.30) and (A.31) in (A.34) convert it into the following:𝐹_±𝛽₂𝑑₂=𝑁_±cos2𝛽₂𝑑₂−𝜙₃−𝜓_±+𝑃cos2𝛽₁𝑑₁∓𝜙₁cos2𝛽₂𝑑₂−𝜙₃−𝑄sin2𝛽₁𝑑₁∓𝜙₁sin2𝛽₂𝑑₂−𝜙₃, (A.36) where 𝑁_±=𝑠₂𝑠₃𝑆₂₃𝐴_±cos𝜙₂²+𝐵_±sin𝜙₂²,𝑃=𝑛₁²+𝑛₂²𝑠₁𝑠₃𝑆₁₃,𝑄=2𝑛₁𝑛₂+𝑘₁𝑘₂𝑠₁𝑠₃𝑆₁₃, (A.37) and from (A.2)–(A.4), (A.6), and (A.7), it follows that 𝑆₁₃≡𝒩₋₁𝒩₊₁𝒩₋₃𝒩₊₃=𝑛₀−𝑛₁²+𝑘²₀𝑛₀+𝑛₁²+𝑘²₀×𝑛₂−𝑛₃²+𝑘₂−𝑘₃²×𝑛₂+𝑛₃²+𝑘₂+𝑘₃^21/2,𝑆₂₃≡𝒩₋₂𝒩₊₂𝒩₋₃𝒩₊₃=𝑛₁−𝑛₂²+𝑘₁−𝑘₂²𝑛₁+𝑛₂²+𝑘₁+𝑘₂²×𝑛₂−𝑛₃²+𝑘₂−𝑘₃²×𝑛₂+𝑛₃²+𝑘₂+𝑘₃^21/2. (A.38)

Substitution of the above (A.17), (A.20), and (A.35) into (A.8) finally yields the required expressions 𝑎₄²=4𝒩₊₁𝒩₊₂𝒩₊₃𝐿₊+𝑀cos2𝛽₁𝑑₁−𝜑₊+𝐹₊𝛽₂𝑑₂,𝑐₄²=4𝒩₊₁𝒩₊₂𝒩₊₃𝐿₋+𝑀cos2𝛽₁𝑑₁−𝜑₋+𝐹₋𝛽₂𝑑₂, (A.39) where the quantities 𝐿_±,𝑀 and 𝐹_±(𝛽₂𝑑₂) are, respectively, given by (A.26), (A.18), (A.36), and their accompanying constituents.

A knowledge of both |𝑎₄|² and |𝑐₄|² is sufficient to determine the reflectance 𝑅 from (38). But for determining the transmittance 𝑇 from the same formula, it is necessary to know not only |𝑎₄|² but also the transmission coefficients product𝑡₁²𝑡₂²𝑡₃²=4𝒩₊₁𝒩₊₂𝒩₊₃16𝑛²₀𝑛₁²𝑛₂² (A.40) which is obtained from (6) for coefficients 𝑡_𝑚,𝑚=1,2,3.

B. Mathematical Justification of the Device Averaging

The general (43)–(59) have been derived for the case of monochromatic irradiation at wavelength 𝜆 = const. But experimentally, one deals with nonmonochromatic conditions when (𝜆₀−Δ𝜆)<𝜆<(𝜆₀+Δ𝜆), which is generated by the spectrophotometer with an operating slit of finite width ∼Δ𝜆⋘𝜆₀. In such a case, the device indicates some average value. Our subsequent task is to prove that the average value of 𝐹_±(𝛽₂𝑑₂) given by (52)–(59) is negligibly small for the 4M structure with a thick substrate such that𝛽𝜂≡²𝑑₂𝛽₁𝑑₁=𝑛₂𝑑₂𝑛₁𝑑₁≫1. (B.1)

Denoting 𝑥=𝛽₁𝑑₁≡(2𝜋/𝜆)𝑛₁𝑑₁, it is easy to see that 𝑋₁≤𝑥≤𝑋₂ where𝑋_1,2=𝛽₀𝑛₁𝑑₁1∓Δ𝜆𝜆₀with𝛽₀=2𝜋𝜆₀. (B.2)

Now, the average value of 𝐹_±(𝛽₂𝑑₂) can be defined as𝐹_±𝛽₂𝑑₂=1Δ𝑋^𝑋2_𝑋1𝐹_±(𝑥)𝑑𝑥, (B.3) whereΔ𝑋≡𝑋₂−𝑋₁=2𝛽₀𝑛₁𝑑₁Δ𝜆𝜆₀≪1, (B.4) and in accordance with (52) and (B.1)𝐹_±(𝑥)=𝑁_±cos2𝜂𝑥−𝜙₃−𝜓_±+𝑃cos2𝑥∓𝜙₁cos2𝜂𝑥−𝜙₃−𝑄sin2𝑥∓𝜙₁sin2𝜂𝑥−𝜙₃. (B.5)

Transformations of the trigonometric functions appearing in (B.5) give the following results: cos2𝜂𝑥−𝜙₃−𝜓_±=cos𝜓_±+𝜙₃cos2𝜂𝑥+sin𝜓_±+𝜙₃sin2𝜂𝑥,cos2𝑥∓𝜙₁cos2𝜂𝑥−𝜙₃=12cos2𝜂−1𝑥+±𝜙₁−𝜙₃+cos2𝜂+1𝑥+∓𝜙₁−𝜙₃≈cos±𝜙₁cos2𝜂𝑥−𝜙₃=cos𝜙₁cos𝜙₃+cos2𝜂𝑥cos𝜙₁sin𝜙₃sin2𝜂𝑥,sin2𝑥∓𝜙₁sin2𝜂𝑥−𝜙₃=12cos2𝜂−1𝑥+±𝜙₁−𝜙₃−cos2𝜂+1𝑥+∓𝜙₁−𝜙₃≈sin∓𝜙₁sin2𝜂𝑥−𝜙₃=±sin𝜙₁sin𝜙₃∓cos2𝜂𝑥sin𝜙₁cos𝜙₃sin2𝜂𝑥. (B.6)(B.7)(B.8)

Approximate equality in (B.7) and (B.8) corresponds to the condition 𝜂≫1 that holds true for thick substrates (see (B.1)).

Substitution of (B.6)–(B.8) into (B.5) gives𝐹_±(𝑥)=𝐶_±cos2𝜂𝑥+𝑆_±sin2𝜂𝑥, (B.9) where (see (53)–(59) for 𝑁_±, 𝑃 and 𝑄)𝐶_±=𝑁_±cos𝜓_±+𝜙₃+𝑃cos𝜙₁cos𝜙₃∓𝑄sin𝜙₁sin𝜙₃,𝑆_±=𝑁_±sin𝜓_±+𝜙₃+𝑃cos𝜙₁sin𝜙₃±𝑄sin𝜙₁cos𝜙₃. (B.10)

By integration of (B.9) in accordance with (B.3), we arrive at the desired result for the averaged value of 𝐹_±(𝛽₂𝑑₂):𝐹_±𝛽₂𝑑₂=𝐶_±cos2𝛽₀𝑛₂𝑑₂+𝑆_±sin2𝛽₀𝑛₂𝑑₂×sin𝜂Δ𝑋𝜂Δ𝑋⟶0as𝜂Δ𝑋≫1. (B.11)

The similar procedure of averaging applied to the terms 𝑀cos(2𝛽₁𝑑₁−𝜑_±) entering into (43) gives the following result:𝑀cos2𝛽₁𝑑₁−𝜑_±=𝑀cos2𝛽₀𝑛₁𝑑₁−𝜑_±×sinΔ𝑋Δ𝑋⟶𝑀cos4𝜋𝜆₀𝑛₁𝑑₁−𝜑_±asΔ𝑋≪1. (B.12)

Therefore, from (B.11) and (B.12), it follows that for the 4M structures with a thick substrate the averaged contribution to (43) from 𝐹_±(𝛽₂𝑑₂) is negligibly small and can be omitted but the terms 𝑀cos(2𝛽₁𝑑₁−𝜑_±) should be taken at the center wavelength 𝜆₀.

In conclusion, it should be emphasized that the above mathematical justification is only an approximate estimation based on the additive averaged contributions to (43) for the reflectance and transmittance optical spectra.

Acknowledgment

The authors wish to acknowledge the Russian Foundation for Basic Research for financial support (Grant no. 09-03-00777-a).