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

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.


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 thinfilm optics and optical measurements has been accumulated in numerous scientific publications; well-known books and reviews may be quoted as an example [1][2][3][4][5][6][7][8][9][10][11][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 thinfilm performance; first, we refer to biomedical and ecological applications [15][16][17][18][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][2][3][4][5][6][7][8][9][10][11][12][20][21][22][23][24][25][26][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 threemedia 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 fourmedia 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 socalled 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.

Formulation of the Problem
We shall restrict our consideration to normal incidence of light from a transparent input medium of the real refractive index n 0 .In the general case, the film under examination (numbered by 1) has optical losses so that its refractive index is complex and dispersive: n 1 (λ) = n 1 (λ) − ik 1 (λ) with the real refractive and absorptive indices n 1 (λ) and k 1 (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 n 2 (λ) = n 2 (λ) − ik 2 (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 d 1 on a semiinfinite substrate and (b) the four-media structure (4M structure) appropriate to a single film of thickness d 1 on a substrate of finite thickness d 2 .Media 0 and 3 may be considered respectively an input medium (non absorbing one with real index n 0 ) and an output medium (absorbing one with complex index n 3 (λ) = n 3 (λ) − ik 3 (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.
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 (k 2 1 n 2 1 ) on a transparent substrate (k 2 = 0) has the following form: where x = exp(−αd 1 ), α = 4πk 1 /λ, ϕ = 4πn 1 d 1 /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)) 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][27][28][29][30][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 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 d 2 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]: Here, the items l 2 13 , m 2 13 , p 2 13 , q 2 13 , t 2 13 , and u 2 13 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 l 2 13 , m 2  13 , 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 "stepby-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 l 2  13 , m 2 13 , 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 Ta 2 O 5 on a quartz-glass substrate (n 0 = n 3 = 1, n 1 = 2.21, n 2 = 1.472, k 1 = k 2 = k 3 = 0, d 1 = 1 μm, and d 2 = 500 μm) 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 Ta 2 O 5 (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 ≡ n 2 d 2 /n 1 d 1 , 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 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.
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 n 2 d 2 n 1 d 1 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.
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.

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 where m = 1, 2, . . .and n 0 = n 0 since k 0 ≡ 0 for a nonabsorbing input medium.Each mth interface is characterized by the Fresnel reflection and transmission coefficients r m and Heavens [5] represents the Fresnel reflection coefficient r m as a sum of its real and imaginary parts where in accordance with ( 5) and ( 6), we have with the squared moduli of the refractive indices ( 5) being equal to Unlike Heavens' approach, for our analysis, it is more convenient to apply for r m instead of (7) the exponential form The squared modulus ρ 2 m and phase θ m of the Fresnel reflection coefficient (7) or (10) are defined by the following relations tan where we have introduced the following quantities 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, Yet, the exponential form (10) does not display explicitly this property.Indeed, from (12), it follows that in the lossless situation tan θ 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 quantity where the sign of g m is inserted as From formulae ( 12) and ( 14) explained by Figure 3 it follows that where the new phase angle φ m is defined by the following relation: Substitution of relation ( 17) into (10) yields the desired exponential form of the Fresnel reflection coefficient From ( 18) and ( 19), it evidently follows that in the lossless situation, when φ m = 0, the Fresnel coefficient r m = s m ρ 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.

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 .
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 m z), where the propagation constant consists of the amplitude and phase constants defined as Let us write down the electromagnetic fields inside the two layers under examination (i) for the (m − 1)th layer located at International Journal of Optics (ii) for the mth layer located at Imposing on the electromagnetic fields ( 22) and ( 23) at interface point z = z m−1 the following matching conditions: we couple the complex amplitudes in the (m − 1)th and mth layers by the matrix relation where t m is the Fresnel transmission coefficient (6) for the mth interface at z = z m−1 .The coupling matrix appearing in ( 25) (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 NMstructure depicted in Figure 4.As a result, we obtain from (25) the following relation: where E − N−1 = 0 for a semiinfinite output medium and the resultant coupling matrix is The reflectance and transmittance (defined as ratios of the reflected and transmitted power to the incident power) immediately follow from (27) as 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 that Let us apply ( 26)-( 29) to the special cases appropriate to the 3M structure (Figure 1(a)) and the 4M structure (Figure 1(b)).

Reflectance and Transmittance of the 3M Structure.
For the 3M structure shown in Figure 1(a) the coupling matrix product equals where (21) for m = 1) and the Fresnel coefficients r 1 and r 2 have the exponential form (19).

Reflectance and Transmittance of the 4M Structure.
For the 4M structure shown in Figure 1(b) the coupling matrix product has the form with the elements a 4 and c 4 being equal to 21) for m = 1, 2) and the Fresnel coefficients r m are of exponential form (19), namely, r m = ρ m exp(iφ m ) with ρ m = s m ρ m and φ m = s m φ 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)) From (37), after some laborious transformations, we obtain the desired expressions where the terms |a 3 | 2 and |c 3 | 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 (ii) the terms due to the interference and absorption in layer 2 of optical thickness From a comparison of (33)-( 34 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 Both requirements (42) are needed for the conversion 4M → 3M, as applied to the transmittance T, while the reflectance conversion is immediately fulfilled if n 3 = n 2 , 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.

Physical Form of Expressions for the
Reflectance and Transmittance  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 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 International Journal of Optics (ii) the losses parameters L ± due to optical absorption (see (A.26)-(A.27)) The first term in (52) for F ± (β 2 d 2 ) 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.

Particular Expressions Taking Account of the Device Averaging.
It is the functions F ± (β 2 d 2 ) that take into account the fast oscillations generated by interference in a thick substrate (when n 2 d 2 n 1 d 1 ) 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 socalled 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 ± (β 2 d 2 ) ∼ 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 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 (n 2 , k 2 ) of a thick substrate and its finite thickness (d 2 d 1 ) but fully disregard interference effects in the substrate.
The angles φ 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 As follows from (70), these angles may be called the losses angles because in a lossless situation they vanish tan φ 1 = tan φ 2 = 0 when 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.
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(−2α 2 d 2 ) → 0. Therefore, for thin films on a highly absorbing substrates, there is no point in using formulae for the 4M structure.

Slightly Absorbing Films on Transparent Substrates. are characterized by the following requirements:
(73) Then, from ( 46)-( 47) and ( 48) or (66), it, respectively, follows that Requirements (73) modify the losses angles φ 1 and φ 2 so that (70) take the following small-losses form In terms of ( 74) and ( 76), (45) assumes the form 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 (tan φ 2 ) 2  1 and (tan ψ) 2  1.The amplitude M of interference oscillations follows from formulae (65) and (44) in the following form: (i) for the 3M structure (ii) for the 4M structure International Journal of Optics The phase angles ϕ ± of interference oscillations for the 3M and 4M structures given by ( 65) and ( 45) can be represented in a common form where (i) for the 3M structure (ii) for the 4M structure 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 (ii) for the 4M structure ( 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.

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, . 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(2β 1 d 1 − 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.

Reflectance Spectrum.
Let us write down the reflectance spectrum (60) or (63) in the following form: We consider instead of R(λ) the frequency spectrum R(x) where x ≡ β 1 d 1 = (ω/c)n 1 d 1 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: 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 (k 1 = 0, α 1 d 1 = (2π/λ)k 1 d 1 = 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 In this simplified case, from (70), it follows that the losses angles of interfaces 1 and 2 are equal to 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, where | tan ϕ| is not small because of noticeable losses in a highly absorbing substrate.Then, (85) for the reflectance assumes the simplified form 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) Formulae ( 88)-( 90) have been used to compute the reflectance spectrum R(λ) for a non-absorbing film (n 0 = 1, n 1 = 2.3, k 1 = 0) of thickness d 1 = 1 μm deposited on a titanium substrate (n 2 = 1.89, k 2 = 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 (n 2 = 1.89, k 2 = 0).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 form As follows from (91), the absorption of a substrate indeed shifts the spectrum R(x) along the frequency axis x ≡ β 1 d 1 = (ω/c)n 1 d 1 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 TiO 2 effect of the dispersion function n 1 (λ) becomes noticeable for λ < 600 nm. Figure 6(a) contains the curve n 1 (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]) 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 d 1 = 1 μm on a titanium substrate (n 2 = 1.89, k 2 = 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 n 1 = 2.3.From comparison of the two curves, it follows that an increase in n 1 (λ) at short wavelengths leads to a rise in number of oscillations for both the spectra.

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 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 (ii) for the 4M structure Position of maxima and minima in the spectrum (93) is found from the condition ∂T(x)/∂x = 0 (with x = β 1 d 1 ) which gives the following equation to find them: 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: n 0 = n 3 = 1, n 1 = 2.21, k 1 = 0.01, n 2 = 1.472, k 2 = 0 and d 1 = 1 μm.
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 n 2 (see Figure 1(a)) yields substantially higher values of T(λ) because of neglecting influence of medium 3 with index n 3 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%.
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 (n 2 = 1.472, k 2 = 0, d 2 = 500 μm) 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   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 n 1 (λ i ) and k 1 (λ i ) at points λ i (i = 1, 2, . ..) of an 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 n 1 (λ i ) and k 1 (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, where λ is measured in μm.Thickness of the film d 1 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 d 1 = 0.824 μm 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.

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 13), (15), and (18) to explicitly write down parameters of the Fresnel reflection coefficient (19), that is, 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 ρ 1 = s 1 ρ 1 and φ For subsequent transformations it is necessary to employ the following quantities made up on the basis of (A.2)-(A.4),namely,

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 (λ 0 − Δλ) < λ < (λ 0 + Δλ), which is generated by the spectrophotometer with an operating slit of finite width ∼ Δλ ≪ λ 0 .In such a case, the device indicates some average value.Our subsequent task is to prove that the average value of F ± (β 2 d 2 ) given by ( 52)-( 59) is negligibly small for the 4M structure with a thick substrate such that (B.11) The similar procedure of averaging applied to the terms M cos(2β 1 d 1 − ϕ ± ) entering into (43) gives the following result: (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 ± (β 2 d 2 ) is negligibly small and can be omitted but the terms M cos(2β 1 d 1 − ϕ ± ) should be taken at the center wavelength λ 0 .
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.
caused by interference only in the thin film Ta 2 O 5 .This stems from the fact that International Journal of Optics

Figure 2 :
Figure 2: Reflectance (R) and transmittance (T) spectra of the Ta 2 O 5 -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.

Figure 3 :
Figure 3: Illustration of the phase angles θ m and φ m for representing the Fresnel reflection coefficient in the exponential form r m = ρ m exp(iθ m ) = s m ρ m exp(is m φ m ).

Figure 4 :
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 z m−1 specified by the Fresnel reflection coefficient r m = s m ρ m exp(is m φ m ).
) and (40), it follows that both terms (|a 3 | 2 , |c 3 | 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 (|a 4 | 2 , |c 4 | 2 ), as seen from (39), prove to be multiplied by the factors exp(±2α 2 d 2 ) 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 3 f a (β 2 d 2 ) and ρ 3 f c (β 2 d 2 ) 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 ρ 3 ≡ |r 3 | ∝ | n 2 − n 3 |.

Figure 5 :
Figure 5: Reflectance spectrum of a transparent film on a metal substrate (solid line) and on a nonabsorbing substrate (dashed line).

Figure 6 : 9 TFigure 7 :
Figure6: (a) Experimental curve n 1 (λ) for an oxide film TiO 2[27] and (b) the reflectance spectrum R(λ) calculated for such a film (solid line) and for a nondispersive film with n 1 = 2.3 (dashed line) both deposited on a titanium substrate.

6 TFigure 8 :
Figure8: Experimental transmittance spectrum of an oxide film Ta 2 O 5 on a quartz-glass substrate (thin line) and the spectrum T(λ) calculated from (93) for the 4M structure (thick line) by using d 1 = 0.824 μm, n 1 (λ) and k 1 (λ) in the form of (97) which are obtained from the experimental spectrum.

2 A. 1 .
(b)) with a double film (of comparable values of n 1 d 1 and n 2 d 2 ) on semiinfinite substrates.Appendices A. Derivation of Expressions for |a 4 | 2 and |c 4 | General Relations.Let us begin with employing (11), (
Appendix A is devoted to expressing the quantities |a 4 | 2 and |c 4 | 2 obtained above in the form of ( 4.1.General Expressions.