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.

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 [

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 [

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 [

Although most theoretical approaches generally begin with studying a system of multiple layers [

Section

We shall restrict our consideration to normal incidence of light from a transparent input medium of the real refractive index

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

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

The main task of theoretical analysis is a derivation of mathematical expressions for the spectra of reflectance

In Swanepoel’s notation [

When using (

A distinguishing feature of expressions (

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

Certain expressions for the spectra

Expressions (

Formulae (

Figure

Reflectance (

Our subsequent task consists of deriving the analytical expressions for

Consider any two adjacent layers of the multilayer structure under consideration, say, the

Heavens [

Unlike Heavens’ approach, for our analysis, it is more convenient to apply for

In the absence of optical absorption (

From formulae (

Illustration of the phase angles

Substitution of relation (

From (

Let us consider the

Schematic geometry of the multilayer structure consisting of

We shall describe light propagation in the

for the

for the

Imposing on the electromagnetic fields (

To express the input electric fields

The reflectance and transmittance (defined as ratios of the reflected and transmitted power to the incident power) immediately follow from (

Therefore, in order to obtain

Let us apply (

For the 3M structure shown in Figure

The general formulae (

For the 4M structure shown in Figure

The general formulae (

From (

the terms due to the interference and absorption in layer 1 of optical thickness

the terms due to the interference and absorption in layer 2 of optical thickness

From a comparison of (

It is very important to note that both the additional terms (

Both requirements (

Hence, our further transformations will deal with the 4M structure, as a general one, to reduce the reflectance

Appendix

In (

(i)

The first term in (

It is the functions

Thus, when taking into account the device averaging, (

Accordingly, by means of these quantities the spectra

The analogous simplified form of

In such a case, (

Expressions (

The angles

It is extremely remarkable that (

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

an arbitrary film on a highly absorbing (metal) substrate

a slightly absorbing film on a transparent (dielectric) substrate.

Let us consider modifications of the above expressions for

are characterized by the only requirement

Hence, the reflectance

are characterized by the following requirements:

Requirements (

Formulae (

Th

for the 3M structure

for the 4M structure

Th

for the 3M structure

for the 4M structure

Th

for the 3M structure

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 (

All the relations obtained above are valid for the general case when each

For this reason, below we take into account only the main contribution to the dependencies

Let us write down the reflectance spectrum (

Positions of maxima and minima in the spectrum (

An essential simplification of (

In this simplified case, from (

Then, (

Formulae (

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

As seen from Figure

As follows from (

According to our experimental data, for amorphous oxide films TiO_{2} effect of the dispersion function

(a) Experimental curve _{2} [

The transmittance spectra (

for the 3M structure

for the 4M structure

Position of maxima and minima in the spectrum (

This equations is identical to (

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 (

The results of calculation are given in Figure

Transmittance spectrum of a thin dielectric film (

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 (

Experimental transmittance spectrum of an oxide film

Unlike the above cited papers, we have applied our newly derived (

The spectrum

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 (

The crucial point of our theory is an introduction of the device averaging procedure into (

Thus, the reflectance and transmittance spectra represented in the simplified (with the device averaging) form (

Let us begin with employing (

for interface 1 with

for interface 2 with

for interface 3 with

For subsequent transformations it is necessary to employ the following quantities made up on the basis of (

It is convenient to rewrite (

We now pursue further transformations beginning with the interference terms (

Using (

Substitution of (

Using relations (

From (

Insertion of (

The use of (

Substitution of (

These functions are defined in (

Introducing the following denotation

Substitution of the above (

A knowledge of both

The general (

Denoting

Now, the average value of

Transformations of the trigonometric functions appearing in (

Approximate equality in (

Substitution of (

By integration of (

The similar procedure of averaging applied to the terms

Therefore, from (

In conclusion, it should be emphasized that the above mathematical justification is only an approximate estimation based on the additive averaged contributions to (

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

_{2}thin film prepared by sol-gel process on a soda-lime glass

_{2}films deposited by off-axis target sputtering