## Abstract

In the present paper we determine the optical constants and thicknesses of multilayer thin film stacks, in the visible and near infrared ranges. These parameters are derived from the transmittance and reflectance spectra measured by a spectrophotometer, for several angles of incidence. Several examples are studied, from a simple single layer structure up to a 22-layer dielectric filter. We show that the use of a large number of incidence angles is an effective means of reducing the number of mathematical solutions and converging on the correct physical solution when the number of layers increases. More specifically, we provide an in-depth discussion of the approach used to extract the index and thickness of each layer, which is achieved by analysing the various mathematical solutions given by a global optimization procedure, based on as little as 6 and as many as 32 variable parameters. The results show that multiple incidences, lead to the true solution for a filter with a large number of layers. In the present study, a Clustering Global Optimization algorithm is used, and is shown to be efficient even for a high number of variable parameters. Our analysis allows the accuracy of the reverse engineering process to be estimated at approximately 1 nm for the thickness, and 2 10^{−3} for the index of each layer in a 22-layer filter.

©2012 Optical Society of America

## 1. Introduction

The determination of the complex refractive index (refractive index *n* and extinction coefficient *k*) of thin films is of great importance in the design of any optical interference filter. Successful manufacturing of such filters is often directly related to the accuracy of the complex refractive index and thickness (*d*) measurements, since even a small difference in refractive index can lead to drastic changes in the filter's spectral response (reflectance R and transmittance T). Moreover, the index of thin films, which differs from that of bulk materials, is strongly dependent on the deposition process. Therefore, the determination of the complex refractive index and thickness, which can be considered to be a reverse engineering problem, is clearly necessary.

In general, it is possible to accurately estimate the index of a single-layer with a simple optical dispersion [1–3] and in most cases, only the transmittance at normal incidence is used [4–6]. However, as observed by Tikhonravov *et al.* [7], if in addition to the use of transmittance data, reflectance data can be included, the imaginary part of the film’s index is determined with improved accuracy. The index determination procedure used generally consists in minimising an error function calculating the difference between experimental and theoretical spectral values. The variable parameters include the layer thickness, and the parameters describing the real and imaginary parts of the refractive index. Systematic measurement errors which could cause bad index determinations should be avoided whenever possible. Tikhonravov *et al.* [8] pointed out that an accuracy of 5% in the optical characterization of thin film could be obtained with 0.2% - 0.4% systematic errors of spectrophotometric measurements. Suyong Wu [9] also studied the index determination method for diminishing the passive impact of systematic spectral errors.

In the case of multilayer stacks, it is also possible to determine the indices and thicknesses of thin films from spectral measurements. The use of a single incidence is not recommended as several designs (or solutions) can provide the same spectral response. In mathematical terms, many solutions exist at local minima, and the best global solution may not necessarily be the correct solution. This phenomenon is amplified when the spectral measurements used in the analysis are biased by systematic errors.

Dobrowolski *et al*. [10] were probably the first authors to introduce a modern approach using curve fitting on spectra recorded at multiple angles of incidence. They showed that the sensitivity with which these parameters is determined can be improved by making use of a suitable combination of measured quantities, and performed reverse engineering on a 7-layer ZrO_{2}/MgO stack. Multiple incidence analysis was then widely used in thin film reverse engineering for nearly 30 years [11–17]. In a recent publication, Tikhonravov [18] used reverse engineering for filters having between 1 and 15 dielectric layers. The use of specific, variable angle reflectance and transmittance spectra enabled s-polarization and p-polarization measurements to be achieved with a photometric accuracy better than 0.3%, over the wavelength range between 400 nm and 1000 nm, at all incidence angles from 7° to 40°. This level of accuracy made it possible to estimate the refractive indices of single layer and multilayer structures. The multi-angle approach thus allowed very accurate curve fitting to be achieved from 380 nm to 1100 nm, on a 15-layer mirror.

In the present paper, we also investigate the use of single and multiple incidence angle spectral analysis for reverse engineering, but our study focuses on the analysis of multiple solutions given by a global optimization approach. We show that local optimization is inadequate in the case of a complex design including more than 20 layers.

Under such conditions, commercial software such as TFCalc [19], Filmwizard [20], Essential McLeod [21], Optilayer [22], must be used with considerable care, and be restricted to application of a ‘simple’ local optimization by means of a sequential optimization of thickness and indices.

When a single incidence angle is used, numerous different ‘mathematical’ solutions can be found. With multiple incidence angle analysis, a reduced number of possible solutions are found - all of which are similar and provide a well-fitted curve. A global optimization program can then be used to find the optimal solution, and the distance between all ‘acceptable’ solutions provides a good indication of the accuracy of the indices and thicknesses of the optimized solution. We present reverse engineering investigations of three structures: a single-layer thin film (Ta_{2}O_{5}), a resonant structure (7-layer, Ta_{2}O_{5}/SiO_{2}), and a notch filter (22-layer, Ta_{2}O_{5}/SiO_{2}). Measurement are performed from 350nm to 1800nm at several different incidence angles (normal-quasi normal-20°-30°-45°), and for two orthogonal states of polarization. In the following, we assume that the layers are homogeneous, and that the index of a given material is constant, whatever its location inside the stack.

## 2. Experimental and computational procedures used for index determination

In this present study, the complex refractive index of a thin film was estimated from its reflectance and transmittance values. The experimental reflectance (Rexp) and transmittance (Texp) were measured under a given wavelength (*λ*) and incidence angle (θ), as shown in Fig. 1
, using a commercial spectrophotometer Perkin Elmer 1050. Three optical accessories are available: a transmission module used to measure transmittance at normal incidence, a specific module used to measure both reflectance and transmittance at quasi normal incidence (8°), and a universal reflectance accessory used to measure reflectance under controlled polarization conditions, at incidence angles ranging from 8° to 60°. This spectrophotometer can be operated at wavelengths ranging from UV to Near Infrared.

The theoretical reflectance (Rcal) and transmittance (Tcal) values were calculated using the matrix method [23], taking the backside reflection into account. These calculated values depend on the angle of incidence (θ), the wavelength under consideration (*λ*), the polarisation state (pol), the refractive index of the substrate, and the complex refractive index (*n*, *k*) and thickness (*d*) of each layer. The index of the BK7 glass substrate was deduced from the transmittance of a bare substrate under normal incidence. The unknown quantities remaining to be determined are thus the complex refractive index and the thickness of each layer.

There are several optical dispersion models which can be used to describe the complex refractive index of a thin film layer. As reported in our previous study [24], the Tauc-Lorentz (TL) model [25], which has been widely used for many amorphous materials [26–28], is the most appropriate for the analysis of thin Ta_{2}O_{5} film. It was derived from the Tauc joint density of states. The standard Lorentz form for the imaginary part of the dielectric function (*ε*_{2}) of a set of oscillators [29] is given by:

*E*is the band gap,

_{g}*E*

_{0}is the peak transition energy,

*C*is a broadening parameter and

*A*is a factor representing the optical transition matrix elements.

The real part of the complex dielectric function (*ε*_{1}) can be retrieved from the Kramers-Kronig relation [30], thus introducing another free parameter *ε _{∞}*.

Since

where*h*is Planck constant and

*c*is the speed of light, Eq. (1) can be rewritten in order to express

*ε*

_{2}and

*ε*

_{1}as a function of

*λ*.

Since

the refractive index*n*and the extinction coefficient

*k*can then be deduced.

Six parameters are thus required in order to fully characterise a single-layer (*E _{g}*,

*A*,

*E*,

_{o}*C*,

*ε*and

_{∞}*d*). In the case of multilayer filters, the same dispersion model was used for a SiO

_{2}layer. So for a 7-layer film, 17 parameters are required, and for a 22-layer film 32 parameters are required (

*E*

_{g}_{1},

*A*

_{1},

*E*

_{o}_{1},

*C*

_{1}and

*ε*

_{∞}_{1}for Ta

_{2}O

_{5},

*E*

_{TL}_{2},

*A*

_{2},

*E*

_{o}_{2},

*C*

_{2}and

*ε*

_{∞}_{2}for SiO

_{2}, in addition to the thickness of each layer

*d*

_{1},

*d*

_{2},

*d*

_{3}…

*d*

_{i}).

An error function (EF) was used to denote the difference between the calculated and the experimental reflectance or transmittance values:

*N*is the number of wavelengths. For each value of j, S

_{w}_{j,cal}and S

_{j, exp}are respectively the computed and experimental values.

*X*is a vector containing the free parameters derived from the laws governing the refractive index

*n*and the extinction coefficient

*k*, and ΔS

_{j}is the uncertainty on the values of reflectance or transmittance (in this paper ΔS

_{j}= 1). There are four types of EF, namely EF

_{0°}at normal incidence, EF

_{8°}at quasi normal incidence, EF

_{θ}at a specified single oblique incidence, and EF

_{mi}at multiple incidences:

_{θ}is the number of different oblique incidence (in this paper N

_{θ}= 3).

As a consequence of the relatively large number of variables required to compute the EF, a simple local non-linear least-squares optimisation technique is inefficient, and a global optimization procedure is required. In the global optimization process, EF is considered to be the objective function for which the global minimum must be found. Our laboratory has considerable experience in the use of the Clustering Global Optimization (CGO) method [24,31], for the determination of the complex refractive index and thickness of thin films. The underlying principle of this method, which has demonstrated its efficiency in solving thin-film design problems, is recalled below [32–34].

CGO methods can be viewed as a modified form of the standard Multistart procedure, in which a local search is performed from several starting points distributed over the entire search domain. A drawback of the Multistart technique is that when a large number of starting points is used, the same local minimum may be identified several times, thereby leading to an inefficient global search. Clustering methods are designed to avoid this inefficiency, through careful selection of the points from which the local search is initiated.

As shown in Fig. 2 , the CGO algorithm can be concisely described as follows:

- (1) Consider
*m*points with a uniform distribution, in an initially*n*-dimensional space*S*, and add them to the current sample*C*. Then refine this distribution to a smaller selection of only*p*points in*C,*having the best merit function. - (2) Apply the clustering procedure to these
*p*points. If all of the points belong to a cluster, go to step (4). Let*p*be the number of non clustered points. - (3) A local search is applied to these
*p*points. If a new local minimum is found, go to step (1): with*m*new starting points surrounding the local minimum. - (4) Find the local minimum having the smallest value.

As step (3) allows a new starting process to be initiated in the vicinity of the previous local minimum, the local procedure can find solutions outside the starting interval. This characteristic of the algorithm led us to refer to it as a “global optimization procedure”, even though the global aspect is not demonstrated.

Several solutions can be found at different incidence angles, with each solution described by a set of thicknesses and indices *n* and *k,* which are wavelength dependent. In order to express the difference between two solutions sol1 and sol2, the 'distances' of *n*, *k* and *d* are defined, using the root mean square distances expressed by:

*N*is the number of layers.

_{d}## 3. Results and discussion

Our investigation was carried out on three filter designs: a single-layer thin film (Ta_{2}O_{5}), a resonant structure (7-layer, Ta_{2}O_{5}/SiO_{2}), and a notch filter (22-layer, Ta_{2}O_{5}/SiO_{2}). These multilayer films were composed of alternate high and low index layers, and were all deposited onto a 170µm thick BK7 substrate, using the ion assistance electronic beam deposition technique. The experimental reflectance and transmittance values were measured with the spectrophometer, at 2 nm intervals, over the wavelength range from 350 nm to 1800 nm. With a 170µm thickness, the internal transmittance of the BK7 glass can be considered to be unity over the whole spectral range. Although the spectrophotometer's 'universal reflectance accessory' allows reflectance measurements to be made at angles as high as 60°, we observed that optimal results were obtained when the angle of incidence was limited to a maximum of 45°. Three oblique angles: 20°, 30°and 45°, were thus used, under both s and p polarizations, to measure the films' reflectance spectra. We also considered the films' transmittance at normal incidence, and both their reflectance and transmittance at quasi normal incidence. Finally, when using a multiple incidences optimization process, we sought to minimize an error function based on 9 different sets of measurements: 7 reflectance and 2 transmittance spectra.

#### 3.1 Single-layer (Ta_{2}O_{5})

For each 'single' incidence angle θ, referred to as S.I.θ, we used our optimization procedure, to determine the index and thickness of thin film and selected the best solution corresponding to an error function EF of S.I.θ. In order to verify that the solution derived at this S.I.θ was also valid for the other incidence angles, the spectra at other incidence angles were then calculated, using the parameters of this solution. By comparing the calculated and experimental spectra, we were able to calculate the average EF for all the incidence angles. This average value was then compared with EF_{mi}, the error function obtained through the multiple incidences (M.I.) optimization process described above.

The optimized error functions for a Ta_{2}O_{5} single-layer film, at different S.I.θ and M.I., are shown in Fig. 3
. It can be seen that all of the S.I.θ and M.I. provide very similar EFs, with a value below 0.4. This means that, in all cases, a good fit was found between the experimental and calculated spectra and that, in all likelihood, they found the same solution.

The distances between the solutions given by S.I.θ and M.I. are plotted in Fig. 4 . It can be seen that all the S.I.θ and M.I. found similar solutions, because the solutions' distances are small. Thus, for a simple single-layer case, the M.I. is not really necessary, and a S.I.θ (even T at S.I.0°) is sufficient to determine the layer's index and thickness.

As shown in Fig. 5(a)
, the experimental and M.I. optimized spectra for the Ta_{2}O_{5} single-layer are in good agreement over the whole spectral region. This is confirmed by the small value of EF_{mi}, which indicates a very small difference between the calculated and experimental data. The refractive index and extinction coefficient as a function of wavelength calculated by the solution of M.I. are shown in Fig. 5(b). These values are consistent with our knowledge of this thin film material.

In conclusion, for the case of a single-layer, we have demonstrated that both the S.I.θ and the M.I. can provide a satisfactory solution, with an accuracy of better than 1nm for the layer's thickness, and approximately 5.10^{−3} for its refractive index.

#### 3.2 Resonant filter: 7-layer, Ta_{2}O_{5}/SiO_{2} structure

The usually approach for the index determination of a thin film, consisting of only a S.I.0°, is not optimal for a complicated multilayer. In the following, we present a 7-layer filter designed to achieve resonances under total reflection [35]. When the indices and thicknesses were determined at S.I.0°, a very small EF_{0°} equal to 0.333 was obtained. However, when this solution was transposed to S.I.45° at pol *s*, an obvious mismatch was observed between the calculated and measured values, with an EF greater than 1.3. Figure 6
provides an illustration of the good match achieved for S.I.0°, together with the deviations observed at S.I.45°. In particular, there is a notable mismatch between the experimental and calculated values between 350 nm and 400 nm.

M.I. optimization, as a compromise method, attempts to minimize the discrepancies for all the incidence angles. Moreover, the number of solutions provided by the global optimization is dramatically reduced, thus confirming the notion that M.I. optimization is a beneficial strategy. The numerical results provided in Fig. 7
demonstrate that the M.I. error function (EF_{mi}) is smaller than any of the average EF obtained through the S.I.θ optimization process.

This conclusion is different to that found for the single-layer analysis. When the number of variable parameters increases (indices and thicknesses) to describe a complex design, a greater quantity of information is required in order to find a robust solution. The spectra measured at several incidence angles, and for different polarizations, are thus helpful in this context.

The distances between the solutions given by S.I.θ and M.I. are plotted in Fig. 8
. This figure shows that maximum errors of: 0.017 for *n*(Ta_{2}O_{5}), 0.016 for *n*(SiO_{2}), 0.755 10^{−3} for *k*(Ta_{2}O_{5}), 0.771 10^{−3} for *k*(SiO_{2}), and 2.8 nm for the thickness would be incurred, if only a S.I.θ is used.

The experimental and M.I. optimized spectra for the Ta_{2}O_{5}/SiO_{2} 7-layer filter are plotted in Fig. 9(a)
, showing that very good fits are obtained for all incidence angles, as could be expected with the small EF value produced by the M.I. optimization. The corresponding refractive indices as a function of wavelength, and thicknesses for the 7-layer optimized by the M.I., are plotted in Fig. 9(b). For the spectral range under consideration, the absorption of the filter materials is negligible.

#### 3.3 Notch filter: 22-layer, Ta_{2}O_{5}/SiO_{2} structure

In this example, we attempt to reverse engineer a more complex structure. This filter, composed of 22 alternate Ta_{2}0_{5}/SiO_{2} layers, has the optical function of a notch filter centred at 430 nm, with a rejection of approximately 56% at S.I.0°. Some of the layers are very thin, i.e., they have a thickness in the range 10 - 15 nm.

Our aim is to show that M.I. optimization makes it possible to find the ‘true’ solution, to illustrate the limitations of different S.I.θ optimizations, and to evaluate solutions given by an optimization process made at two different angles of incidence.

### 3.3.1 First approach: Optimization at S.I.0°

The S.I.0° optimization is poorly adapted to the analysis of this filter: the S.I.0° optimization provided many different designs that could produce exactly the same spectral transmittance at S.I.0°, and even with fixing the indices, the S.I.0° optimization could still find several different sets of thicknesses, which led to a low EF value. Figure 10 plots the experimental and calculated transmittance curves for 2 sets of solutions, for which the indices are strictly identical, but which (as an example) have a 14 nm difference in thickness for layer 15. It can thus be seen that, for the purposes of a reverse engineering problem, it is impossible to find a unique and right solution for the layer thicknesses on the basis of a S.I.θ measurement alone.

### 3.3.2 M.I. (0°-8°-20°-30°-45°) Optimization

M.I. optimization keeping both indices and layer thicknesses as variables (32 parameters) gave also many solutions. We extended the study of solutions from the best one, leading to the smallest EF_{mi} (equal to 0.647), to other solutions providing a 5% increasing of the EF value. Within this range, the M.I. optimization provided 10 solutions. We have studied the 10 best ones in order to determine whether or not they were nearly identical.

In Fig. 11 , the distances between solution 1 and the other 9 solutions are plotted. For the thicknesses, all of the solutions have distances of less than 1.1 nm, and index differences of less than 0.002. This means that the 10 alternative solutions could be merged into a single “physical solution”, with a thickness accuracy of approximately 1.1 nm. We are firmly convinced that this design is the true solution to the reverse engineering problem.

Using these index parameters and thicknesses, the calculated spectra are in good agreement with the experimental spectra, as shown in Fig. 12(a)
, with a corresponding to the small EF_{mi} (0.647). Figure 12(b) plots the values of the indices and thicknesses of this 22-layer design.

### 3.3.3 S.I.8° and S.I.45° investigations

M.I. optimization led to the ‘physical solution’ (*n*, *k*, *d*) of the 22-layer filter. Two S.I.θ reverse engineering cases justify a more detailed investigation:

- - The 8° angle of incidence recordings, as these include both transmittance and reflectance measurements with a dedicated module for accurate measurements. It should be noted that in the case of an almost lossless (non-absorbing) filter, only a small quantity of ‘new’ information is contributed by the transmittance measurements.
- - The 45° angle of incidence recordings, as these correspond to the highest angle of incidence used in the present study. At such an angle of incidence, the s and p spectra are substantially different, and the exploitation of both polarizations could considerably simplify the search for the optimal solution.

S.I. optimization, in which all of the layer indices and thicknesses are variables (32 parameters), led to many solutions with an equivalent EF_{S.I.} (approximately 0.55 for S.I.8° and 0.57 for S.I.45°). The distances between the S.I. solutions and the physical solution – given by M.I. optimization - are plotted in Fig. 13(a)
. It can be clearly seen that with a thickness distance of 18.5 nm for S.I.8° and 7.6 nm for S.I.45°, the S.I. investigations do not lead to conclusive results.

### 3.3.4 Investigations using spectral data recorded at two different incidence angles

The question now arises as to whether all of the measurements, made at different angles of incidence, should be taken into account, or whether a two-angle analysis is sufficient for the successful reverse engineering of our 22-layer sample.

We thus analysed data recorded at 8° (R and T) and 30° (Rs and Rp). Three numerical solutions provided an equivalent EF_{8° + 30°} of approximately 0.60. Unfortunately, the thickness distances between these solutions and the physical solution were respectively 3.0 nm, 3.6 nm and 7.0 nm (see Fig. 13(b)).

When spectra recorded at 8° (R and T) and 45° (Rs and Rp) were used, a better result was found: three different solutions with an equivalent error function value were obtained (EF_{8° + 45°} = 0.70). The thickness distances between these solutions and the physical solution do not exceed 1.6 nm, as shown in Fig. 13(b), and these solutions can be merged with the result found using M.I. optimization. This means that the reverse engineering problem for a dielectric filter can also be solved using spectra recorded at just two angles of incidence, in particular using quasi-normal incidence, together with the most oblique angle of incidence. However, in order to minimize the influence of possible measurement errors, under certain configurations, we recommend the use of more than two angles of incidence for the reverse engineering of a dielectric filter. The use of four or five different angles of incidence appears to be lead to a good compromise between accuracy and computing time.

#### 3.4. Comparison of the material optical properties of the 3 structures

As the 3 filters described in this study were deposited using the same ion-assisted electronic beam technique, the index *n*(Ta_{2}O_{5}) of their layers could be compared. The M.I. optimized values are compared in Fig. 14
, showing that these 3 structures have index distances below 0.01, which is compatible with our knowledge of the index reproducibility of the ion-assisted technique.

## 4. Conclusion

It is crucial for the optical engineer to be able to determine the optical constants and thicknesses of manufactured multilayer thin film filters, as such a process allows the specimen’s deviations from the initial design to be evaluated. Several studies have already demonstrated the advantages of multiple incidence measurements. In the present paper, we focus on a global optimization approach and the analysis of different potential solutions. We show that for the analysis of a single layer, multiple incidence angles are not necessary. When the complexity of a filter design increases, the single incidence angle method becomes less reliable. For a 7-layer stack, a thickness accuracy of the order of 3 nm is achieved when single incidence is used. In the case of a 22-layer notch filter, including very thin layers, we show that single incidence is inefficient. Conversely, the multiple incidences approach - combined with a global optimization algorithm and careful analysis - leads to a reverse engineered solution with a thickness accuracy of 1.1 nm and an index accuracy of 2 10^{−3}, with respect to the physical solution. We also show that when computational resources are limited, the multiple incidences study can be reduced to a two-angle optimization using normal (or quasi normal) incidence, combined with measurements at a very oblique angle of incidence.

As a conclusion, the reverse engineering for complex thin film designs relies on the successful implementation of several critical points:

- - for multilayer designs, several spectral measurements, measured at different incidence angles and polarization states, are needed. We refer to this as multiple incidences optimization. Of course, in the case of systematic errors, a bias can arise in the index and thickness determination,
- - an efficient global optimization procedure makes it possible to find several solutions, having acceptable error function values,
- - all of these potential solutions should be carefully studied, in order to avoid non-physical solutions, and to verify that the remaining 'realistic' solutions can be merged into a single one.

When these three conditions are satisfied, the final solution for the true opto-geometrical parameters is then realistic.

## Acknowledgements

This program has been supported by the French “Agence National de la Recherché” (ANR PNANO SEEC).

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