Refractive index determination of SiO 2 layer in the UV/Vis/NIR range: spectrophotometric reverse engineering on single and bi-layer designs

In this paper we use spectrophotometric measurements and a Clustering Global Optimization procedure to determine the complex refractive index of SiO 2 layer from 250 nm to 1250 nm. A special commercial optical module allows the reﬂection and transmission measurements to be made under exactly the same illumination and measurement conditions. We compare the index determination results obtained from two different single layer SiO 2 samples, with high and low index glass substrates, respectively. We then determine the refractive index of SiO 2 for a bi-layer design in which the ﬁrst deposited layer is Ta 2 O 5 . The corresponding solutions are discussed and we show that the real part of the complex refractive index obtained for a bi-layer is slightly different to that found for a single layer investigation. When SiO 2 is included inside a thin ﬁlm stack, we propose the use of an index determination method in which a bi-layer is used for the real part of the complex refractive index, and single layer determination is used for the imaginary part of the refractive index in the UV range. [DOI: http:/


INTRODUCTION
The determination of the complex refractive index (refractive index n and extinction coefficient k) of thin films is of great importance for the design and production of any optical interference filters.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 filters spectral response (reflectance R and transmittance T).Moreover, the index of thin films, which differs from that of bulk materials, strongly depends on the deposition techniques and conditions.The index of a single layer can also change in a multilayer structure, because the deposition conditions, especially temperature, vary with time [1].Therefore, the determination of the complex refractive index and thickness, which can be considered to be a reverse engineering process, is clearly necessary.
The determination of these optical parameters and thicknesses is generally carried out by means of ellipsometric and spectrophotometric analysis.Due to the complicated nature and high cost of the ellipsometer, more readily available and affordable spectrophometer has attracted considerable attention.
In the case of spectrophotometric analysis, the envelope method, using the envelopes of the oscillating transmittance or reflectance spectra, was developed many years ago.This method does not require any postulate or assumptions with respect to the optical dispersion model.Manifacier [2] and Swanepoel [3,4] have improved this method, in order to determine the optical parameters and thickness of a nonabsorbing or weakly absorbing single layer, deposited onto a transparent or weakly absorbing substrate.An alternative solution for a weakly absorbing layer consists in analysing the spectral wavelengths corresponding to reflectance extremes, at several oblique angles of incidence [5].An error minimization method [6] and a midpoint method [7] have also been proposed.However, the accuracy of this approach still strongly depends on the definition of these envelopes.In addition, this method does not work for film having significant dispersion and/or strong absorption.The other technique based on spectrophotometric analysis is the curve fitting method, which requires an optical dispersion model.This model is able to integrate a considerably greater quantity of spectral information, such as transmittance and reflectance, or spectra at multiple oblique incidences, into the index determination process [8].In addition, it is also able to manage complicated cases, such as strongly absorbing and complex multilayer filters.
As SiO 2 is the mostly commonly used low index film in many types of multilayer interference coatings, the determination of its refractive index, which can be essential to the successful production of a high quality multilayer filter, should be carefully investigated.In the present paper, we study three different configurations for the determination of the refractive index of SiO 2 , using the curve fitting method in the UV/Vis/NIR range.A SiO 2 layer was deposited onto a low index Corning 7980 Fused Silica substrate [9] and a high index S-LAH66 substrate [10].In this present study, a bi-layer Ta 2 O 5 /SiO 2 was also deposited onto a fused silica, following a procedure proposed in [11] where a bi-layer was deposited in order to overcome the problem of less contrast of interference fringes.Then, from spectrophotometric measurements at quasi-normal incidence, we tried to determine the complex refractive index and thickness for these three configurations, using a Clustering Global Optimization algorithm.We discuss this method, the results, the accuracy and the limitations of these three configurations.The layers were assumed to be homogeneous.Some authors [12] have pointed out that by taking the inhomogeneities into account, the index determination can be substantially improved if the fit between the theoretical (homogeneous model) and experimental data is poor.In the case of a single layer, the inhomogeneities can be easily detected, by comparing the minimum reflectance values with that of the bare substrate.

THEORETICAL ANALYSIS: INFLUENCE OF THE SUBSTRATE'S REFRACTIVE INDEX ON THE SiO SINGLE LAYER INDEX DETERMINATION
The refractive index determination of a single layer, by means of spectrophotometric measurements can be performed easily under certain conditions: • The internal transmittance of the substrate should be very close to 100%.Otherwise, it is very difficult to distinguish the absorption produced by the layer from that produced by the substrate.A 1 mm thick UV grade fused silica substrate classically presents an internal transmittance in excess of 99.9% at wavelengths above 250 nm.
The same transmittance should be found for a S-LAH66 substrate at wavelengths above 420 nm.
• The layer should not be opaque.When the transmittance approaches zero, many different solutions (n, k) can fit the experimental reflectance data.
• The films inhomogeneities should be evaluated.In the case of a transparent or weakly absorbing film, these can be checked by measuring the reflectance at wavelengths (λ) for which nd = pλ/2 where p is an integer.
• The accuracy of measurements is of prime importance.A systematic measurement error leads to an incorrect index determination, depending on the sensitivity of the layers index to the photometric measurements.This last point deserves further discussion.In the case of a transparent film, its transmittance (or reflectance) is characterized by interference fringes, the extreme values of which can be expressed as [3]: (2) where n s is the substrate refractive index, T HW and T QW are the T values reached for a half wavelength combination: nd = pλ/2, and for a quarter wavelength combination: nd = (2p + 1)λ/4, respectively (where p is an integer).
Clearly, the T Q W values are the signature of the layers index, and shifts in the values of T QW lead to shifts in the value of the layers refractive index.In the case of a single SiO 2 layer, depending on the wavelength, the layer index lies in the approximate range: n =1.45∼1.5.Using Eq. ( 2), one can extract the layer index n as a function of n s and T QW .Figure 1 illustrates the relationship among the layer index n (plotted values), T QW (vertical axis) and the substrate index (horizontal axis).We then define the layer index sensitivity S (n,n s ) as the layer index shift corresponding to a small variation of T QW , for a given substrate index, as expressed by Eq. ( 3): As shown in Figure 1, for a substrate index equal to 1.45 a variation in T QW of approximately 0.005 (from 0.93 to 0.935) leads to a shift in layer index of 0.02 (from 1.47 to 1.45), and S(1.45, 1.45) = 4.For comparison, S(1.45, 1.78) = 8.
The inaccuracy of the photometric measurements is generally ∆T = 0.001 [13,14], which leads to an uncertainty in the layer index of approximately 0.004 for the case of a fused silica substrate (n s = 1.45) and 0.008 for a S-LAH66 substrate (n s = 1.78).This outcome therefore favors the use of a low index substrate, although the low contrast of interference fringes can make it difficult to determine the thickness.This will be illustrated in Figure 3.

EXPERIMENTAL SETUP AND SPECTRAL MEASUREMENTS
We used a commercial Perkin Elmer lambda 1050 spectrophotometer with a specific module developed by OMT Solutions, as shown in Figure 2. The sample is placed horizontally, with the same measurement spot used for both reflectance (R) and transmittance (T) at quasi-normal angle of incidence (8 • ).The incident beam presents a divergence below 1 degree, and a calculation considering a perfected collimated beam is sufficient in almost cases.The illuminated zone on the sample is a square of 2mm side.The switchover between R and T configurations is performed automatically by means of a 180 • rotation of the detection system.An integrating sphere measures the reflected/transmitted flux from 250-2500 nm.Both the incident beam and the detection system are similar for the R and T configurations, such that it makes sense to estimate the samples absorbance (A) using: In the present study, three samples were deposited using the ion assistance electronic beam deposition technique.This technology is able to produce very high-density layers, with stable optical properties close to those of bulk materials.We deposited a single SiO 2 layer with a thickness of approximately 300 nm onto fused silica (sample S1) and S-LAH66 (sample S2) substrates.Then, a bi-layer Ta 2 O 5 /SiO 2 film was deposited onto fused silica (sample S3).We restricted the experimental spectral range to 250-1250 nm for the samples with a fused silica substrate, i.e., S1 and S3, and to 420-1250 nm for the S-LAH66 substrate, i.e.S2, and used a 2 nm measurement interval.The experimental reflectance (Rexp), transmittance (Texp) and absorbance of these samples are plotted in Figure 3.
The theoretical values of bare SiO 2 and S-LAH66 substrates are also shown.
As shown in Figure 3(a) and 3(b), for sample S1 the spectral  oscillations are barely visible, and |THW-TQW| lies in a range close to 0.5%, whereas the oscillations are clearly visible for sample S2, with |THW-TQW| lying in a range close to 6%.This difference in oscillation amplitudes is due to the different contrasts in refractive index between the substrate and the layer.The spectral responses of S3 are characterized by many oscillations, which in fact contain 'considerable information' related to the bi-layer design.In some spectral regions, its transmission is greater than that of the bare substrate, thus correspond- ing to the local production of an anti-reflection effect, and in other regions the opposite effect is produced.
Figure 3 shows that the absorbance is not detectable for sample S2, since the measurements begin at 420 nm on the S-LAH66 substrate.In the case of samples S1 and S3, their absorbance typically decreases when the wavelength increases from 250 nm to a certain cutoff wavelength λ S .Above λ S , the absorbance is not detectable.From these absorbance curves, λ S1 can be estimated to be approximately 330 nm, i.e., shorter than λ S3 (about 450 nm), even in the presence of measurement noise of approximately 0.2% over the range [250-700 nm], and 0.5% at wavelengths above 750 nm.The value of λ S is directly related to the lowest energy gap of the materials inside a thin film stack.As the bi-layer includes SiO 2 and Ta 2 O 5 , and the energy gap of Ta 2 O 5 is known to be below that of SiO 2 [15,16], the starting absorption wavelength of S3 is longer than that of S1.At short wavelengths, the absorbance of S3 is contributed mainly by the Ta 2 O 5 .It can thus be concluded that k (SiO 2 ) is undetectable for λ > λ S1 , whereas k (Ta 2 O 5 ) is undetectable for λ > λ S3 .

COMPUTATIONAL PROCEDURE FOR THE INDEX DETERMINATION
The complex refractive index of a thin film is estimated from the reflectance and transmittance values, as shown in Figure 4.The theoretical reflectance (Rcal) and transmittance (Tcal) were calculated using the matrix method [17], taking the backside reflectance into account.They depend on the measurement wavelength, the refractive index of the substrate n s , the complex refractive index (n, k), and the thickness (d) of the layer.n s was assumed to be known.Accordingly, the complex refractive index and thickness of the layer need to be determined.
There are several optical dispersion models describing the laws of the complex refractive index.The most frequently used is Cauchy, which is simple and performs well for a transparent thin film.As wavelengths as short as 250 nm were used in this study, the thin SiO 2 -Ta 2 O 5 film exhibits absorption, and the Tauc-Lorentz model (TLM) [18] is then more suitable.TLM, which has been widely used for many amorphous materials [19]- [21], was derived from the Tauc joint density of states.The standard Lorentz form for the imaginary part of the dielectric function ( 2 ) of a set of oscillators [22] is given by: where E g is the band gap, 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 ( 1 ) can be retrieved from the Kramers-Kronig relation [23], thus introducing another free parameter ∞ .
Since where h is Planck constant and c is the speed of light, Eq. ( 4) can be rewritten in order to express 2 and 1 as a function of λ. Since the refractive index n and the extinction coefficient k can then be derived.
Six parameters are thus required in order to fully characterise a single layer (E g , A, E 0 , C, ∞ and d), and twelve parameters are needed for a bi-layer.An error function (EF) was used to denote the difference between the calculated and experimental reflectance or transmittance values: where N w is the number of wavelengths.For each value of j, R j and T j are respectively the computed values of reflectance and transmittance, whereas R j,exp and T j,exp are the experimental values of reflectance and transmittance, and d is the thickness of the film.X is a vector containing the free parameters derived from the laws governing the refractive index n and the extinction coefficient k. ∆R j and ∆T j are respectively the uncertainties on the values of reflectance and transmittance (in this paper ∆R j = ∆T j = 1, which makes sense with our measurement system where reflectance and transmittance are measured with the same accuracy).
As a consequence of the relatively large number (12) of variables required to compute the EF, a simple local nonlinear 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.As reported in our previous study [24], the Clustering Global Optimization (CGO) algorithm [25,26] performs perfectly 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 [27]- [29].
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 Figure 5, the CGO algorithm can be described as follows: (1) Consider m points with a uniform distribution, in an initially n-dimensional space, and add them to the current sample CS.Then refine this distribution to a smaller selection of only p points in CS, 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.
Taking a simple 1-dimensional function as an example, the functioning of the CGO can be more easily understood, as shown in Figure 6.Each step is described: (1) The aim is to find the global minimum after comparing all the local minima of a function.In the present drawn case, three minima exist (x = 5.1, x = 8.2 and x = 11).
(3) A local descent is applied to the best p evaluated points.Two local minima are thus detected (x = 5.1 and x = 8.2). (5) A new starting interval is imposed around the new minimum.When all local searches lead to the already detected solutions, the global minimum is found by comparing all of the local minima, and the algorithm is then stopped.
The efficiency of CGO depends on the starting interval: with a small starting interval it is difficult to jump to another minimum.With a given density grid, a broader starting interval leads to a larger step between evaluated points.Under such conditions, the program may be unable to detect very sharp minimum peaks.A good compromise, which can be found with the users experience, is mandatory when a high number of variables (typically > 20) is used, and one can expect to find all of the minimum values in cases where the number of variable parameters is less than 20.Of course, for problem where solutions are not perfectly known, one does not have proof that the global minimum is detected by CGO.By changing either starting interval or grid density or number of selected points, one may check that the global solution remains constant, which is a beginning of proof.

RESULTS AND DISCUSSION
CGO provides a unique solution, with an excellent EF value (below 0.1) for both S1 and S2 problems.The number of starting points is set to m = 1000, and the number of p minimum is set to 10.These values are chosen from our experience on an optimization problem with 6 variables describing n,k and d.
The optimized parameters E g , E 0 , C, A, ∞ and d allow n and k to be computed.
The corresponding values for samples S1 and S2 are plotted in Figure 7. Concerning n, the values are nearly identical, with differences smaller than 10 −4 .Concerning k, it appears that for wavelengths in the range [420-450 nm], SiO 2 is absorbant for S2.We are convinced that if S2 is slightly absorbing, this is due to the internal transmittance of the S-LAH66 substrate, which begins to be lower than 1.Under this assumption, k cannot be detected on a S-LAH66 substrate.Conversely, the fused silica substrate allows k to be determined over the range [250-330 nm].We determined the value of k to be 10 −3 at λ = 260 nm.For wavelengths above 330 nm, k is smaller than 2 •10 −4 .We consider this to be the minimum threshold value that can be extracted from our spectrophotometric determination.Moreover, for values of k of the order of 10 −4 , other 13010-5  techniques such as Cavity Ring-down Spectroscopy [30] are better adapted.
For the S3 -bi-layer design investigations, we tested two configurations: Config.(a): all parameters, i.e., thicknesses, real and imaginary parts of the complex refractive index (12 parameters) were made variable.This configuration confers a high flexibility to all of the parameters.CGO provides one solution (sol1) for this configuration.
Config.(b): this configuration is proposed as a consequence of the difficulty in determining k(SiO 2 ), since it is likely that most of the absorption is produced by the Ta 2 O 5 layer.We then chose to set k(SiO 2 ) to the value obtained from the S1 optimization.All of the other parameters remained variable.CGO provides two solutions (sol2 and sol3) for this configuration.
The EF values of the three solutions are very similar, i.e., EF ≈ 0.32.For the purposes of illustrating the curve-matching achieved with this method, Figure 8 shows a plot of the fit between the experimental and calculated spectra of sol1.As suggested by their similar EFs, the fits of sol2 and sol3 are also satisfactory.For the refractive index of the Ta 2 O 5 layer, the three solutions lead to a satisfying value, consistent with the single layer determination described in our previous publication [26].
Concerning the SiO 2 , its refractive index computed from the three solutions is shown in Figure 9.It can be seen that sol1 and sol2 are similar in terms of the index values.sol3 is slightly different, but should be eliminated since n = f (λ) is not a 'physically' acceptable value, and has an oscillatory behavior between 400 nm and 700 nm.However, as can be seen in Figure 9, the index value given by the bi-layer is not exactly the same as that given by the single layer determination.Both indices are the same at λ = 300 nm (n = 1.506), but differ when the wavelength increases.At λ = 1000 nm, the single layer optimization leads to an index of n = 1.470, whereas a bi-layer determination leads to n = 1.454.We are convinced that this difference is real.In the case of a bi-layer, the SiO 2 layer is deposited immediately after the Ta 2 O 5 layer, and the chamber is already under steady-state conditions (continuous regime).Moreover, a hypothetical transition layer with a thickness in the range 1-5 nm, between the layers of Ta 2 O 5 and SiO 2 , could also provide an explanation.An index difference of approximately 0.01 is plausible, and would be in agreement with other values reported in the literature [31]- [33].

CONCLUSION
In this paper we have determined the complex refractive index of single SiO 2 layers on different substrates and for a bilayer design.We obtained a similar value of refractive index for single layers, and this could be determined over the range from 250 to 1250 nm using a SiO 2 substrate.The extinction coefficient is no longer detectable at wavelengths above 330 nm, when k is less than 2 •10 −4 .The use of a global algorithm is an efficient means for determining the refractive index of SiO 2 on a bi-layer design including a first layer of Ta 2 O 5 .The SiO 2 index determinations of single layers and bi-layer show a very small difference (in the 0.01 range) for the visible and NIR wavelength range.When SiO 2 is used in a multilayer stack, we propose to use bi-layer results to determine n, the real part of the complex refractive index, and a single layer on a fused silica substrate to determine k, the imaginary part of the complex refractive index.

Fig. 1 .
Fig. 1.Layer index as a function of T QW and the substrate index, i.e., n = f(T QW , s n FIG. 1 Layer index as a function of T QW and the substrate index, i.e., n = f (T QW , n s ).

1 OMTFIG. 2
FIG. 2 OMT module for transmittance (a) and reflectance (b) configurations.The moving part, consisting of three reflecting mirrors, is indicated by the dashed lines.

Fig. 3 .
Fig.3.Experimental reflectance (a), transmittance (b) and absorbance (c) of sample S1, S2, S3, and theoretical values for a bare SiO 2 substrate and a bare S-LAH66 substrate in (a) and (b), as a function of wavelength.

FIG. 3
FIG. 3 Experimental reflectance (a), transmittance (b) and absorbance (c) of sample S1, S2, S3, and theoretical values for a bare SiO 2 substrate and a bare S-LAH66 substrate in (a) and (b), as a function of wavelength.

( 4 )
Perform a new local search with new starting intervals in the vicinity of the two local solutions: a new local minimum is detected (x=11).The interesting fact is that this new solution does not belong to the initial starting interval x ∈ [4-8].

Fig. 6 .
Fig. 6.Example of a 1-dimensional function illustrating the functioning of CGO.

FIG. 6
FIG. 6 Example of a 1-dimensional function illustrating the functioning of CGO.

Fig. 7 .Fig. 8 .FIG. 8
Fig. 7. Refractive index (n) and extinction coefficient (k) for the samples S1 and S2 as a function of wavelength.FIG.7 Refractive index (n) and extinction coefficient (k) for the samples S1 and S2 as a function of wavelength.

Fig. 9 .
Fig. 9. Refractive index (n) as a function of wavelength, for the SiO2 layer in the S3 bi-layer, obtained from sol1, sol2 and sol3, compared with the refractive index found for the single S1 SiO2 layer.