1. Introduction

The absorption level in optical filters based on dielectric stacks is critical for most applications and should remain as low as possible. A few decades ago, methods were developed to determine low absorption values in materials and thin films. For instance, laser calorimetry and photoacoustic measurements were shown to be sensitive enough to measure absorption levels down to 10⁻⁶ [1]. However, such techniques provide extinction coefficient (k) values only at a single wavelength (λ) and the cost per measurement is high. Using standard spectrophotometry, the evaluation of low k values in quasi-transparent films is very challenging [2]. For example, in a comparison study of different optical characterization methods [3], Arndt et al. showed that the refractive index (n) of Sc₂O₃ films could be evaluated with a precision of typically 1%, while the k values were too small (k < 10⁻³) to be accurately determined by these techniques. Furthermore, round robin measurements performed on Ta₂O₅ films by 14 laboratories around the world that were presented during the 2004 Topical Meeting on Optical Interference Coatings (OIC) also emphasized the difficulty of precise k evaluation [4].

The determination of the optical constants of thin films from spectrometric measurements in the UV-VIS-NIR range generally relies on the development of an optical model including various dispersion equations that represent n(λ) and k(λ) for each layer over a broad wavelength range [5]. Most of the n(λ) and k(λ) functions (e.g., Lorentz, Drude, Tauc-Lorentz, etc.) are Kramers-Kronig consistent and were developed on theoretical grounds, which ensures the physical validity of the model. Once the model is built, the film thicknesses and the dispersion function parameters are fitted using an iterative regression procedure until the difference between the model and the experimental data is minimal.

In the context of the evaluation of k, this approach is effective as long as the model matches well the experimental data and the k values are relatively large (A≥1%, k ≥ 10⁻²). However, in the spectral range where absorption (A) becomes increasingly small (A<1%, k < 10⁻²), which is typically the case above the band edge of dielectric materials, a slight mismatch between the model and the experimental data will cause a large error in the evaluation of k values. This mismatch may result from two causes: (i) measurement error (noise, spectral bandwidth, beam collimation, sample alignment, etc.); and (ii) modeling error due to sample non-idealities (film thickness non-uniformity, interfaces and inhomogeneities, anisotropy, etc.).

Using modern double-beam spectrophotometers, measurement error on transmission (T) or reflection (R) spectra is typically in the range of 0.1% [6]. Such error in T corresponds to k error in the range of 10⁻⁴ and 10⁻⁵ for coatings that are 0.1 and 1.0 micron thick, respectively. Modeling error caused by sample non-idealities is more difficult to quantify since it is sample dependent. However, in the case of inhomogeneous coatings, it is not unusual to see a difference of 1% or more between the modeled and experimental data, which means that the modeling error can be an order of magnitude higher than the measurement error itself.

In the present work, we propose a new method named TRACK (Transmission, Reflection, Absorption Combination for K evaluation) that significantly decreases the modeling error and thus enables one to measure k down to 10⁻⁵ with high accuracy for optical coatings with thicknesses in the one micron range. The principle, the requirements and the technical aspects of the method are detailed in section 2. We present examples of how the method can be practically applied to: (i) substrates (section 3.1); (ii) high quality optical coatings deposited on complex substrates (section 3.2); and (iii) laboratory-type coatings deposited on glass presenting various non-idealities (section 3.3). We also compare the results obtained at two wavelengths (410 and 550 nm) from the TRACK method to those obtained by the Laser Induced Deflection (LID) technique in the case of 1 micron thick TiO₂ films (section 4).

2. Methodology and requirements

2.1. Principle

In essence, the method is based on the multi-angle collection of R and T data in p polarized light as depicted in Fig. 1. First, the thickness and n values of the film are obtained from the modeling of R and T data acquired at an angle of incidence (AOI) of 10°. Second, another set of R and T data is collected at an AOI of 65° (see next section for details about AOI optimization) and combined together to yield the A data:

 

Fig. 1 Diagram of the methodology used for the evaluation of the film thickness and the optical constants of coatings. The same methodology can be applied for the determination of the optical constants of substrates.

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In the next step, k values are directly computed from the A spectrum at each wavelength using a two layer (substrate/film) optical model, given the thickness and refractive index of both layers as well as the extinction coefficient of the substrate. Note that the bare substrate should be characterized beforehand using the same methodology as for the coated sample in order to include its optical constants in the model. The thickness of the substrate can be easily obtained with a micrometer.

Compared to the modeling of R and T data only, the analysis of A presents several advantages: (i) since interference fringes in R and T spectra nearly compensate each other for quasi-transparent samples, A data are not sensitive to film non-idealities such as thickness non-uniformity, interfaces and refractive index inhomogeneities, anisotropy, etc.; (ii) A data are also little affected by most instrument limitations and errors: spectral bandwidth, beam collimation, incidence angle offset; (iii) consequently, a very simple two layer optical model can be used to compute k values without any negative impact on the method accuracy; finally (iv), A spectra can be smoothed in order to reduce instrument noise without losing important information since they are free of interference fringes or any sharp feature. These advantages are demonstrated through various examples in section 3.

2.2. Optimization of angle of incidence

In this section, we explain the choice of the two angles of incidence (10° and 65°) used in the TRACK method. In particular, we show that measurements performed close to the Brewster angle in p polarized light are key for obtaining high quality interference-free A spectra.

Figures 2(a) and 2(b) present the calculated T, R and A spectra for different hypothetical samples: i) a bare glass substrate; ii) a SiO₂ coating on glass; iii) an Al₂O₃ coating on glass; and iv) a TiO₂ coating on glass. All the coatings are assumed to be 1 micron thick and to present identical k values in the range of 10⁻⁵ over the spectral range of interest. The angle of incidence is fixed to 10° and the light is p polarized. Figure 2(a) shows that choosing a low angle of incidence leads to interference fringes with high amplitude in R and T spectra. Since the fringes are directly related to the film thickness and to the refractive index contrast between the film and the substrate, this configuration will maximize the sensitivity for both parameters.

 

Fig. 2 Transmission, reflection and absorption spectra simulated for different film materials in p polarized light at two angles of incidence. Spectra calculated at near-normal incidence (10°) and near-Brewster incidence (65°) are shown in (a–b) and (c–d), respectively. Film materials with low (SiO₂: 1.46), medium (Al₂O₃: 1.77) and high (TiO₂: 2.28) refractive index values at 550 nm are exemplified. Spectra obtained for a typical bare glass substrate are also shown for reference.

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In Figure 2(b), the simulated absorption spectrum corresponding to the SiO₂ coating is relatively featureless and shows an increase of A values compared to the bare substrate due to additional absorption in the film. For films with higher refractive indices, especially the TiO₂ coating, one observes the presence of residual oscillations in the A spectra, which indicates that the interference fringes in R and T spectra don’t compensate very well. The reason for this phenomenon is that reflected light is less absorbed than transmitted light. This is evidenced by the fact that the maxima of R in Fig. 2(a) occur at the same wavelength as minima of A in Fig. 2(b). Since reflection increases for high index coatings, the mismatch between R and T increases as well. Obviously, the presence of oscillations in the A spectra is not ideal for the determination of k, since a more complex optical model would be required to obtain a good fit of the experimental data. In addition, the reflection of the TiO₂ coating is so high in the region above 700 nm that absorption values are below those obtained for the bare substrate. This means that the sensitivity to film absorption is decreased and hence larger error on calculated k values will occur.

The issues mentioned above can be solved by optimizing the angle of incidence. This is exemplified by Figs. 2(c) and 2(d), where the AOI is set to 65° while all other parameters are unchanged. This angle is close to the Brewster angle (θ_B) of the films, in particular for the TiO₂ coating (θ_B = 66.3° for n₅₅₀ = 2.28), leading to minimized reflection and maximized transmission in p polarized light (Fig. 2(c)). This results in strongly reduced residual interference fringes in the A spectra and nearly identical absorption values for all coatings, independently of their refractive index (Fig. 2(d)). Therefore, this “near θ_B” configuration will lead to more accurate k values.

In principle, one should measure R and T spectra at exactly θ_B in order to completely eliminate the interference fringes in the A spectrum. However, this would require to know the refractive index of the film beforehand and to adjust the AOI at each wavelength due to material dispersion. In addition, when films with high refractive indices are analyzed, θ_B reaches high values (e.g., germanium films: n ≈ 4.0, θ_B ≈ 80° in the IR range), leading to a large projected area of the light on the sample surface (factor of 1/cos(AOI)). This is not desirable, firstly because surface contamination and defects will have more impact at very large AOI, and secondly because in some cases the projected area may exceed the sample dimensions, which will invalidate the measurement. For these reasons, we believe that it is more practical to restrict the AOI to 65°, which yields good results in most situations (film refractive indices in the 1.4–2.5 range). At any rate, one should ensure that the amplitude of the residual fringes in A spectra are much lower than the absolute A values (as in Fig. 2(d)), so that their impact on the calculated k values is negligible.

2.3. Optimization of film thickness

Since spectrophotometers are typically accurate to ≈ 0.1% [6], the film thickness should be chosen so that A ≥ 0.1%. The minimum required thickness can be estimated using the Beer-Lambert law:

Therefore, the minimum film thickness that fulfills the condition A ≥ 0.1% is given by:

As an example, Eq. (6) can be used to predict that the films simulated in Figs. 2(d) with expected k ≈ 5 × 10⁻⁵ should be deposited with a minimal film thickness of ≈ 1000nm in order to determine k with good accuracy at λ = 600nm. Since it is not always possible to predict the k range beforehand, the alternative strategy would be to deposit the thickest possible film, i.e. typically a few microns. However, the technique employed to fabricate the films may also restrict the available thickness range depending on the film growth rate, and a compromise has to be made between high absorption values and reasonable deposition time. Note that Eq. (6) can also be used as a tool to verify the validity of the measured k values, given the film thickness.

2.4. Choice of substrate and cleaning

The substrate onto which the coatings are deposited should be transparent over the wavelength range of interest, with absorption levels equal or lower than that of the films to be analyzed (typically ≤ 1% of absorption). Fused silica is the best substrate since it presents low absorption values over a wide spectral range (UV to NIR), although the cost may be prohibitive for routine measurements. High quality glass or plastic are cheaper alternatives for absorption measurement above 400 nm, where these materials are usually transparent. The substrate should also be optically flat and present good parallelism (< 5 arcmin), so that the reflected beam does not diverge or deviate. The substrate thickness should be large enough that the surface does not deform under the coating’s intrinsic stress (for instance, microscope 150μm thick glass cover slide are not suitable) and small enough that the beam reflected from the back side is entirely collected by the detector at oblique incidence. In the case of the Cary 7000 UMS instrument, the maximum substrate thickness recommended by the manufacturer is 5 mm. For routine absorption measurements, we found that high quality Corning glass (Gorilla®, Eagle®) yield excellent results in the visible spectral range. Examples of absorption measurements performed on Gorilla glass and optical plastic substrates are presented in section 3.

The substrate should be free of dust and contaminants in order to avoid any film delamination and limit the proportion of scattered light. In most cases, efficient cleaning can be achieved simply by gloved-hand washing the substrate with Windex®, rinsing it in distilled water, and drying it with nitrogen. In this study, this procedure was systematically applied to glass substrates, while plastic substrates were clean enough out of the box that they didn’t require such treatment.

2.5. Instrumentation and data acquisition

Although the idea of combining R and T data in order to derive A seems trivial, such measurements are practically difficult to carry out. Indeed, the method requires that the sample is exactly at the same AOI and the same position with respect to the light beam for both R and T collections, so that interference fringes in R and T spectra cancel each other and no residual oscillations are visible in the resulting A spectrum. In order to fulfill this requirement, we employed the Cary 7000 UMS instrument from Agilent Technologies, which is a double beam spectrophotometer equipped with rotating sample and detector stages. Using this instrument, R and T data can be acquired at various angles of incidence without moving the sample (from 5° to 85° for R and from 0° to 180° for T). In addition, the Cary 7000 UMS is equipped with a motorized polarizer, and the spot size and beam collimation can be manually adjusted using various aperture masks (1°, 2° or 3° of beam divergence). When the polarizer is in use, the accessible wavelength range of the instrument is limited to 250 to 2500 nm.

In order to measure R and T with the highest accuracy (error < 0.1%), it is critical that the instrument is well stabilized with minimal thermal drift. This can be achieved by maintaining the instrument always powered up with the halogen lamp turned on and the deuterium lamp turned off (the D2 lamp is more costly and doesn’t produce much heat). In addition, when the instrument is not operated for a few hours, we perform another stabilization procedure that consists in acquiring data from a dummy glass sample using the same instrument setup (wavelength range, slew rate, number of scans, etc.) as the one employed for the actual measurement. This procedure may be repeated several times until the drift observed from one measurement to another is within instrument noise. Note that this step is not necessary in the case of single wavelength measurements (see section 4) since the delay between the baseline and the sample measurement is shorter in this case, and therefore instrument drift is less critical.

Instrument and sample alignment is another aspect that should be controlled for accurate R and T measurements. In the case of the Cary 7000 UMS, instrument alignment is ensured by properly calibrating the sample and detector stages using built-in routines. The vertical tilt of the sample stage may also be manually optimized if necessary. Sample alignment can be easily checked by placing a laser pointer facing the sample compartment and by verifying the position shift of the beam reflected from the sample when the sample stage is rotated by 180°.

The collection of R and T spectra was performed using the following parameters and settings: the acquisition time per data point was fixed to its minimum value (33 ms); the largest aperture mask (3° beam divergence) of the instrument was used in order to maximize the signal to noise ratio; the slit width and the wavelength step were set to 4 nm and 1 nm, respectively, which are reasonable values to resolve interference fringes of 1 micron thick coatings. In order to reduce noise and instrument error in A data, R and T spectra collected at AOI=65° were averaged over 10 cycles. However, 1 cycle was sufficient for R and T spectra acquired at AOI=10°. This configuration leads to a total measurement time below 30 min per sample when scanning over a wavelength range of 250–1000 nm.

2.6. Software and computation of k from A

Modeling of R and T data was carried out using the WVASE32 software (J.A. Woollam Co.), from which we obtained the film thickness and the refractive index. Note that the k values determined from the analysis of R and T spectra in WVASE32, which are presented for comparison purpose in section 3, are not required for the TRACK method. A detailed description of the various optical models employed in WVASE32 for each of the presented samples is provided in section 3. Since WVASE32 does not allow one to analyze absorption data, the computation of k values from A was performed in another program written in Python. This custom application makes use of the tmm Python package that simulates light propagation in multilayer thin and/or thick films using the Fresnel equations and transfer-matrix method [7]. The lmfitPython package was used for fitting the k values at each wavelength (λ) of the A spectrum [8]. The optical model used for the computation of k consisted in a simple two-layer stack (substrate/film). The input parameters of the model were: (i) the film and substrate thicknesses; (ii) the film and substrate refractive indices at λ; (iii) the substrate extinction coefficient at λ; (iv) the angle of incidence (65°); and (v) the light polarization state (p).

3. Practical examples

3.1. Complex substrate

In this section, we demonstrate how the TRACK method can be applied to the determination of optical constants of substrates. For this practical example, we chose a CR-39 plastic substrate (allyl diglycol carbonate, 1.7 mm thick) that is commonly used in eye-ware applications. In order to increase the scratch resistance of ophthalmic lenses, the CR-39 is usually dip-coated to produce a “hard coat”, i.e., a coating that is a few micron thick and that displays optical properties close to those of the substrate while being mechanically harder. The presence of the hard coat makes the optical analysis of this type of substrate more challenging and we will see below how these difficulties can be handled using the TRACK approach.

The R and T spectra of the CR-39 substrate with hard coat presented in Fig. 3(a) were measured at an AOI of 10°. Interference fringes with amplitude around 1% are visible on both spectra and are caused by the hard coat. Using an optical model consisting of two layers (CR-39 and hard coat), we were able to fit relatively well the R and T spectra, and the difference between the modelled and experimental data did not exceed ±0.1% (see Fig. 3(b)), except for the spectral range below 400 nm where the instrument noise was higher (±1% difference). The CR-39 dielectric function (ε) was modelled as a function of light energy (E) using the following equations [9, 10]:

 

Fig. 3 Results obtained for the CR-39 substrate with hard coat using the TRACK method: (a) T and R spectra measured at an AOI of 10° in p polarized light and modelled data; (b) Difference between experimental and modelled data; (c) Dispersion curves calculated from the parametric model; (d) T and R spectra measured at an AOI of 65° in p polarized light; (e) A spectrum calculated from T and R data; (f) Final dispersion curves of the CR-39 substrate: n is obtained from the parametric model and k is directly calculated from the A spectrum, using n and the film thickness as input parameters.

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Figure 3(d) shows the R and T spectra obtained at an AOI of 65°. Since that AOI is close to θ_B, the average reflection is lower and the amplitude of interference fringes is smaller compared to the measurements performed at 10°. The absorption spectrum (A) is directly derived from R and T, as shown in Fig. 3(e). No residual interference fringes are visible in the A spectrum, and one obtains a smooth curve except for values between 500 and 800 nm that are noisy due to the sensitivity limit of the instrument (≈ 0.1%). The slight step around 720 nm is caused by the change of grating in the instrument’s monochromator.

Figure 3(f) presents the optical constants of the CR-39 substrate. The refractive index is obtained from the Cauchy model and the extinction coefficient is calculated for each wavelength of the absorption spectrum. Since no interference fringes are visible in the A spectrum, one doesn’t have to take into account the hard coat layer in the optical model, as long as the optical constants of the hard coat are close to those of the CR-39. In Figure 3(f), the shape of k is obviously very similar to the shape of the A spectrum. The k values below 350 nm are not reliable since absorption is nearly 100% in this range and the dynamic range of the detector is not high enough to provide accurate T values. Note that a thinner substrate could be used in order to determine k with better precision in the UV region. The k values between 500 and 800 nm are noisy but still more realistic than those obtained from the dispersion equations (compare to Fig. 3(c)). In addition, the rise of k around 900 nm is very well defined and can be ascribed to the presence of water in the plastic substrate. This example shows that the TRACK method is efficient to determine the dispersion curves of complex composite substrates.

3.2. Thin film on complex substrate

In this section, we use the TRACK method to evaluate the optical constants of a high refractive index thin coating (TiO₂ ≈ 400nm) deposited on a plastic substrate similar to that characterized in the previous section. The TiO₂ coating was deposited using an industrial e-beam coater (BOXER pro, Leybold Optics GmbH) equipped with an ion-assist gun, which produces high quality films with uniform thicknesses. However, the presence of the hard coat on the CR-39 substrate makes the optical analysis more difficult for such samples.

Figure 4(a) shows the transmission and reflection spectra obtained at an AOI of 10°. R and T spectra display a combination of large interference fringes caused by the TiO₂ coating and smaller fringes caused by the hard coat. The optical properties of the substrate and the hard coat were fixed, while the dispersion curves of the TiO₂ layer were modelled by Eqs. (7)–(9). Two Gaussian oscillators were used in Eq. (8) in order to take into account the coating absorption in the UV range, and the amplitude of the two poles in Eq. (9) was set to zero. The difference between the experimental data and the model is within ±1%, as displayed in Fig. 4(b), and the error is mainly due to the presence of the hard coat, as observed from the small residual interference fringes. The TiO₂ film thickness derived from the model was 392 nm. The dispersion curves of the TiO₂ coating obtained by fitting the parameters of Eqs. (7)–(9) are presented in Fig. 4(c). In addition, we plot the n and k values resulting from a point by point fit of the R and T spectra, which is a method commonly used to assess the validity of the optical model. In this method, all the parameters of the optical model are fixed (e.g., the film thickness, the optical properties of the substrate, etc.), and only n and k of the film are fitted at each wavelength of the R and T spectra. In the ideal situation where the optical model represents perfectly the sample, n and k values calculated from the point by point fit method and those obtained from the parameterized dispersion curves should overlap. From Figure 4(c), we see that the discrete and parameterized n values are relatively close to each other, although some artifacts appear in the case of the point by point fit values. On the contrary, the discrete and parameterized k values are very different: the parameterized values are below 10⁻⁵ from 470 nm onward, while the discrete values are generally higher, especially in the infrared range. Therefore, we believe that n values determined from R and T spectra are accurate within a few percents, while k values are globally inaccurate.

 

Fig. 4 Results obtained for the TiO₂ coating on top of the CR-39 substrate with hard coat using the TRACK method: (a) T and R spectra measured at an AOI of 10° in p polarized light and modelled data; (b) Difference between experimental and modelled data; (c) Dispersion curves calculated from the parametric model as well as the discrete point by point fit; (d) T and R spectra measured at an AOI of 65° in p polarized light; (e) A spectra calculated from T and R data for both TiO₂-coated and uncoated substrates; (f) Final dispersion curves of the TiO₂ coating: n is obtained from the parametric model and k is directly calculated from the A spectrum, using n and the film thickness (392 nm) as input parameters.

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Figure 4(d) presents the R and T spectra measured at an AOI of 65°. As mentioned previously, measuring under near θ_B conditions and using p polarized light yields a strong reduction of the interference effect. The resulting absorption spectrum is displayed in Fig. 4(e), along with the absorption spectrum obtained for the bare CR-39/hard coat substrate (Fig. 3(e)). Again, no residual interference fringes are visible in the A spectrum. The difference between the two A spectra is obviously caused by the absorption occurring in the TiO₂ film. In Figure 4(f), k values are calculated for each point of the A spectrum, taking into consideration the film thickness and the n values obtained from R and T spectra at an AOI of 10°. There are several points that one may note when comparing the results from Figs. 4(c) and 4(f): first of all, the k values calculated from A are in a more realistic range compared to those obtained from R and T separately, and there is no value below 10⁻⁵; secondly, the k values derived from A follow a smoother trend in contrast to the point by point fit of R and T, which is more erratic; thirdly, the evolution of k versus wavelength in Fig. 4(f) follows a trend that is typical of wide band gap semiconductors such as TiO₂, for which absorption gradually falls off below the band gap energy (≈ 3.4 eV, 365 nm [11]). From these results, we can conclude that k values obtained from A at an AOI of 65° are more accurate than those derived from R and T separately at an AOI of 10°.

3.3. Complex film on glass substrate

In this section, we consider the case of a weakly absorbing optical coating, deliberately presenting two non-idealities, namely refractive index inhomogeneity across the film thickness and film thickness non-uniformity along the sample surface. These non-idealities may be encountered in coatings prepared using non-optimized processes for preliminary research purposes. In the present situation, the coating is made of silicon nitride (Si₃N₄) deposited by plasma-enhanced chemical vapor deposition (PECVD, [12]) in a small-scale laboratory-type vacuum system. The substrate is a 50 × 50mm, 1mm thick Gorilla glass (Corning Inc.). The thickness non-uniformity of the coating is related to the small size of the deposition system along with the non-uniform distribution of gas precursors in the chamber during the process. The refractive index inhomogeneity is related to the temperature increase at the substrate level due to plasma exposure during the deposition process, leading to progressive film densification. Figure 5(a) presents the refractive index profile of the film obtained from optical characterization (see below for details). Figure 5(b) is a photograph of the sample showing the thickness non-uniformity.

 

Fig. 5 Non-idealities of a sample Si₃N₄ coating: (a) Refractive index inhomogeneity along the film thickness evaluated from the modeling of the R and T spectra at AOI=10° and p polarization; (b) Film thickness non-uniformity evidenced by the presence of Newton’s rings visible on the sample surface (dimensions are 50 × 50mm).

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In the following, we propose to use the TRACK method in order to determine the thickness and optical constants of the Si₃N₄ coating. In particular, we intend to demonstrate that accurate k values can be obtained by this method even though we use a very simple optical model that doesn’t take into account the various film non-idealities. Figure 6(a) shows the R and T spectra of the sample measured at an AOI of 10°, along with the curves obtained from the optical models. Two optical models were designed to represent the sample. The first one is purposely very simple, consisting of a glass substrate (n and k are fixed), on top of which we place the Si₃N₄ layer whose refractive index is defined by the Cauchy formula (Eq. (10)). The extinction coefficient of the layer is represented by the Urbach equation, which is an exponential tail modelling the absorption below the band edge of Si₃N₄:

 

Fig. 6 Results obtained for the non-ideal Si₃N₄ coating on top of a glass substrate using the TRACK method: (a) T and R spectra measured at an AOI of 10° in p polarized light, and data generated from two models (simple Cauchy/Urbach model and complex general oscillator model including two film non-idealities); (b) Difference between experimental and modelled data for the Cauchy/Urbach and Genosc models; (c) Dispersion curves calculated from the Cauchy/Urbach model and the Genosc model; (d) T and R spectra measured at an AOI of 65° in p polarized light; (e) A spectra calculated from T and R data for both Si₃N₄-coated and uncoated substrates; (f) Final dispersion curves of the Si₃N₄ coating: n is obtained from the Cauchy equation, and k is directly calculated from the A spectrum, using n and the film thickness as input parameters.

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In the next step, we build a second model that is a refined version of the previous one. The dispersion curves of the film are represented by physically-correct Kramers-Kronig-consistent oscillators, film surface roughness is added on top of the stack, the refractive index of the film is graded, and film thickness non-uniformity is also included in the model. The imaginary part of the film dielectric function is modelled by the Tauc-Lorentz oscillator:

Figure 6(c) shows a direct comparison of the dispersion curves obtained with the two models (Cauchy/Urbach versus General Oscillator model including the two non-idealities). One can see that the n values obtained from the Cauchy model slightly deviate from the more accurate ones calculated from the Genosc model (n_Cauchy = 1.93 vs n_Genosc = 1.95 at 550 nm). This 1% error would be acceptable for most optical applications such as antireflective coatings, broad-band reflectors, narrow-band pass filters, etc. Indeed, such error can be corrected during the deposition process by using appropriate monitoring techniques (e.g., turning point monitoring for quarter-wave stacks) without impacting the overall filter performance [16]. Extinction coefficient values from the two models are relatively similar in the UV range where the material is more absorbing, but they strongly differ in the visible and near-infrared ranges.

Figure 6(d) displays the R and T spectra of the Si₃N₄ coating measured at an AOI of 65° in p polarized light, and Fig. 6(e) shows the A spectrum derived from R and T, along with the A spectrum of the bare glass substrate that was characterized beforehand using the same methodology as in section 3.1. Figure 6(f) presents the k dispersion that was calculated from A using the approximate n and thickness values obtained from the simple Cauchy model. One can make several observations regarding the k spectrum. Firstly, k values are less noisy than those obtained from the TiO₂ coating in section 3.2. This is related to the higher thickness of the Si₃N₄ film (≈ 1750nm vs ≈ 400nm for TiO₂), which increases the absorption difference between the coated and uncoated substrates (see Figs. 4(e) and 6(e)), leading to higher sensitivity on the k values. The second observation is that the shape and values of the k spectrum derived from A are very similar to those obtained from the Genosc model (see Figs. 6(c) and 6(e)). Since the Genosc model is expected to give accurate values of n and k, according to lower MSE values, this is a good indication that the k values derived from A are accurate as well. One noticeable difference occurs around 400 nm, where the k values obtained from the Genosc model present an abrupt change of slope (discontinuity at $E_{\mathit{TL}}^g$ in Tauc-Lorentz equation), while the k values calculated from A display a more regular decrease that seems more realistic for this type of amorphous material.

The conclusion of these results is that one can assess the dispersion curves of complex non-ideal films using the TRACK method. Even more interesting is that one doesn’t have to build a complex model including all the film non-idealities to reach this goal. Indeed, we showed that it is sufficient to use a simple Cauchy/Urbach dispersion model to obtain approximate values of the film thickness and refractive index, and then calculate k directly from the absorption spectrum using these approximate values as input parameters.

4. Comparative study

In the last section of this study, we try to evaluate the absolute accuracy of the TRACK method. In order to do so, we compare our results with those obtained by Laser Induced Deflection (LID), which is a highly sensitive technique capable of measuring film absorption values in the 10⁻⁴ range [17, 18]. The films under investigation were 1 μm thick TiO₂ films fabricated in the same deposition system as in section 3.2. Substrates were 1 mm thick Gorilla glass slides for TRACK measurements, plus a 6 mm thick quartz substrate specifically required for LID analysis. All samples were processed during the same deposition run, ensuring consistent optical properties and film thickness among them. LID measurements were performed at the Leibniz Institute of Photonic Technology (IPHT, Jena, Germany) using the same methodology as that described in reference [18].

Since the LID technique measures absorption at a single wavelength, we modified our measurement methodology with the Cary7000 spectrophotometer in order to obtain R and T values at specific wavelengths rather than continuous spectra. In general, single wavelength measurements performed with a spectrophotometer are more accurate than spectroscopic scans for different reasons: firstly, there is no change of filter, grating and slit position in the instrument between the baseline measurement and the sample measurement, which provides better reproducibility; secondly, the elapsed time between the baseline and sample measurements is usually shorter, which virtually eliminates any instrument drift artefact; thirdly, since only one data point is acquired, it is possible to use longer acquisition times (up to a few seconds) and perform multiple repetitions under the same conditions in order to increase both the signal to noise ratio and measurement reproducibility.

In the present case, measurements were performed in p polarized light at 410 and 550 nm that corresponds to the regions of high and low absorption for TiO₂ material, using an angle of incidence of 65°. We used a 4 s acquisition time and each baseline and sample measurement was repeated 10 times in order to reduce instrument error and obtain standard deviation values. In addition, this procedure was repeated 5 times including sample mounting and dismounting between each run in order to average possible film variations or defects on the sample surface. Using this method, the whole measurement at a single wavelength took around 30 min and a total number of 50 points were obtained for both R and T. Absorption values are summarized in Table 1 along with LID results. As expected, the absorption values at 410 nm are an order of magnitude higher than those measured at 550 nm due to the proximity of the TiO₂ band gap at ≈ 365 nm [11]. The values obtained by spectrophotometry for the films deposited on the two Gorilla glass substrates are very close to each other, which confirms the uniformity of the optical properties from one substrate to the other. Absorption values obtained by LID for the film deposited on quartz are slightly different compared to those determined by spectrophotometry, especially at 410 nm.

Table 1. Comparison of the absorption and extinction coefficient values obtained at 410 nm and 550 nm for three TiO₂ films using the TRACK and LID methods.

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The methodology to calculate the extinction coefficient of the films from absorption values at single wavelengths was similar to that described in previous sections. The film thickness and refractive index were obtained from spectral measurements performed at an angle of incidence of 10° in p polarization, as shown in Table 2. The film thickness difference measured between the quartz and the Gorilla glass samples is ≈ 6%, possibly due to the difference of substrate geometries. Refractive index values are nearly identical for all samples (less than 2% variation). The calculated k values obtained from both LID and spectrophotometric measurements are presented in Table 1 and present good consistency for the two techniques (around 1% variation at 550 nm). Surprisingly, k values obtained at 410 nm are less consistent, even though the film absorption is higher and therefore better sensitivity is expected. This could be explained in part by the larger uncertainties observed for both methods at this wavelength. In particular, the response of the Cary 7000 UMS instrument is noisier at 410 nm compared to 550 nm, due to lower detector sensitivity and greater absorption of the polarizing element. Still, we cannot rule out the possibility that systematic error occurred at this wavelength, either on the LID or TRACK side. All together, these results indicate that the level of accuracy obtained with the TRACK method is in the range of 10⁻⁴ at 410 nm and 10⁻⁵ at 550 nm for a 1 micron thick film.

Table 2. Film thickness, refractive index and diffuse transmittance values measured at 410 nm and 550 nm for a bare Gorilla substrate and three TiO₂ films deposited during the same run.

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Furthermore, in order to verify the assumption that scattering is negligible compared to absorption (S ≪ A, thus A=100%−R−T), we measured the diffuse transmittance T_d of the TiO₂ films deposited on Gorilla glass at 410 and 550 nm. These measurements were performed using a Lambda1050 spectrophotometer (Perkin Elmer Inc.) equipped with a 150 mm diameter integrating sphere accessory (Labsphere Inc.). The methodology for diffuse transmittance measurements was based on the ISO13696 standard [19] and the following equation was used:

5. Conclusion

In the present work, we demonstrate the benefits of using the TRACK method for the evaluation of low k values in optical thin films. To conclude, we provide a few recommendations for the reader who would be interested in applying the TRACK method to his samples. Regarding the instrumentation, a spectrophotometer with rotating sample stage and detector is mandatory in order to acquire R and T data at the same AOI without moving the sample. To the best of our knowledge, at the time of this publication, only two manufacturers on the market provide this type of accessory (Agilent and OMT Solutions). The TRACK method also requires high-quality samples, i.e., substrates that are flat, smooth and transparent in the spectral range of interest (A < 1%), and films that are thick enough to show measurable absorption (A > 0.1%) with low haze level (S ≪ A). Regarding the software, the options to analyze the A data in order to extract the k values are presently rather limited. We believe that this feature could be easily integrated in the existing software packages (e.g., WVASE32, OptiLayer, Essential Macleod, etc.). Alternatively, we are willing to provide the Python macro that was used in this study on special request. As a final word, we strongly encourage the optical thin film community to use the TRACK method for the routine analysis of optical coatings and the accurate evaluation of their optical constants.