1. Introduction

The extinction coefficient k (immediately related to the absorption coefficient: α=4πk/λ), along with the refractive index n, are the two main optical parameters that are necessary for optical coating design. In refractive optics, but also in reflective optics based on dielectric materials, transparent materials are required. However, materials are not fully transparent, except vacuum, so that k of any material is nonzero. Effective optics require materials with small k. The k of a film material can be derived from photometric [1,2] as well as ellipsometric measurements [3,4]. Yet, when k is small, the dependence of reflectance and/or transmittance on k often becomes smaller than the uncertainties of reflectance and/or transmittance measurements. The same problem is found when trying to measure a small k through ellipsometry, since small values of k for a rather transparent film material are below the detection limit of ellipsometry. Then one is often in the situation that k of the material is too small to be measured but too large to be ignored. This may be the case in the design of optical coatings, which needs accurate values of n and k of each material in the coating, but k of materials that are referred to as transparent is often unknown or inaccurate.

The k of bulk transparent materials, such as glass or monocrystals, even though it may be very small, such as 10⁻⁶ or smaller, it can be measured by transmittance since the tiny intrinsic absorption can be made measurable by using thick pieces of the material [5,6]. For thin films grown at a temperature far from the bulk growth temperature, such as far from the crystallization temperature for crystals or from the glass transition temperature for glasses, even though the film k may be typically orders of magnitude larger than that of the aforementioned bulk materials, yet a measurable absorption often requires a thick film. Preparing such thick films may not be always possible. Furthermore, for many materials, k (also n) depends on the film thickness (for instance, for SrF₂ [7]). Hence k should be ideally measured for film thicknesses similar to the specific application, which can be often too small to produce significant absorption, and this results in that spectrophotometry and ellipsometry cannot detect such small values of k. Direct measurement of k with a spectrophotometer requires that absorption in the target film is large enough, which is difficult to obtain for relatively thin films and small k. This is often addressed by measuring transmittance as a function of wavelength and by using an optical model of the material, such as, for example, Cauchy, Sellmeier or Tauc-Lorentz. This assumes that the material must follow such model, which is neither guaranteed nor predictable.

There is a set of techniques in optical spectroscopy called modulation spectroscopy that is involved in measuring and interpreting the variations in optical spectra that are produced by changing the measurement conditions in some way [8]. This involves several procedures that are useful to measure, for instance, k of a material. Aspnes [8] classified modulation spectroscopy techniques into external and internal. Hence modulation spectroscopy techniques provide the derivative of some optical response function through a modulation originated either in an external physical stimulus or in the measurement conditions (internal). Various external modulation techniques have proved successful to calculate k, such as thermoreflectance, piezoreflectance, electroreflectance, photoreflectance or rotoreflectance [9]. The interpretation of measurements with these techniques may require a deeper understanding of the effect of the stimulus on the electronic bands of the material, which may not be available with many materials.

Regarding internal modulation techniques, the derivative of the spectra with respect to wavelength has been implemented in spectrophotometers to measure reflectance [10,11], as well as transmittance derivatives (even second derivatives [12]); the latter has been implemented in a spectrophotometer to measure the absorption coefficient of transparent crystals [13]. However, such measurements not only include the derivative of the optical function, but it also involves the derivative of the source background spectra, optics transmittance, and detector sensitivity. The removal of the undesired components from the desired one may not be straightforward. A technique called dual-beam spectroscopy has been implemented to measure absorption differences over two samples of the same material with different lengths [14–16]. The samples are inserted successively between a source and a detector, and the signal difference measured by the detector provides the difference in transmission losses of the two samples. This procedure ignores interferences and hence it is not applicable for the case of films.

Among nonoptical procedures, calorimetry takes advantage of sample heating originated in the absorption of light by the material [13,17–20]. Again, a transparent thin film will absorb little energy so that it can be difficult to detect the resulting small temperature increase and to discriminate between heating generated at the film and at the substrate. For such discrimination, the thermal dynamics of film and substrate must be determined, and extra parameters like the specific heat capacity or the derivative of the refractive index versus temperature [21] of the material must be known. Laser calorimetry can provide intense radiation to produce a larger temperature increase; however, lasers are often available at some particular wavelengths, which is not suitable to cover extended spectra.

All enumerated procedures involve either limited sensitivity, or extra knowledge of other physical properties or models of the material or the instrument. Many of these procedures have been used to measure bulk material properties, but are not applicable where interference in a thin film is not negligible.

A new purely optical procedure conceived for films is presented here, which consists in measuring the differential transmittance in two areas of a sample that differ in the target material film thickness. Differently to the aforementioned dual-beam procedure, measurements are performed at a specific angle of incidence and polarization, so that interferences between light reflected at the outer and at the inner interface of the film are avoided. This enables a direct measurement of k when the relative position of the light beam and the target is made to cross between two target areas of different thicknesses with a specific modulation, and measurement is performed with a lock-in amplifier. Section 2 develops the equations that support the present procedure. Section 3 describes the experimental equipment used to measure the differential transmittance of thin films of AlF₃ deposited at 250°C in the far ultraviolet (FUV) and to obtain k from those measurements. Section 4 discusses the dependence of the uncertainty of k obtained with the present procedure on the uncertainty of the main measurement parameters.

2. Procedure to obtain k from differential transmittance measurements

Let us have a sample consisting in a substrate which is coated on one area with a thin film of thickness x₁ and with no film on area 2, as depicted in Fig. 1. Let a collimated light beam impinge on the sample. The transmittance (ratio of transmitted and incident intensities) in area 2 (only substrate, no film) is given by [22]:

 

Fig. 1. Ray trace for a beam crossing a sample with two areas: a) one uncoated area and one coated area with a film of thickness x₁; b) the two coated areas have thicknesses x_1a and x_1b, with x_1a< x_1b; we name x₁= x_1b-x_1a. The beam is p polarized and impinges on the film at the Brewster angle θ_B of the film material; that is why there is no ray reflected between the incidence medium and the film material.

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It was assumed that the entrance and exit media are the same. The substrate thickness was considered much larger than the beam coherence length, so that the different contributions from reflections back and forth between the two substrate interfaces were added incoherently. Transmittance in area 1 (substrate + film) is given by [22]:

The latter is straightforward from Eq. (1) by just replacing the single transmittance and reflectance terms at the incidence medium/substrate interface (T₀₂ and R₂₀) with the intensity transmittance and reflectance of the incidence medium/film/substrate structure (T₀₁₂ and R₂₁₀).

T₀₁₂, apart from the ratio of entrance and exit refractive indices, involves the intensity ratio that comes from t₀₁₂, the amplitude transmittance ratio of the entrance medium / film / substrate system with two interfaces:

Equation (3) assumes coherent superposition of the different reflected/transmitted terms at the interfaces since the film thickness is typically thinner than the beam coherence length. r_ij and t_ij stand for the well-known amplitude reflectance and transmittance Fresnel coefficients, respectively, for light impinging from i to j media. The following is satisfied in the above assumption that the entrance and exit media are the same:

β_i is the phase term in layer i, with i = 1 (film) or 2 (substrate):

We will assume that the film material is rather transparent, so that k₁<<n₁. This means we can consider that there exists a Brewster angle θ_B, given by n₀tanθ_B= n₁, at which, for p polarized light, it is satisfied that r₀₁= 0, t₀₁= 1, $\frac{{{N_0}}}{{\textrm{cos}{\theta _0}}} = \frac{{{N_1}}}{{\textrm{cos}{\theta _1}}}$, r₁₂= r₀₂, t₁₂= t₀₂, and r₂₁₀= r₂₀. Hence:

Using Eq. (6), Eq. (2) at θ_B can be simplified to:

The normalized differential transmittance between areas with and without film is given by:

When ΔT<<T, which is expected to be satisfied in most cases of interest for the present procedure, we can approximate:

To obtain Eq. (12) it was assumed that n₁> n₀sinθ₀, which, for air or vacuum as the entrance medium, it is immediately satisfied at least for all materials with n₁> 1, such as the dielectric materials in the far UV to the infrared.

At θ_B, n₁= n₀tanθ_B is satisfied, which, applied to Eq. (12), it results in:

In the approximation to obtain Eq. (10), it was assumed that ΔT/T and, equivalently, Re(β₁) are small. The case that the material has small intrinsic absorption but the film thickness is large enough as to make Re(β₁) to not be small is also covered with this approach if one retains the logarithm term in Eq. (9) up to Eq. (15).

Expression (14), and the more general (15), is fundamental to the present research. We have obtained a relation that makes the transmittance difference over two areas proportional to k of the film, which enables calculating the latter. It requires knowing the light wavelength, the film Brewster angle, and the film thickness, but it does not require knowing any other physical parameter or to have an optical model of the material. Since measurements must be performed at the material Brewster angle, the n of the film material must be known at each wavelength at which we need to measure k. The n must be measured previously; it can be made with standard ellipsometry, which would be adequate to measure n of a small-k material, but not adequate to measure such small k.

The present procedure of calculating k in principle does not need to know the n of the substrate. If we wanted to obtain k with parameters away from θ_B and p polarization, such as either measuring differential transmittance or using standard photometric measurements, n of both film and substrate would be required to calculate k from such measurements. Measuring at the Brewster angle with p polarization suppresses the interference across the film, which results in the simplification of the present procedure that does not need to know the n of the substrate. In spite of this, below it will be shown that a way to minimize various uncertainties in k calculation is by using a substrate with n close to the film material.

In fact, the transmittance through the area with no film must also be measured for normalization purposes to apply Eq. (14) or (15). But this is easier to measure, because T is not small, versus the small film absorption in most cases of interest for the present procedure. In practice, we do not exactly need to measure ΔT and T, but the ratio between the difference of transmitted intensities I_t0 (light intensity through the substrate) and I_t1 (intensity through the film and substrate) over the former:

In order to make k proportional to a single measurement, ΔT, or more precisely ΔI_t, can be directly measured with a modulated beam alternately crossing between sample areas with and without film, so that the use of a lock-in amplifier will provide such direct measurement; hence in little time many differential measurements can be averaged out. This is an important detail for the present procedure, which differs from measuring the transmittance of a coated and an uncoated sample with a spectrophotometer, since in the latter, the two measurements are performed with a relatively large time lag, which may result in a drift of the spectrophotometer background, which also limits the number of measurements that can be averaged out.

The light beam can be made to alternately cross between the two sample areas either as a linear or as a rotational movement. The linear movement would be made by alternately moving the sample back and forth in a linear way in a direction parallel to its surface. The rotational movement would be made by rotating the sample through an axis perpendicular to the sample surface and placed at the border between the two areas, for a preferentially circular sample; the light beam would impinge such as in the center of the sample radius. In either case, the linear or circular motion is used as the reference of the lock-in amplifier. In the example given in Section. 3 we used the linear movement.

All the above can be also applied to a sample with two areas coated with the same material but with different thicknesses over the two areas, x_1a and x_1b. In that case, Eqs. (14) and (15) keep valid by just making x₁= x_1b- x_1a.

A similar procedure to calculate a small k₁ can be nominally derived by measuring the differential reflectance of the same sort of sample. Operating in the same way that above, the following relation was obtained:

3. Application of the procedure: k of AlF₃ films

The procedure is exemplified with an AlF₃ film. A LiF substrate was coated on one area with an AlF₃ film and the other area was left uncoated. An AlF₃ film was deposited by evaporation with the substrate heated at 250°C; the sample was left to cool down in vacuum before it was measured. The coating was deposited in a 75-cm diameter, 100-cm height, oil-free high-vacuum deposition chamber pumped with a cryo pump; the chamber is located in an ISO6 clean room. The base pressure was 5 × 10⁻⁶Pa and it increased up to 3 × 10⁻⁵Pa during deposition. The deposition rate was 0.7 nm/s. Film thickness was monitored in situ with a quartz crystal monitor and measured a posteriori with a Sopralab GES5E spectroscopic ellipsometer at 74°, which resulted in a thickness of 37.3 nm. Figure 2 plots ellipsometry measurement and fitting; the optical constants of AlF₃ in the ellipsometry range were modeled with a Lorentz oscillator.

 

Fig. 2. Ellipsometry measurements and fitting of the AlF₃ film deposited on a Si wafer.

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The differential transmittance of the sample in the FUV was measured with a reflectometer that has a grazing-incidence, toroidal-grating monochromator, in which the entrance and exit arms are 146° apart. The monochromator covers the 12.5–200-nm spectral range with two Pt-coated diffraction gratings that operate in the long (250 l/mm) or in the short (950 l/mm) spectral subranges. The light source was a deuterium lamp. To obtain p-polarized radiation, a commercial Rochon polarizer was used to cover the spectral range of wavelengths longer than 130 nm; a home-made Al/MgF₂ coating transmittance polarizer [23] enabled covering the 120-130 nm range. The detector was a channel-electron-multiplier with a CsI-coated photocathode. Since the FUV is absorbed by normal air, the reflectometer used to measure differential transmittance operates under vacuum. The monochromator and polarizer are in high vacuum chambers and the reflectance measurement chamber is in ultra-high vacuum (UHV).

The reflectometer allowed for measuring the differential transmittance over the two sample areas. This was carried out by making the sample to linearly oscillate with a sample manipulator. The beam crossed the border between the two sample areas at a rate of 0.2 Hz; this small frequency was due to the simple oscillation system that we could implement in our vacuum operating FUV reflectometer. The signal was preamplified and then entered into the signal channel of the lock-in amplifier. The typical integration time was approx. 4 min per wavelength. Larger signal frequency could be obtained for instance with sample rotation; for measurements in the visible and adjacent ranges, considerably shorter measurement times can be expected due to the availability of more intense sources, more efficient optics, and more sensitive detectors than in the FUV. The manipulator motion rate was supplied to the lock-in amplifier as the reference signal. For normalization purposes, the intensity transmitted at the noncoated sample area was measured with the standard static procedure.

Differential transmittance should be measured at or close to the AlF₃ film Brewster angle, θ_B. Since precise data of the refractive index of AlF₃ films deposited on a hot substrate were not available, we used data of MgF₂ films [24], a material that is expected to have close FUV optical properties to AlF₃ [25–29]. Figure 3 plots n of 250°C-hot deposited MgF₂ films [24] and the corresponding θ_B to select the angles at which to apply the procedure. Figure 4 provides the measured normalized differential transmittance vs. wavelength and k calculated therefrom using Eq. (14).

 

Fig. 3. The refractive index n of MgF₂ films deposited by evaporation at 250°C [24] and the corresponding Brewster angle θ_B, with n = tanθ_B (the entrance medium is assumed to be vacuum).

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Fig. 4. The measured normalized differential transmittance and k obtained from these measurements using Eq. (14).

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4. Dependence of measured k on the various sources of uncertainty

One requirement of the present procedure is that we need to know n of the film in order to measure the differential transmittance at the Brewster angle. Figures 5 and 6 plot calculations that evaluate the effect of not accurately knowing the refractive index of the film material n and hence of θ_B of the film (both connected through n = tanθ_B, where n₀, the refractive index of the entrance and exit media, is assumed to be 1) on the normalized differential transmittance and hence on k calculated using Eq. (14). The calculation of deviations was performed for various examples: a 40-nm thick MgF₂ film deposited at 250 °C [24] (λ=160 nm, Fig. 5) and a 100-nm thick SiO₂ film deposited at 300°C [30] (λ=553 nm, Fig. 6). Calculations were made assuming two different substrates, MgF₂ [31] and SiO₂ glass [32]. Calculations were performed by arbitrarily changing the film n (and hence its Brewster angle) in a range containing the real value (the remaining parameters were left unchanged), followed by ΔT/T exact calculation, from which k was calculated using Eq. (14). The sample was assumed to be coated with the film on one area and with no film on the other area.

 

Fig. 5. Calculations at 160 nm of the dependence of ΔT/T (black line) and k of the film obtained with Eq. (14) (red line) on the uncertainty in the incidence angle of measurement (with respect to θ_B) (a) and, equivalently, on the uncertainty in n of the film material (b) for a film deposited on two different substrates: MgF₂ [31] (solid lines) and SiO₂ [32] (dashed lines). MgF₂ deposited by evaporation at 250°C was chosen as the film material [24]. Transmittance difference was evaluated over two areas: one with a 40-nm thick film on the substrate and the other one with the bare substrate. The curves for the two substrates cross at null deviation.

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Fig. 6. Calculations at 553 nm of the dependence of ΔT/T (black line) and k of the film obtained with Eq. (14) (red line) on the uncertainty in the incidence angle of measurement (with respect to θ_B) (a) and, equivalently, on the uncertainty in n of the film material (b) for a film deposited on two different substrates: MgF₂ [31] (solid lines) and SiO₂ [32] (dashed lines). SiO₂ deposited by evaporation at 300°C was chosen as the film material [30]. Transmittance difference was evaluated over two areas: one with a 100-nm thick film on the substrate and the other one with the bare substrate. The curves for the two substrates cross at null deviation.

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The two films and materials were selected for film and substrate to have either close or more different refractive indices. In Fig. 5, the dependence of k on the uncertainty in n (or equivalently in θ_B) is considerably smaller for MgF₂ on MgF₂ than for MgF₂ on SiO₂. A smaller uncertainty is also found in Fig. 6 for SiO₂ on SiO₂ compared with SiO₂ on MgF₂. k deviation due to the uncertainty in n seems to slow down for the substrate with n closer to the film material, which suggests the use of a substrate material with n as close as possible to the film material. The dependence on the refractive index difference between substrate and film is further discussed below. In the example presented in Section 3, the substrate material was LiF for two reasons: it has a small FUV refractive index similar to MgF₂ (and hence to AlF₃) and it extends the substrate transparency deeper in the FUV.

Let us look in more detail at the uncertainty dependence on n difference between film and substrate. Figure 7 evaluates the dependence of k uncertainty for a 40-nm thick MgF₂ film on a transparent substrate as a function of the latter’s n; k is calculated with Eq. (14) at λ=160 nm. The source of uncertainty is a given inaccuracy in the measurement angle (expressed as Δθ_B), which is set at ±1°, either by not precisely knowing the film n (and hence, not knowing the exact θ_B) or by an uncertainty in angle positioning. The plot shows that the relative uncertainty of the film k (k_f) turns zero close to but not exactly at the common refractive index of film (n_f) and substrate (n_s), and it increases away from such common n. This non-perfect coincidence is attributed to the k difference between film and substrate. To confirm this, Fig. 7 also displays the same calculation except that k of the film was reduced to one tenth of it, i.e., k_f/10, so that it is much closer to substrate k = 0. Indeed, a close-to-null k uncertainty was obtained for film and substrate common n. However, Δθ_B had to be reduced to ±0.1° in order to scale down k uncertainty. The latter is due to that k absolute uncertainty vs. angle uncertainty remains of the same order after reducing k of the film, so that the relative k uncertainty proportionally increases in the latter case. Summarizing, a substrate that matches the film refractive index is generally recommended to apply the present procedure, which turns increasingly necessary for a decreasing film k. Furthermore, the need to precisely operate close to the Brewster angle also remains increasingly necessary with decreasing film k.

 

Fig. 7. Dependence of the relative uncertainty of calculated k for a given uncertainty in the angle of measurement with respect to θ_B, expressed as Δθ_B, as a function of n of a transparent substrate coated with a 40-nm thick MgF₂ film. λ=160 nm. (a): Δθ_B =±1°. (b): Δθ_B= ±0.1°. k of MgF₂ film (k_f) is used in (a), but it is reduced to its tenth, k_f/10, in (b). The green line highlights the case of n_s= n_f.

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We have seen that the relative k uncertainty is larger for a small k. Let us look at its dependence on two sources of uncertainty: the measurement angle and the presence of a small proportion of the cross polarization. Figure 8 evaluates the relative uncertainty of k versus k for a + 0.1° inaccuracy in the measurement angle with respect to θ_B. The example considers a thin film of SiO₂ deposited at 300 °C [30] on an SiO₂ glass substrate [32] at λ=553 nm wavelength. Two film thicknesses were considered: 40 and 100 nm. The plot displays that the uncertainty is minimal for a range of k of ∼0.003. The relative uncertainty grows at smaller k because absorption in the film decreases, which results in that any source of inaccuracy, in this case the measurement angle, affects transmittance more than such small absorption. The growth of the uncertainty at larger k is due to the assumption in Eq. (10), which starts deviating when ΔT/T grows; if we recalculate k with Eq. (15), hence using –ln(1-ΔT/T) instead of ΔT/T (dashed lines), the curves dramatically flatten for k > 0.003. The larger film thickness results in larger k uncertainty; on the other hand, a larger film thickness makes it easier to measure transmittance difference for small k, so that the film thickness must be selected as a tradeoff between measurable absorption and k uncertainty.

 

Fig. 8. Dependence of the relative uncertainty of calculated k as a function of k for a given uncertainty in the measurement angle with respect to θ_B, expressed as Δθ_B, (a) and for the presence of a certain proportion of s polarization (b) for a thin film of SiO₂ deposited at 300°C on an SiO₂ glass substrate at 553 nm wavelength. Two film thicknesses were considered: 40 and 100 nm. The specific Eq. (14) or (15) used in the calculation is indicated.

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Another source of uncertainty is the presence of some proportion of the cross polarization instead of pure p polarized light. Figure 8 also considers the same film and substrate that in the previous example, where we restore the correct θ_B and we replace the angle inaccuracy with the presence of 1/200 proportion of light with s polarization, so that the degree of polarization is given by (I_p-I_s)/(I_p+ I_s) = 0.99. Again, we obtained that the lowest uncertainty is obtained for a range of k of ∼0.003. The growth of Δk/k for small k is attributed again to the small absorption in the film compared with the modified transmittance due to the presence of some light with s polarization. The growth of Δk/k for larger k is also attributed to the approximation of Eq. (14). If k is recalculated with Eq. (15) (dashed lines), the curves dramatically flatten once more for k > 0.003.

The usability of the presented technique requires the availability of uniform substrates with small transmittance variation through their surface, so that such transmittance variations are not wrongly assigned to k of the film. The requirement of substrate uniformity to obtain a certain k accuracy can be easily derived from Eq. (14). Unfortunately, the providers of optics blanks do not characterize them in terms of transmittance uniformity. Regarding refractive index uniformity, the ISO 10110 standard categorizes class 5 optics as those with maximum n variation over the optics within ±0.5 × 10⁻⁶, from which transmittance variation could be estimated. The number of bubbles and inclusions in glass is another feature that influences substrate uniformity. Hence high-quality optics with fewest defects are demanded for high-power laser applications, and this could be a hint for suitable substrates. Regarding this, there are standard glass qualities with reduced number of bubbles/inclusions and with reduced defect diameter; hence there is a commercial grade reaching as low as 2 maximum inclusion/bubbles per 100 cm³ volume of the material, with a maximum diameter that can be limited to 0.06 mm [33]; such low number would enable reaching a level of zero bubbles and inclusions in a typical substrate volume. Nonuniformity can also come from defects on the surface, either from the polishing or the cleaning processes or they can be generated in a dirty storage. All this must be minimized for the present procedure to apply. Nevertheless, it must be said that such uniformity requirements are similarly needed for k evaluation using the common transmittance measurements with a standard spectrophotometer, since substrate defects would also affect such transmittance measurements towards evaluating the effect of a small-absorption film.

The above examples display the dependence of k on specific sources of uncertainty. Let us evaluate the uncertainty level of k that could be reached with an experimental setup. To perform general error propagation would be cumbersome. Instead, we have analyzed the uncertainties for a specific example, which may give a hint of the range of acceptable parameter uncertainties. We focused on the example presented in Fig. 6: a 100-nm thick SiO₂ film deposited on one area of an SiO₂ glass substrate and no coating on the other area, with optical constants from [30] and [32], onto which p polarized light with λ=553 nm wavelength impinges over the sample at SiO₂’s-film Brewster angle (56.3797°). We exactly calculated transmittance and let the relevant parameters vary around the central one until the obtained ΔT/T resulted in an erroneous contribution to k of 10⁻⁵ as per Eq. (14). ΔT stands for the differential transmittance for the modified parameter (film n, angle, degree of polarization, wavelength) or the absolute difference between the transmittance calculated with the correct parameters and the one calculated with the modified parameter (substrate n nonuniformity, Scratch/Dig). The results are plotted in Table 1.

Table 1. Uncertainty range of various parameters that will lead to ΔT/T that contributes to Δk in 10⁻⁵ for the case of a 100-nm thick SiO₂ film deposited on an SiO₂ glass substrate, with optical constants from [28] and [30], onto which p polarized light with λ=553 nm impinges over the sample at SiO₂’s-film Brewster angle (56.3797°)

View Table

It can be seen that contributions to Δk of ∼10⁻⁵ are compatible with feasible uncertainty limits of substrate defects, film n, angle precision and/or angle spread, polarization, and wavelength. Regarding uncertainty in the degree of polarization, at 0.998 it would require the use of polarizers with high extinction ratio. The angle error or spread requirement (the latter gets somewhat relaxed because the uncertainty given by higher angles tend to be compensated with the one given by lower angles) to within ±0.1° is also demanding and, to implement it in a spectrophotometer, it may need the use of extra aperture stops to reduce angle spread. Even though some of the parameter uncertainty ranges in Table 1 are demanding, they can be considered feasible. In addition to this example, other film thickness combinations on the two sample areas which maintained the 100-nm thickness difference between them were also calculated; we obtained similar results to those plotted in Table 1, with some variations due to interferences, since the latter are not fully avoided when the setup is not exactly operating at the film Brewster angle with full p polarization.

In order to obtain the combined uncertainty, various of the above sources contribute in a negligible proportion: 1) substrate n, since it was shown above that substrates with n uncertainty in the 0.5 × 10⁻⁶ are available; 2) film n, since, for instance, ellipsometry can measure it to within approx. 0.001. If we add the other contributions to uncertainty (scratch/dig, measurement angle, degree of polarization, and wavelength) in quadrature, we get a combined uncertainty in k of 2 × 10⁻⁵. This is a remarkably low number, which favorably compares the present procedure with ellipsometry, which would have difficulties to determine k of a film below 0.001 or even below 0.01.

From Eq. (14) we can see that the dependence of k measurement on substrate nonuniformity is inversely proportional to the film thickness, so that the uniformity requirement on the substrate can be relaxed by using a thicker film, although the difficulties of using thick films were noted in Section 1 and can be seen in Fig. 8. Conversely, substrate uniformity requirement proportionally increases with wavelength. In fact, the requirement is proportional to the ratio of wavelength to thickness; in this respect, optical coatings have typical thicknesses that are proportional to wavelength (such as for quarterwave layers), which turns the substrate uniformity requirement relatively independent of wavelength.

Conclusions

A new procedure to measure k of a film with small absorption has been presented. It consists in measuring the normalized differential transmittance of a sample between two areas coated with two different thicknesses of the target film material; in its simplest form, one sample area can remain uncoated. When measured at the film-material Brewster angle with p polarization, the normalized differential transmittance is proportional to k of the film material. The proportionality factor only depends on wavelength, film Brewster angle (which depends on the refractive index of the thin film), film thickness, and numerical constants. The differential transmittance can be measured with a single measurement using a modulation that makes the beam to oscillate with respect to the sample and a lock-in amplifier; such measurement must be normalized by also measuring the standard transmittance through one of the sample areas. The procedure could be implemented in a spectrophotometer that enables a relative oscillation between the light beam and the sample; the oscillation can be carried out both linearly or rotationally. The procedure is exemplified by measuring k in the FUV of an AlF₃ film deposited by evaporation on a 250°C-hot LiF substrate.

The application of the new procedure requires substrates with good transmittance uniformity. It also requires knowing n of the film beforehand, which can be measured with standard techniques, such as ellipsometry, and it can be performed on the same sample. In contrast, instruments like ellipsometers and spectrophotometers are hardly sensitive to a small k in the film, so they are inadequate to measure a small k. The present procedure requires to be complemented with some characterization technique to measure n of the film, such as ellipsometry, and, if required, also to measure both n and k at other spectral ranges where k does not satisfy to be small enough as to apply the present procedure.

Various sources of uncertainty in the calculated k have been exemplified and discussed: n, Brewster angle, wavelength, and the degree of polarization of the incident beam. The low-k uncertainty limit with the present procedure was evaluated for one example, and the conditions were found to reach a k uncertainty of 2 × 10⁻⁵. The deviation of k obtained with the present procedure due to the uncertainty in n of the film can be reduced by choosing a substrate with similar n to the one of the film.