Sophisticated Attenuated Total Reflection Correction Within Seconds for Unpolarized Incident Light at 45°

Abstract

The most common mid-infrared (MIR) attenuated total reflection (ATR) accessory has a nominal angle of incidence of 45° and does not have a polarizer. A spectrum recorded with such an accessory does not hold enough information for the sophisticated ATR correction of MIR spectra with strong peaks, which are often strongly affected by refractive index changes due to anomalous dispersion. Here we show that a 45° ATR spectrum recorded without a polarizer and the polarization angle for the same ATR Fourier transform infrared spectroscopy system provide enough information to determine the ATR s-polarized spectrum. Further analysis with an improved non-iterative Kramers–Kronig analysis immediately yields the complex refractive index function. The analysis is about two orders of magnitude faster than iterative formalism and runs within seconds on a typical office PC. The effectiveness of our advanced ATR correction formalism is showcased through its application to water, employing diamond, ZnSe, and Ge ATR crystals, along with two distinct ATR accessories. Additionally, the formalism is applied to octadecane spectra. Potential sources of errors such as incidence angle spread, dispersion of the polarization angle, and the influence of reflection at the air/ATR crystal interface are investigated by simulations.

Keywords

Internal reflection attenuated total reflection infrared ATR-IR ATR correction Malus’ law

Introduction

Attenuated total reflection (ATR) accessories are used with many Fourier transform infrared spectroscopy (FT-IR) systems because they require less effort to record a reproducible spectrum than recording a transmission spectrum with KBr pellet preparation. However, ATR introduces changes in peak positions, intensities, and shapes. So-called ATR correction is the two-step process of (1) converting an ATR spectrum to the complex refractive index function, $\hat{n} = n + i k$ and (2) converting from $\hat{n}$ to conventional transmission or absorbance spectra. Step 2 is well established, so this paper is concerned with Step 1.^1–5

The differences between ATR and transmission spectra (or reflectance absorbance and transmittance absorbance spectra) go beyond the need for intensity correction, particularly for lower wavenumbers.^6–11 The effective thickness, which is the thickness that produces absorbance values equivalent to those obtained from transmission experiments, increases inversely proportional to the wavenumber.¹² Electromagnetic theory correspondingly demonstrates that ATR absorbance depends on the index of refraction and the penetration depth, which is the thickness at which the evanescent wave is reduced to 1/e of its original intensity. The penetration depth is itself influenced by the index of refraction. Consequently, the absorbance band maxima are shifted toward the maxima of the index of refraction, and the band shapes can undergo significant changes.^7–11 One way to mitigate this issue is to use high-index ATR crystal materials such as Si and Ge and to employ high angles of incidence.^2,13 However, this approach reduces the penetration depths and overall intensities, thereby decreasing the sensitivity of the method. In addition, although effects such as peak shifts and band shape changes are reduced, they remain present to some extent.² On the other hand, this approach helps prevent the use of the ATR method outside of its specifications, ensuring that ATR spectra remain internal reflectance spectra. What do these specifications entail? The primary requirement is that the angle of incidence (AOI) must exceed the critical angle, α _c :²

α_{c} (\tilde{ν}) = \arcsin \frac{n_{2} (\tilde{ν})}{n_{1} (\tilde{ν})}

(1)

where

n_{2} (\tilde{ν})

is the refractive index function of the sample and

n_{1} (\tilde{ν})

that of the ATR crystal. Since

n_{2} (\tilde{ν})

and

n_{1} (\tilde{ν})

depend on the wavenumber, the critical angle is also a function of the wavenumber.

If, for example, the crystal used in ATR spectroscopy is diamond or zinc selenide (ZnSe), both of which have a refractive index of approximately 2.4, and the AOI α _i amounts to 45°, then the refractive index of the sample being analyzed must be <1.7. This means that certain materials, such as charcoal, cannot be analyzed using this combination of crystal and AOI because the resulting spectra are influenced mainly by the refractive index rather than the absorption index.¹⁴ The reason is that according to Fresnel's equations below the critical angle, a fraction of the incident light is transmitted into the sample and the reflectance is therefore generally <1. Accordingly, –log R would be positive, although nothing is absorbed. Additionally, if a fraction of the light that is transmitted into the sample is absorbed, because the absorption index is different from zero, it usually scarcely alters –log R, at least for not-too-strong oscillators (this builds the foundation of infrared [IR] refraction spectroscopy, see Mayerhöfer et al.¹⁴). Even when the refractive index of a sample is below the critical angle in the transparency region between mid-IR (MIR) and visible (Vis) light, there can still be problems caused by dispersion. Absorption causes a change in the refractive index of the sample. Therefore, overlapping weak bands or a medium-strong band such as a C=O vibration can result in a spectral range where the critical angle is larger than the incidence angle chosen for an ATR accessory. In this case, some of the radiation in that range is transmitted into the sample without being absorbed. Unfortunately, it is only possible to measure and quantify the deviation from total reflection, but not whether this deviation is solely due to absorption. This means that absorbance values in ranges with a higher critical angle than the actual AOI can seem to be much higher than they actually are.^2,11 This is not an issue when spectra are evaluated based on electromagnetic theory, as reflection losses due to transmission are automatically taken properly into account.² The only requirement is that the polarization state of the incident light must be known for correction. This is the real challenge at present since most accessories do not have a polarizer.

Theoretical Considerations

Recently, Azam et al.¹⁵ developed a technique for determining the polarization angle by comparing an experimental spectrum of water to a calculated one. By utilizing known optical constants, Fresnel's equations, and Malus’ law, the polarization angle can be computed. According to Malus’ law, the reflectance in dependence of the polarization angle φ is given by:

R = R_{s} co s^{2} φ + R_{p} si n^{2} φ

(2)

where R_s and R_p are the reflectances with s- and p-polarized incident light. Before we discuss how to calculate R_s and R_p, we want to point out an important implication of Malus’ law, which is that the form of Eq. 2 renders it impossible to differentiate between distributions with regard to the polarization angle and a single angle, e.g., naturally polarized light consists of waves for which every polarization angle is present, each with equal probability. Since the polarization angle can be represented as a non-polar vector (which reflects itself in the fact that the cosine and the sine of the polarization angle are squared), the average of such vectors can always be represented as a single vector. In the case of naturally polarized light, the uniform distribution of polarization angles leads to the fact that the reflectance for naturally polarized light is R_n= 0.5 R_s+ R_p . Therefore, for naturally polarized light, the polarization angle φ = 45°. In the same way, as mirrors and other optical elements alter the polarization distribution, this altered distribution will cause a reflectance that can still be described according to Malus’ law with a corresponding single polarization angle φ. Accordingly, we will speak in the following of an individual polarization angle, keeping, however, in mind that it is actually a polarization distribution of unknown nature that we are referring to. Note that since mirrors and other optical elements are usually made of materials with low dispersion in the wavenumber range a spectrometer is operating,¹⁶ we can assume that this polarization angle is fairly constant in this wavenumber range. Yet, this simplifying assumption is not a requirement for the following, and the technique of Azam et al. can also be used to fit a wavenumber-dependent polarization angle

φ (\tilde{ν})

, for which the following relations will stay valid.

According to Fresnel's equations, the reflectances R_s and R_p can be calculated according to²

\begin{matrix} R_{s} = \frac{n_{1} \cos α_{i} - \sqrt{n_{2}^{2} - n_{1}^{2} si n^{2} α_{i}}}{n_{1} \cos α_{i} + \sqrt{n_{2}^{2} - n_{1}^{2} si n^{2} α_{i}}} \\ R_{p} = \frac{n_{1} \sqrt{1 - (n_{1}^{2} / n_{2}^{2}) si n^{2} α_{i}} - n_{2} \cos α_{i}}{n_{1} \sqrt{1 - (n_{1}^{2} / n_{2}^{2}) si n^{2} α_{i}} + n_{2} \cos α_{i}} \\ R_{i} = {| r_{i} |}^{2} \end{matrix}

(3)

where R_s and R_p are the complex reflection coefficients.

It can be shown based on Eq. 3 that for an AOI α _i = 45°, a special relationship between R_s and R_p is established according to which $R_{p} (45 \circ) = R_{s}^{2}$ (45°) (in general, peak positions do not agree for R_s and R_p!). We can use this relationship to cast Eq. 2 in the following form:

R (45 \circ) = R_{s} (45 \circ) co s^{2} φ + R_{s} (45 \circ)^{2} si n^{2} φ

(4)

Accordingly, we find after some arithmetic that:

R_{s} (45 \circ) = \frac{1}{2} (- co t^{2} φ + cs c^{2} φ \sqrt{co s^{4} φ + 4 R (45 \circ) si n^{2} φ})

(5)

This means that if we know the polarization angle, we can calculate the s-polarized spectrum from the experimental spectrum R (45°) by employing Eq. 5. Once we know R_s (45°), we can apply a Kramers–Kronig analysis as derived by Bertie and Lan¹⁷ for s-polarized incident light,

ϕ (\tilde{ν}) = - \frac{1}{π} ℘ \int_{0}^{\infty} \frac{{\tilde{ν}}^{'} \ln (R (45^{\circ}, {\tilde{ν}}^{'}) / R_{0} (45^{\circ}, \tilde{ν}))}{{\tilde{ν}}^{' 2} - {\tilde{ν}}^{2}} d {\tilde{ν}}^{'}

(6)

where

ϕ (\tilde{ν})

is the phase,

℘

stands for the Cauchy principal value, and R₀ (45°, ṽ) is the reference spectrum without a sample. In practice, this integral is numerically calculated, and the phase has to be corrected by two additional terms to account for singularities¹⁷:

\begin{aligned} ϕ (\tilde{ν}) = & - d \frac{2}{π} [\sum_{j}^{} \frac{{\tilde{ν}}_{j} \ln [R (45 \circ, {\tilde{ν}}_{j}) / R_{0} (45 \circ, \tilde{ν})]}{{\tilde{ν}}_{j}^{2} - {\tilde{ν}}^{2}}] \\ + π - 2 \arctan [\sqrt{n_{1} {(\tilde{ν})}^{2} si n^{2} 45 \circ - n_{\infty}^{2}} / (n_{1} (\tilde{ν}) \cos 45 \circ)] \\ - \frac{800^{2} c m^{- 2}}{{\tilde{ν}}^{2}} d \frac{2}{π} [\sum_{j}^{} \frac{{\tilde{ν}}_{j} \ln [R (45 \circ, {\tilde{ν}}_{j}) / R_{0} (45 \circ, {\tilde{ν}}_{\infty})]}{{\tilde{ν}}_{j}^{2} - {\tilde{ν}}_{\infty}^{2}}] \end{aligned}

(7)

n_∞ is the refractive index of the sample at a wavenumber well above the band with the highest wavenumber in the MIR, e.g.,

{\tilde{ν}}_{φ} = {\tilde{ν}}_{\infty} = 4000 {cm}^{- 1}

(a good approximation for non-colored materials is n_D²⁰). It is important to state that the second term of the correction factors only holds for s-polarized light. For light that is neither s- nor p-polarized, this correction factor would take on an extremely complicated form that has not been derived until now (to our best knowledge). On the other hand, this correction factor is vital and without it, the errors in the phase would be so large that the complex refractive index function could not be determined. This seems to be the reason a direct correction of ATR spectra recorded with unpolarized light by a Kramers–Kronig analysis as detailed above has not been possible. Furthermore, the correction factor ends the need for Kramers–Kronig relational iteration, which was apparently a workaround for not having the correction factor.

Note that we have added, compared to the formula provided by Bertie and Lan, a semi-empiric correction factor for the last term, 800² cm^–2/ ${\tilde{ν}}^{2}$ , which derives from dispersion theory (hence the factor ${\tilde{ν}}^{- 2}$ ; it reminds us of a term of the damped harmonic oscillator model away from a resonance with the resonance wavenumber set to zero, cf. Eq. S1 in the Supplemental Material). The correction factor S²= 800² cm^–2 could be seen as an oscillator strength that actually varies only slightly from material to material, increasing slightly with increasing strength of the oscillators of the material under consideration. In addition, the factor decreases with both, the AOI and the refractive index of the ATR crystal. For germanium, e.g., the optimal value for the correction factor is S = 0, regardless of the AOI. This means that the last term in Eq. 7) can be omitted (please keep in mind that the correction factors due to the singularities can anyway, on principle, never be exact as long as the correct optical constants functions are unknown¹⁸). Once the phase is calculated, the complex refractive index function can be computed by (this formula is again only valid for s-polarized light!):

{\hat{n}}_{2} (\tilde{ν}) = n_{1} (\tilde{ν}) \sqrt{si n^{2} 45 \circ + co s^{2} 45 \circ {(\frac{1 + \sqrt{R_{s} (45 \circ, \tilde{ν})} \exp [i ϕ (\tilde{ν})]}{1 - \sqrt{R_{s} (45 \circ, \tilde{ν})} \exp [i ϕ (\tilde{ν})]})}^{2}}

(8)

Note that we explicitly take into account the wavenumber dependence of the refractive index of the incidence medium n₁(

\tilde{ν}

) by using the refractive index function determined by Tatian.¹⁹

Depending on the spectral resolution, the calculation requires about 2–10 s on a typical office PC, which means that it is much faster, up to two orders of magnitude, than an iterative correction procedure used, e.g., in Mayerhöfer et al.²⁰ In contrast to the latter procedure, it usually also works when the condition of total reflection is violated around bands, i.e., when the refractive index of the sample becomes so large that the critical angle is larger than the AOI. For ZnSe and Diamond ATR crystals and an AOI of 45° this happens for refractive indices larger than about 1.7, which is quite often reached, e.g., for C=O or amide I bands. In this case, the so-called “advanced ATR correction” formalisms fail, which are based on a much less sophisticated correction scheme than the one introduced in this work, but do not require knowledge of the polarization angle.

Experimental

Materials and Methods

Two ATR accessories, designated ATR-1 and ATR-2, were used in this work, without temperature controllers. All ATR crystals were single-bounce with 45° faces for the incident and reflected MIR light.

ATR-1: A fully automated FDM ExpertATR (Fiveash Data Management, Inc., USA). The range of effective angle of incidence (AOI) is from 38.8° to 58.8°. Both the AOI and optional polarization are automated with stepper motors, which provides for excellent electromechanical control. The control software changes the AOI and polarization and communicates with the FT-IR workstation software to perform both the background collection and spectrum collection with minimal effort. The entire system can produce multiangle and/or multipolarization ATR data sets of virtually any size. Each spectrum in the resulting data set has a corresponding background spectrum. The polarizer is removable. Polarized spectra have the polarizer in-line and unpolarized spectra have the polarizer removed.

ATR-1-Plate-1: Zinc selenide (ZnSe) (20 mm diameter sampling area).

ATR-1-Plate-2: Germanium (20 mm diameter sampling area).

ATR-2: A PIKE GladiATR (Pike Technologies).

ATR-2-Plate-1: Diamond (3 mm × 2 mm sampling area).

ATR-2-Plate-2: Germanium (3 mm diameter sampling area).

ATR-2 has a nominal AOI of 45° and does not have a polarizer.

Note that ATR-1 and ATR-2 have different focusing optics to support the much different areas of their ATR crystals. The AOI spread is comparably small and usually clearly below a standard deviation of 1° (Table S1, Supplemental Material). Accordingly, spectra can usually be successfully corrected as if there is no AOI spread at all. This even holds for standard deviations as high as 5° (Figures S22–S25, Supplemental Material). Note that such high standard deviations should be avoided for ZnSe and diamond ATR crystals and 45° AOI, as in this case for many materials a part of the light falls onto the sample under a subcritical angle so that the ATR condition is noticeably violated. This manifests itself in a reflectance in non-absorbing regions < 1 or, equivalently, in baselines > 0 in reflectance absorbance spectra.

Two similar FT-IR systems, designated FT-IR-1 and FT-IR-2, were used. FT-IR-1: A Thermo (Thermo Fisher Scientific) Nexus 670 hosting ATR-1. FT-IR-2: A Thermo Nexus 870 hosting ATR-2. The instruments sit side by side.

Chemicals

Consumer-grade distilled water was purchased from a retailer in Monona, Wisconsin, USA. The Octadecane (99%+) was purchased from Sigma-Aldrich.

Calibrations

There are two calibration methods used in this paper to obtain the polarization angle, a measurement of the net polarization of the ATR, and an FT-IR system in use. It is desirable to have a colorless sample with peaks in the MIR and a known $n_{D}^{20}$ (refractive index at 20 °C, taken at the sodium D line).

Calibration Method I

R(45°) of water was recorded on an ATR accessory without a polarizer.

The polarization angle is then determined with a fit based on Eq. 2, Fresnel's equations, and the water optical constants according to the method of Azam et al.¹⁵

Calibration Method II

An ATR accessory with a polarizer does not require the water optical constants:

R(45°), R_s(45°), and R_p(45°) of water, or any suitable liquid, is recorded.

The polarization angle is then determined with a fit based on Eq. 2.

Analysis

R(45°) is recorded on an ATR accessory without polarization.

R_s(45°) is calculated with Eq. 5 and the polarization angle from the calibration procedure.

The phase spectrum $ϕ (\tilde{ν})$ is calculated with Eq. 7.

With $ϕ (\tilde{ν})$ and Eq. 8, one obtains the complex refractive index function $\hat{n} = n + i k$ , thereby completing Step 1 in the ATR correction process.

Results and Discussion

To check our ATR correction formalism, we recorded s-, p-, and un-polarized ATR spectra of water on FT-IR-1-ATR-1-Plate-1. The experimental spectra are shown in Figure 1. We note that the relation derived from Fresnel's equations that $R_{p} (45 \circ) = R_{s}^{2} (45 \circ)$ is fulfilled with experimental accuracy. The reflectance measured without a polarizer is, as can be expected from theory, usually between R_s and R_p, with the exception of spectral regions where the reflectance is close to unity, and inaccuracies are comparably large.

Usually the optical constant functions of water in combination with Fresnel's equations and Malus's law would be used to determine the polarization angle of the ATR accessory–instrument combination (Eq. 3) in combination with Eq. 4 (Calibration Method I),¹⁵ we used the possibility provided by our accessory–instrument combination, which allows the employment of a polarizer, and determined the polarization angle directly for the sample by Eq. 4 (Calibration Method II). To that end, we recorded the s- and p-polarized spectra and computed the polarization angle by minimizing the residual sum of squared differences between the spectra recorded without a polarizer. By this procedure, we found that the polarization angle equals approximately 39.3°. As mentioned above, the (beam) spread is ±0.2°. Together these parameters are useful for establishing the suitability of the ATR accessory and optics for recording ATR correctable spectra of a given sample (see the table in the Supplemental Material). This will be discussed in more detail in a future paper. A comparison between the experimental and the fitted spectrum is provided in Figure 2.

Except for spectra below about 1000 cm^–1, the experimental and the fitted spectrum are virtually congruent. A further check of consistency is the comparison of the s-polarized experimental spectrum and the one that can be computed from Eq. 5 by employing the determined polarization angle. Again, both spectra are virtually congruent. The remaining deviations are small enough to prove the value of Eq. 5, and, thereby, our fast and sophisticated ATR correction scheme. In this scheme, the last step is to apply the Kramers–Kronig analysis according to Eqs. 7 and 8.

Since there were only very small deviations between the squared s-polarized ATR spectrum and the p-polarized spectrum (Figure 1) it is clear that the combination of the sample, water, and the ATR-FT-IR optical system is sufficient to record R, R_s, and R_p spectra that are highly consistent with respect to the Fresnel equations and Malus’ law and this formalism can reliably use these data to calculate the complex refractive index function for the sample.

We can also expect that the differences between the complex refractive index functions determined from the s-polarized and the p-polarized ATR spectra are very small, since we have established that $R_{s}^{2} = R_{p}$ and that Fresnel's equations (Eq. 3) are at play.

In addition, since the differences between the experimental s-polarized ATR spectrum and the s-polarized ATR spectrum calculated from the spectrum gained without a polarizer based on Eq. 5 are also minor ones, the same can be expected for the deviations for the complex refractive index functions determined from these spectra. These expectations are fully confirmed by the results as illustrated in Figure 3, which shows the complex refractive index functions calculated from unpolarized spectra, as described in this paper, but also from s- and p-polarized spectra (how to obtain the complex refractive index function from p-polarized spectra shall be described in a future paper). This proves that our method of correction is not only fast, but also on the same level in this respect as other sophisticated methods such as iterative formalisms and dispersion analysis.^2,17,21

While internally all results seem to be consistent, a comparison of the determined optical constants functions n( $\tilde{ν}$ ) – n_∞ and k( $\tilde{ν}$ ) seem to reveal a larger discrepancy of about 10% (we have chosen n_∞= 1.315, so that the refractive index curves agree around 4000 cm^–1 with the literature values).²² Since our spectra were taken during summer time without a temperature controller such deviations are not unexpected, since from 1 °C to 50 °C the optical constants can decrease by more than 20%,²³ which is remarkable since the density only changes by about 12%. Therefore, water might not be the ideal compound for calibration (overall, the IR spectrum of water is still not fully understood²⁴). Unfortunately, the optical constant functions of water are not available for different temperatures and most ATR accessories do not allow us to measure and keep constant temperatures. This means that to determine the polarization angle $φ$ by the calibration procedure suggested by Azam et al.,¹⁵ the standard residual sum of squares should not be used since its use implies the absence of systematic errors. Otherwise, changes in the intensities in the ATR spectra of water due to temperature changes will lead to different polarization angles resulting from the calibration procedure. To some degree, a loss function based on a two-trace two-dimensional correlation can provide a remedy.^20,25,26 A temperature controller is planned for future work.

Note that the correction scheme presented in this work can only work if the sample obeys Eq. 1 in the high-wavenumber limit, i.e., if the index of refraction n₂( $\tilde{ν}$ _∞) is smaller than n₁( $\tilde{ν}$ _∞)sin α _c .¹⁷ In addition, the same should hold for the low-wavenumber limit $\tilde{ν}$ _e : n₁( $\tilde{ν}$ _e )sin α _c > n₂( $\tilde{ν}$ _e ). This may exclude a variety of, in particular inorganic, materials if the ATR crystal consists of ZnSe or diamond because then their upper limit of the refractive index at the edges of the spectral range would be 1.7 (for a Ge crystal and 45° AOI the limit would be 2.8). In fact, as can be seen in Figure 3, water is a good test case since it seems to approach a refractive index of 1.7 around 600 cm^–1. More importantly, there is an extremely strong absorption at lower wavenumbers,²² which explains the stronger deviations below 1000 cm^–1 between our results and the literature values, in particular concerning the refractive index function. These are caused by the dispersion of the extremely strong lower wavenumber absorption. In this sense, water is an as pathological example as possible, not only for a Kramers–Kronig relational analysis, because also dispersion analysis would not be able to compensate for resulting problems. Under these conditions, the results resemble, at least in the case of the absorption index function, literature values comparably good above 1000 cm^–1. We also tested the algorithm with different optical constant functions generated by the classical damped harmonic oscillator model and found that our version of Bertie's and Lan's Kramers–Kronig relational formula is indeed very resistant to numerical instabilities and provides results very comparable to those of dispersion analysis. Accordingly, for most of the organic liquids and solids, our formalism should work even when ZnSe and diamond ATR crystals are used. As further examples and proofs, results are shown for octadecane (Figures S1–S3, Supplemental Material; octadecane is an unproblematic example) as well as for water on a different ATR accessory (FT-IR-2-ATR-2-Plate-1, Figures S4–S6, Supplemental Material) and a different crystal (FT-IR-1-ATR-1-Plate-2, Figures S7–S9, Supplemental Material and FT-IR-2-ATR-2-Plate-2, Figures S10–S12, Supplemental Material). We also provide comparisons between the formalism used in this work and the one used in Mayerhöfer et al.²⁰ of the results for the optical constants functions for water and octadecane, Figures S13–S14 (Supplemental Material). The results show slight differences due to the fact that parts of the experimental spectra show slightly negative absorbance that was not corrected (the formalism used in Mayerhöfer et al.²⁰ suppresses corresponding negative values for the absorption index function). We verified for a synthetic sample that the results are identical when there are no negative ranges in the absorbance spectrum; see Figures S15–S16 (Supplemental Material). In addition, Figures S17–S21 (Supplemental Material) compare results with and without the semi-empiric factor 800² cm^–2/ $\tilde{ν}$ ² in Eq. 7.

Since dispersion is usually very small in the transparency region in between the MIR and the ultraviolet–visible (UV–Vis), the incorporation of n_∞ assures that the errors of not extending the integral beyond the upper limit of the wavenumber range $\tilde{ν}$ _f = $\tilde{ν}$ _∞ are very small.²⁷ The same is usually not automatically correct for the range 0 < $\tilde{ν}$ < $\tilde{ν}$ _e , but if dispersion is small around $\tilde{ν}$ _e , the errors are again also small, for the same reasons the errors are small if the Kramers–Kronig relations are applied to gain the refractive index function from the absorption index function.²⁷ As already mentioned, in contrast to the edges of the spectral range under investigation, it is no problem for the algorithm if the condition is violated somewhere in between (Figures S19–S21, Supplemental Material).

One may object that the optical model assuming the ATR crystal as a semi-infinite incidence medium does not reflect the experimental conditions sufficiently well. In fact, Harrick²⁸ has shown that for s-polarized radiation the measured reflectance is given by:

R_{meas, s} = \frac{(1 - R^{2}) R_{s}}{1 - R^{2} R_{s}^{2}}

(9)

Here, R is the reflectance from the interface air–ATR crystal, which is itself described by Fresnel's equations, but with a different AOI (usually this angle is zero for conventional ATR accessories) and R_s is the s-polarized reflectance from Eq. 3. The situation is comparable to the one in UV–Vis spectrophotometry, where the reference measurement is recorded with the pure solvent in the cuvette.²⁹ In the latter case, the effect is practically not relevant for most of the solvents with values of the refractive index function of around 1.5. We assume a similar situation for ATR spectroscopy, and indeed, as shown in the Supplemental Material, if the wavenumber-dependent refractive index function of ZnSe as ATR crystal and poly(methyl methacrylate) as sample are used, the differences between using Eqs. 3 and 9 are marginally and the spectra agree within line thickness. The same is true for the resulting complex index of refraction function (see Figures S26–S31, Supplemental Material). If Ge is used as an ATR crystal, the errors increase by a factor of about four, but they also cumulate in the refractive index function. Accordingly, the derived absorption index function still agrees within line thickness with the original one, which means that the corrected absorbance agrees well with the true absorbance. We nevertheless provide in the following a correction scheme based on the inversion of Eq. 9. If it is applied to the spectrum gained from Eq. 5, it corrects this seemingly insignificant problem, and the result can be fed into the Kramers–Kronig analysis following Eq. 6:

R_{s} = \frac{- 1 + R^{2} + \sqrt{1 - 2 R_{meas, s}^{2} + 4 R_{meas, s}^{2} R^{2} + R^{4}}}{2 R_{meas, s} R^{2}}

(10)

An additional source of error, which is also usually not considered, is that the polarization of the light beam may vary with wavenumber if no polarizer is used. If the polarization ratio, or, equivalently, the ratio of R_p and R_s (R_p/R_s) varies noticeably with wavenumber, then band ratios in a spectrum can change if different instruments/accessories combinations are used (a larger ratio of R_p/R_s at a certain peak maximum means that the corresponding peak intensity is relatively larger in absorbance than for another peak with a lower ratio of R_p/R_s). As already mentioned above, a possible remedy is to assume that the polarization angle is wavenumber dependent and to determine this dependency by extending the method of Azam et al. correspondingly. We carried out such an analysis for FT-IR-1-ATR-1-Plate-1 (the corresponding result can be found in Figure S32, Supplemental Material). Having determined φ(

\tilde{ν}

), we can compute R_s from the wavenumber-dependent form of Eq. 5:

\begin{aligned} R_{s} ( & \tilde{ν}, 45 \circ) = \frac{1}{2} \\ (- co t^{2} φ (\tilde{ν}) + cs c^{2} φ (\tilde{ν}) \sqrt{co s^{4} φ (\tilde{ν}) + 4 R (\tilde{ν}, 45 \circ) si n^{2} φ (\tilde{ν})}) \end{aligned}

(11)

The corresponding results are compared with those from Figure 2 in Figure S33 (Supplemental Material). The comparison of the resulting real and imaginary parts of the complex refractive index function in Figure S34 (Supplemental Material) shows relatively small differences (these differences will certainly vary for different instrument/accessory combinations).

Overall, one should keep in mind that there still are certain restrictions that might then require a more sophisticated method of correction such as dispersion analysis.² Finally, the formulas presented here (and elsewhere in the literature) do not allow us to evaluate p-polarized spectra. We will detail how we evaluated R_p, shown in Figure 3, in an extended future correspondence, where we will also cover measurements for different incidence angles.

Conclusion

Overall, we have developed a fast and sophisticated ATR-spectra correction scheme for spectra gained with unpolarized incident light and a 45° incidence angle. This scheme is based on a simple relation between s- and p-polarized reflectance which allows us to calculate the s-polarized spectrum from the spectrum recorded without using a polarizer if the polarization angle (in dependence of the wavenumber) is known. We have confirmed the validity of this scheme by experiment and by extensive discussion of potential limitations and errors. In our opinion, this sophisticated ATR correction scheme should replace the so-called advanced ATR correction scheme, which is in fact only more advanced than merely multiplying ATR spectra with the wavenumber, and which breaks down if the refractive index around a band causes the critical angle to become larger than the incidence angle.

Supplemental Material

sj-docx-1-asp-10.1177_00037028231219528 - Supplemental material for Sophisticated Attenuated Total Reflection Correction Within Seconds for Unpolarized Incident Light at 45°

Supplemental material, sj-docx-1-asp-10.1177_00037028231219528 for Sophisticated Attenuated Total Reflection Correction Within Seconds for Unpolarized Incident Light at 45° by Thomas G. Mayerhöfer, William D. P. Costa and Jürgen Popp in Applied Spectroscopy

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research and/or authorship of this article: This work was supported by the EU, the Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft, the Thüringer Aufbaubank, the Federal Ministry of Education and Research, Germany (BMBF), the German Science Foundation, the Fonds der Chemischen Industrie, and the Carl-Zeiss Foundation.

ORCID iDs

Thomas G. Mayerhöfer

William D. P. Costa

Jürgen Popp

Supplemental Material

All supplemental material mentioned in the text is available in the online version of the journal.

References

Chalmers

J.M.

Griffiths

P.R.

. Handbook of Vibrational Spectroscopy. Chichester, UK: John Wiley and Sons, 2002.

Mayerhöfer

T.G.

. Wave Optics in Infrared Spectroscopy: Theory, Simulation, and Modelling. Amsterdam: Elsevier, 2024.

Hawranek

J.P.

Neelakantan

Young

R.P.

Jones

R.N.

. “The Control of Errors in I.R. Spectrophotometry. III. Transmission Measurements Using Thin Cells”. Spectrochim. Acta, Part A. 1976. 32(1): 75–84. https://doi.org/10.1016/0584-8539(76)80054-2

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “Removing Interference-Based Effects from Infrared Spectra: Interference Fringes Re-Revisited”. Analyst. 2020. 145(9): 3385–3394. https://doi.org/10.1039/D0AN00062K

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “CaF₂: An Ideal Substrate Material for Infrared Spectroscopy?”. Anal. Chem. 2020. 92(13): 9024–9031. https://doi.org/10.1021/Acs.Analchem.0c01158

Hansen

W.N.

. “Expanded Formulas for Attenuated Total Reflection and the Derivation of Absorption Rules for Single and Multiple ATR Spectrometer Cells”. Spectrochim. Acta. 1965. 21(4): 815–833. https://doi.org/10.1016/0371-1951(65)80039-X

Corujo

M.P.

Sklepari

Ang

D.L.

Millichip

, et al. “Infrared Absorbance Spectroscopy of Aqueous Proteins: Comparison of Transmission and ATR Data Collection and Analysis for Secondary Structure Fitting”. Chirality. 2018. 30(8): 957–965. https://doi.org/10.1002/chir.23002

Grdadolnik

. “ATR-FTIR Spectroscopy: Its Advantages and Limitations”. Acta Chim. Slov. 2002. 49(7): 631–642.

M. Boulet-Audet, T. Buffetau, S. Boudreault, N. Daugey, M. Pezolet. “Quantitative Determination of Band Distortions in Diamond Attenuated Total Reflectance Infrared Spectra”. J. Phys. Chem. B. 2010. 114(24): 8255–8261. https://doi.org/10.1021/jp101763y

10.

Ramer

Aksyuk

V.A.

Centrone

. “Quantitative Chemical Analysis at the Nanoscale Using the Photothermal Induced Resonance Technique”. Anal. Chem. 2017. 89(24): 13524–13531. https://doi.org/10.1021/acs.analchem.7b03878

11.

T.G. Mayerhöfer, S. Pahlow, J. Popp. “The Bouguer–Beer–Lambert Law: Shining Light on the Obscure”. ChemPhysChem. 2020. 21(18): 2029–2046. https://doi.org/10.1002/cphc.202000464

12.

Averett

L.A.

Griffiths

P.R.

Nishikida

. “Effective Path Length in Attenuated Total Reflection Spectroscopy”. Anal. Chem. 2008. 80(8): 3045–3049. https://doi.org/10.1021/ac7025892

13.

Kazarian

S.G.

Chan

K.L.A.

. “ATR-FTIR Spectroscopic Imaging: Recent Advances and Applications to Biological Systems”. Analyst. 2013. 138(7): 1940–1951. https://doi.org/10.1039/c3an36865c

14.

Mayerhöfer

T.G.

Ivanovski

Popp

. “Infrared Refraction Spectroscopy”. Appl. Spectrosc. 2021. 75(12): 1526–1531. https://doi.org/10.1177/00037028211036761

15.

M.S. Azam, M.D. Ranson, D.K. Hore. “Temperature-Dependent Infrared Refractive Index of Polymers from a Calibrated Attenuated Total Reflection Infrared Measurement”. Appl. Spectrosc. 2022. 76: 1254–1262. https://doi.org/10.1177/00037028221094598.

16.

Palik

E.D.

. Handbook of Optical Constants of Solids. Burlington: Elsevier Science, 1997.

17.

Bertie

J.E.

Lan

. “An Accurate Modified Kramers–Kronig Transformation from Reflectance to Phase Shift on Attenuated Total Reflection”. J. Chem. Phys. 1996. 105(19): 8502–8514. https://doi.org/10.1063/1.472635

18.

Plaskett

J.S.

Schatz

P.N.

. “On the Robinson and Price (Kramers–Kronig) Method of Interpreting Reflection Data Taken Through a Transparent Window”. J. Chem. Phys. 1963. 38(3): 612–617. https://doi.org/10.1063/1.1733714

19.

Tatian

. “Fitting Refractive-Index Data with the Sellmeier Dispersion Formula”. Appl. Opt. 1984. 23(24): 4477–4485. https://doi.org/10.1364/ao.23.004477

20.

Mayerhöfer

T.G.

Singh

A.K.

Huang

J.-S.

Krafft

Popp

. “Unveiling Chiral Optical Constants of Α-Pinene and Propylene Oxide Through ATR and VCD Spectroscopy in the Mid-Infrared Range”. Spectrochim. Acta, Part A. 2023. 302: 123136. https://doi.org/10.1016/j.saa.2023.123136

21.

Huang

J.B.

Urban

M.W.

. “Evaluation and Analysis of Attenuated Total Reflectance FT-IR Spectra Using Kramers–Kronig Transforms”. Appl. Spectrosc. 1992. 46(11): 1666–1672. https://doi.org/10.1366/0003702924926970

22.

Bertie

J.E.

Lan

. “Infrared Intensities of Liquids XX: The Intensity of the OH Stretching Band of Liquid Water Revisited, and the Best Current Values of the Optical Constants of H2O(L) at 25 °C Between 15 000 and 1 cm⁻¹”. Appl. Spectrosc. 1996. 50(8): 1047–1057. https://doi.org/10.1366/0003702963905385

23.

Pinkley

L.W.

Sethna

P.P.

Williams

. “Optical Constants of Water in the Infrared: Influence of Temperature”. J. Opt. Soc. Am. 1977. 67(4): 494–499. https://doi.org/10.1364/JOSA.67.000494

24.

Mayerhöfer

T.G.

Pahlow

Ivanovski

Popp

. “Dispersion Related Coupling Effects in IR Spectra on the Example of Water and Amide I Bands”. Spectrochim. Acta, Part A. 2023. 288: 122115. https://doi.org/10.1016/j.saa.2022.122115

25.

Mayerhöfer

T.G.

Richard-Lacroix

Pahlow

Hubner

Popp

. “Smart Error Sum Based on Hybrid Two-Trace Two-Dimensional (2T2D) Correlation Analysis”. Appl. Spectrosc. 2023. 77(6): 583–592. https://doi.org/10.1177/00037028221077310

26.

Mayerhöfer

T.G.

Noda

Pahlow

Heintzmann

Popp

. “Correcting Systematic Errors by Hybrid 2D Correlation Loss Functions in Nonlinear Inverse Modelling”. PLoS One. 2023. 18(4): E0284723. https://doi.org/10.1371/journal.pone.0284723

27.

Mayerhöfer

T.G.

Ivanovski

Popp

. “Infrared Refraction Spectroscopy: Kramers–Kronig Analysis Revisited”. Spectrochim. Acta, Part A. 2022. 270: 120799. https://doi.org/10.1016/j.saa.2021.120799

28.

Harrick

N.J.

. Internal Reflection Spectroscopy. New York: Interscience Publishers, 1967.

29.

Mayerhöfer

T.G.

Mutschke

Popp

. “Employing Theories Far Beyond Their Limits—The Case of the (Boguer–) Beer–Lambert Law”. ChemPhysChem. 2016. 17(13): 1948–1955. https://doi.org/10.1002/cphc.201600114

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

13.10 MB