Fourier-transform infrared (FTIR) spectra can show artifacts, for example, in the vicinity of the overtone when light is doubly modulated. Technical approaches, i.e., modifications to the spectrometer setup, have been devised in order to reduce those artifacts. Elimination of the artifacts was achieved only partly but at the expense of a loss of intensity or an increase in the noise level. We devised a computational demodulation scheme that is capable of almost fully reducing the artifacts with neither a loss of spectroscopic information nor an increase in the noise level. This has been demonstrated for the FTIR absorption spectra in the overtone regions of HCl(g) and .
© 2017 Optical Society of America
Fourier-transform infrared spectroscopy (FTIR) is a universal tool for the analysis of chemical compounds due to their fingerprint characteristic absorption in the infrared spectral range. With the advent of Fourier transformation, recording of a spectrum with high to medium resolution became very fast. Since the pioneering work of Jacquinot and Fellgett  in the early 1950s, Michelson’s interferometric absorption spectroscopy became a routine tool. With the advent of computers and the algorithmic developments of the fast Fourier transform (FFT) by Cooley and Tukey in 1965 , FT spectroscopy increased its applicability to spectral ranges from lower energy (microwave , nuclear magnetic resonance ) to higher energy (UV/VIS [5,6]).
Rovibrational spectra of molecules in the gas phase are recorded for an unambiguous assignment of corresponding transitions . In standard routine analytical applications, medium resolution is often sufficient. Contrary to the obvious advantages of the Fourier techniques, multimodulation artifacts can arise due to backreflection of the IR beam into the interferometer; those artifacts appear at multiples of the energy associated with absorption signals in the fundamental and overtone spectral regions [8–21]. These artifacts with lower intensity at multiples of fundamentals thus overlap in part with overtones, which are anharmonically shifted toward lower energy. Artifacts due to aperture were observed in the past for CO and in the gas phase [8,11,12]. A few technical approaches to reduce or fully eliminate these artifacts by modifying apertures such that no backreflection should occur were proposed in the past [8–10], but those modifications led to an intensity loss and/or to an increase in the noise level.
We report here artifacts for gaseous HCl between 5400 and and between 5800 and , which extend over a broad spectral range and strongly overlap with overtone and combination bands, thereby perturbing the line shapes as well as intensities. It seems that those artifacts are more a characteristic feature of interferometric systems rather than a rare effect. Therefore, we devised a computational scheme to strongly reduce or even eliminate those artifacts numerically by neither loosing intensity nor increasing the noise level. In addition, we propose a device-oriented modification (use of a special mirror and setup with multiple detectors) that would, in principle, allow for a fully computerized automated compensation [19,20].
2. DESCRIPTION OF THE DEMODULATION SCHEME
A. Single Reflection at a Point in Front of the Sample Cell
First, we consider a single reflection between the mirror C2 and the sample cell F (see Fig. 1) causing the double modulation of the total interference signal. The underlying idea of our compensation method for a Michelson-type interferometer uses the fact that the detected interference signal can be decomposed into parts, thereby following the directions outlined by Chamberlain . Thus, the detected interference signal is a superposition of two (or more) interference signals depending on the mirror offset , one single modulated , and one double modulated by the interferometer:
In order to minimize or better fully compensate artifacts, we aim to subtract a correction interference signal, , from the experimental one, , thereby cancelling the term of the double modulation. A compensated interference signal, , is obtained as follows:
is cancelled, thereby generating a new interference signal with newly introduced but much reduced artifacts at double the mirror offset of the original artifacts, which leads to
We can repeat this idea of succinct compensation by introducing higher terms in the mirror offset (e.g., 4-fold, 8-fold, 16-fold,…) resulting in22]. The compensation coefficients are
The correction coefficient depends on the transmittance coefficient related to reflection, . We assume a slow change in in the wavenumber domain and therefore also a small change in the spatial domain of the interferogram. will then be treated as a constant that is independent of the mirror offset . For example, applying the compensation scheme once (), we see that terms Eqs. (5b) and (5c) compensate exactly. The compensation interference signal contains only a single modulated contribution, Eq. (5a), and a 4-fold modulated contribution, Eq. (5d), which has much lower intensity than and . This follows from the fact that Eq. (5a) scales as5d) scales as
Since , the term represented in Eq. (5d) is smaller than the contributions from Eqs. (5a) and (5b). From the discussion above for the case of , we see that every additional iterative compensation leads to a further decrease by a factor of ().
Equation (5) can be written in a more compact form as
Due to alias overlap, the newly produced artifacts at or , respectively, can be folded back into the spectrum. Equation (9) can also be extended to compensate for higher order modulations by replacing and (double modulation) with and (-fold modulation).
B. Two Reflections: One in Front and One behind the Sample Cell
In this subsection, we consider two reflections: one between C2 and F and one between F and G; see Fig. 1. For such a system, the interference signal takes the form
designates the single-modulated interference signal, describes the double-modulated interference signal between C2 and F, whereas is the double-modulated interference signal between F and G (Fig. 1). In the case of , we assume that the beam is reflected back into the interferometer once (thus double-modulated) at the left window of the sample cell before leaving toward the detector. In this case, the beam crosses the sample cell three times, which leads to (), Eq. (12c). For , the beam is reflected back into the interferometer at the right window of the sample cell (double-modulated) before leaving the sample cell toward the detector. In this case, the beam passes the sample cell once. Therefore, (), Eq. (12b). There is no fundamental reason why and should be equal. Therefore, we retain them as separate parameters,
The general compensation coefficient for compensations is, in this case, found to be
Similar to Eq. (5), the first iteration of the compensation scheme yields
The compensation interference signal contains only a single modulated contribution [Eq. (16g)] and 4-fold modulated contributions, Eqs. (16h)–(16j), with lower intensities than , , and ; this follows from similar reasoning as described in Section 2.1.
3. PRACTICAL IMPLEMENTATION
The main aspect of the practical implementation of Eq. (2) is to determine the correction interference function by doubling the frequency. One possibility to achieve this is to remove every second data point of the original interference function . The compressed interference function is also centered around the point of zero path difference, see Fig. 2, which illustrates the compensation procedure for a double-sided single-directional interferogram.
In principle, one can distinguish two ways to obtain the correction interference function . First, one would construct by increasing the resolution: recording an interferogram of -fold increased length more than needed in the end (corresponds to -fold resolution) and one uses only 1/2 of the interferogram for each repetition of the demodulation scheme. This guarantees that the final compensated interferogram has the same resolution as the originally recorded interferogram. Here, we explore a second scheme, depicted in Fig. 2. In a first step, we compress the interferogram to half its length by deleting every second data point. The result is an interferogram of half the original length (from to , Fig. 2). The points marked and are used to mirror the compressed interferogram, thereby constructing of original length, i.e., without loss of resolution, in contrast to the first method described above. Many more possibilities to obtain can be envisaged; this has been explored in great detail .
4. RESULTS AND DISCUSSION
In this section, we present two examples of gas-phase spectra showing the double-modulation artifacts, namely, HCl(g) and in their respective overtone regions [19,20]. First, the correction scheme described in Fig. 2 is applied to the spectrum of HCl(g).
A. Hydrogen Chloride Gas
We present the corrected spectrum for HCl(g) (lower trace) together with the original spectrum (upper trace) recorded between 5400 and in Fig. 3.
The corrected spectrum has been obtained considering double modulation with a constant compensation coefficient . One can see immediately that the artifacts associated with the fundamental band are strongly compensated by our correction scheme, which has been applied once [, Eq. (5)]. Spectral information of the fundamental (line positions and line intensities) and the overall noise level do not change appreciably upon correction.
For illustrative purposes, we discuss the influence of our correction scheme on the absorption in more detail and therefore calculated the interferograms of the fundamental and the first overtone of the rovibrational spectrum of HCl(g) with and without double modulation. We select HCl(g) because of the simplicity of the absorption spectrum in comparison with and the accessibility of the respective spectroscopic parameters (after [23,24]): we have compared our measured spectrum before and after correction with a simulated spectrum of HCl(g) where spectroscopic parameters have been taken from HITRAN  and found good agreement. It is to be expected that absorbances are more sensitive to corrections than are line positions; our analysis has shown that the absorbance changes due to our correlation scheme are very small. We have to simulate two interferograms twice, namely, an interferogram of the background (empty sample cell) and an interferogram of the sample for single and double modulation. In the latter case, we choose the parameters such that the simulated spectrum resembles the measured spectrum shown in Fig. 3. Our simulation without double modulation uses and . The simulation with double modulation uses and . Figure 4 shows the simulated spectrum of HCl(g) with the compensation parameters just described. The upper trace, Fig. 4(a), displays the simulated spectrum without double modulation, the middle trace, Fig. 4(b), shows the spectrum with double modulation and the lower trace, Fig. 4(c), shows the compensated spectrum with single demodulation [; see Eq. (6b)]. With these simulations, we can quantify the change of absorbance upon considering the double modulation. We can also quantify the effect of our compensation scheme with respect to the original spectrum with and without double modulation. Numbers for these comparisons are given in Table 1 for the fundamental and in Table 2 for the first overtone of HCl(g). In order to quantify the effect of our compensation scheme on line intensities represented by , we need to calculate an interferogram with and without double modulation because measured interferograms always contain contributions from double or even multiple modulations. We used wavenumber-independent and , which signify that no light intensity is lost, where is assumed to be about equal to the natural reflection coefficient of KBr-window material of our gas cell. The compensation coefficient depends on and , Eq. (6b).
Table 1 shows the selected -branch [, ]. For the -branch we have , where the upper and lower -states of the rotational transition are indicated by prime and double prime symbols. The corresponding -branch absorbance of the spectrum without double modulation (unperturbed) of the double-modulated spectrum obtained with parameters as described above as well as the absorbance after compensation (demodulated) are shown in columns 2 to 6. If one compares the double-modulated absorbance with the unperturbed ones, we see that the relative change (column 4) decreases by values between and about . No systematic trend can be observed depending on . After compensation (demodulation), the absorbance changes (column 6) are obtained from the deviation of the ratio (demodulated) to (unperturbed) with respect to 1:20) stands for the demodulated or the double-modulated absorbance. It seems that the deviation caused by introducing double modulation (columns 3 and 4) is for some transitions compensated by our scheme, which then results in smaller relative deviations after compensation (columns 5 and 6).
Similar observations hold for the first overtone of HCl(g) and the corresponding numbers are given in Table 2. They refer to the absorbance for the case without double modulation (unperturbed), with double modulation (columns 3 and 4), and after compensation (, columns 5 and 6). We used the same values for , and as for the fundamental of HCl(g). The double-modulated absorbances show relative changes (, column 4) between and without systematic changes with respect to . The values for the changes in absorbances after demodulation have been obtained as described above and are comparable in magnitude to those values obtained for the fundamental absorption of HCl(g). Again, we observe that the artifacts can be compensated completely without affecting the noise-level appreciably.
The question arises whether or not the change in absorbance upon comparing simulations without demodulation (unperturbed), with double modulation, and with demodulation depend on the value of . If one uses a value of that is too large, one overcompensates the artifacts, which leads to unphysical negative absorbances in the spectral regions of the artifacts. Figure 5 shows of HCl(g) for individual transitions of the fundamental band [Fig. 5(a)], the first overtone [Fig. 5(b)], and the artifacts [Fig. 5(c)]. The values have been obtained according to3 (upper trace).
For a particular value of , we determined in the depicted spectral regions of the fundamental [Fig. 5(a)], the first overtone [Fig. 5(b)], and in the region where the artifacts appear [Fig. 5(c)] for to . In case of the fundamental region [Fig. 5(a)], varies with ; some transitions show strong dependence [e.g., ], whereas others seem to be largely independent of [e.g., ]. Maximum values are of the order of 5% for the interval depicted here. In Fig. 5(b), the values of the overtone change with with no strong dependence on the particular transition.
The aim of our compensation scheme is to minimize the signals of the artifacts. In the ideal case and in the absence of noise, we would seek to achieve a value of , which would compensate the artifact exactly. As can be seen from Fig. 5(c), this compensation can be achieved in the range of (vertical dotted lines), however, this condition cannot be fulfilled strictly for all artifacts at the same time. The shaded box marks the region between about one standard deviation of the noise level (roughly ). Outside the region of the vertical dotted lines, over- or under-compensation would take place. The interval for can be chosen according to an acceptable tolerance level for the compensation.
B. Methane Gas
The correction scheme described in Fig. 2 is now applied to the spectrum of . Figure 6 shows the original spectrum (upper trace) of the -tetradecad band of methane between 5800 and [25,26], centered around , and its corrected form (lower trace). The corrected spectrum has been obtained considering double modulation with a constant compensation coefficient . One can see immediately that the artifacts associated with the -pentad band centered around are strongly compensated by our correction scheme, which has been applied once [, Eq. (5)]. Spectral information of the -tetradecad and the overall noise level do not change appreciably upon correction. The broad absorption feature underlying the to transition (about 6000 to ) is due to the -branch artifact of the -pentad. After compensation, this feature is also removed with all other artifacts resulting from double modulation. Figure 7 shows of for individual transitions of the -pentad [Fig. 7(a)], the -tetradecad [Fig. 7(b)], and the artifacts [Fig. 7(c)]. The corresponding experimental absorbances of the -tetradecad and the artifacts are shown in Fig. 6 (upper trace). values have been obtained according to Eq. (21). For a particular value of , we determined in the spectral regions of the -pentad [Fig. 7(a)], the -tetradecad [Fig. 7(b)], and in the region where the artifacts appear [Fig. 7(c)] for to . In the case of the -pentad region [Fig. 7(a)], varies with ; some transitions show stronger dependence [e.g., ]. Maximum values are of the order of 3.5% for the interval shown here. In Fig. 7(b), the values of the -tetradecad change with with no strong dependence on the particular transition. Again, the aim of our compensation scheme is to minimize the signals of the artifacts. In the ideal case and in the absence of noise, we would seek to achieve a value of , see Eq. (21), which would compensate the artifact exactly. As can be seen from Fig. 7(c), this compensation can be achieved in the range of (see vertical lines in Fig. 7), however, this condition can also not strictly be fulfilled for all artifacts at the same time, as discussed in the case of HCl(g). As can be seen from Figs. 7(b) and 7(c), larger values of would lead to considerable overcompensation.
Up to now, we discussed compensation of interference signals in a Michelson-type interferometer, which were double-modulated. Artifacts caused by double modulation can be visualized in the spectra using the mathematical compensation scheme described above. We performed the compensation by processing the interferograms with single demodulation (using , ) to obtain the compensated absorbance spectrum (). The compensation coefficient should be chosen such that overcompensation can be avoided [see, for example, Figs. 5(c) and 7(c)]. Using the original (experimental) and compensated absorbance spectra, a difference spectrum can be calculated according to (see Fig. 8). In this difference spectrum, negative signals indicate artifacts in the corresponding spectral ranges. Positive signals indicate the “true” transitions, which are not artifacts caused by double modulation. Comparing the spectra with the difference spectrum, such artifacts can be identified easily. As shown above, a proper choice of allows for a complete reduction of the artifacts.
For HCl and , as discussed above, a single iteration of the compensation scheme () was sufficient because our artifacts were compensated to an acceptable degree caused by the smallness of . counts the number of compensations and a convenient is chosen by inspection until the introduced higher order artifacts are sufficiently compensated. Every iteration of the compensation scheme introduces higher order artifacts, which are systematically lower in intensity. Depending on the magnitude of relative to the noise level, more than one iteration step might be necessary for an acceptable degree of compensation until the magnitude of higher order artifacts introduced is lower than the noise level.
So far, we considered artifacts caused only by double modulation. However, it would be possible to test the experimental spectra also for -fold modulation by considering only every th data point to generate ; see Section 3. One would generate a compensated absorbance spectrum with, for example, , with the respective change in Eq. (9). A graphical representation of the difference spectrum with respect to the experiment would show negative signals for the artifacts caused by the -fold modulation.
As an outlook, we mention briefly some ideas about an automatic procedure that might be used to determine the compensation coefficient by experiment. It has been demonstrated above that the key parameter for compensating artifacts caused by double modulation is , see Eq. (10), which has been determined by an iterative numerical procedure so far. As we will show here, can be determined, in principle, experimentally. By measuring a reflection interferogram, the coefficient can be calculated using the ratio of the intensity of the main interferogram and the intensity of the reflection interferogram. Such a reflection interferogram could be measured by introducing mirrors designed with open channels parallel to the focused beam (see Fig. 1 for C2 and C3 horizontal channels and for C1 and C3 vertical channels) and parallel to the collimated beam (see Fig. 1 for the C2 vertical channel and for the C1 horizontal channel). Placing a secondary detector H in the vicinity of one of the mirrors C1 to C3, where the reflected intensity seems to be highest, allows us to measure reflection interferograms.
Such mirrors with two channels that meet at the surface of the mirror, see Fig. 9, were used in laser spectroscopy. The main and the reflection interferogram can be detected simultaneously. For simplicity, we assume that (1) the reflection arises after passing the sample cell and (2) the detected reflection interferogram does not contain a contribution from double modulation (since ). The two detected interferograms (using two detectors) take the form24)], whereas the second case with reflection in front of the sample cell, is given by the ratio of to weighted by the coefficient [see Eq. (25)]. However, the intensity of the detected reflection interference signal must be corrected by the illuminated areas (by the reflected light) of the mirrors placed in front of the secondary detectors and the diameter of the holes, respectively, and by the gain of the secondary detectors in relation to the main detector. This would offer a complete automation of this compensation method.
Zurich University of Applied Sciences (ZHAW) (ASF-2014/1).
We are grateful to Dr. Hans Hollenstein, ETH Zürich, for help and discussions. We thank also Dr. Li Chen and Prof. Alec M. Wodtke, University of Göttingen, for providing us with FTIR spectroscopic data on to test our compensation scheme. JS wants to thank his academic teacher, Prof. Martin Quack (ETH Zürich), for continuous support.
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