## Abstract

Fourier transform infrared (FTIR) spectrometers have been widely used as very important analytical tools in various fields. Owing to the Jacquinot Stop (J-Stop), high throughput is a widely recognized advantage inherent to Fourier transform interferometers. However, there is a fundamental trade-off between the throughput and spectral resolution, which is primarily affected by the size of the J-Stop. So far, no effective optimization methods have been provided to break the trade-off. In this paper, we introduce a numeric technique of the digital J-Stop, which has been experimentally validated using the FTIR spectra collected from a commercial spectrometer. The result shows that the throughput can be increased by ~12 times, while the spectral resolution is also improved. In this way, the signal-to-noise ratio (SNR) gets improved by ~3 times.

© 2017 Optical Society of America

## 1. Introduction

Fourier transform infrared (FTIR) spectrometers have been widely used for the qualitative and quantitative analysis of different substances in various fields, including the food industry, pharmaceutical industry, chemical industry, security industry, environmental protection industry, especially for the weak-signal measurements [1–5]. Compared with the dispersive counterparts, the Jacquinot advantage, also called the throughput advantage, is one of the two widely recognized advantages inherent to FTIR interferometers’ operating principles [5–8]. By replacing the slit with the Jacquinot Stop (J-Stop), this advantage can provide more than ~60 times higher throughput for the same resolving power and similar instrument size [3]. In this way, the spectral signal-to-noise ratio (SNR) can be improved significantly, where the detector noise is dominant.

The finite J-Stop has negative effects on the width-broadening and peak-shifting [9–11]. Unlike a perfectly collimated beam, the beam entering the interferometer has a finite divergence angle. The divergence angle increases with the diameter of the J-Stop, when the collimating optics is constant. The divergent beam will undergo different paths in the interferometer, which causes the non-uniformity of the optical path difference (OPD). As a result, the FTIR spectral peaks are broadened and shifted towards lower wavenumbers.

There is a fundamental trade-off between the throughput and spectral resolution, which comes from the size of the J-Stop. While a large J-Stop can provide high throughput, the width-broadening and peak-shifting appear obviously, especially when the ideal spectral resolution is high. At present, it is the conventional choice to limit the J-Stop to an affordable range [3]. In this way, the throughput of FTIR spectrometers get decreased dramatically, which brings a worse spectral SNR. Therefore, there is an urgent need to increase the throughput without impairment to the spectral resolution.

In this paper, we demonstrate a numerical technique of the digital J-Stop to recover the FTIR spectra with ~12 times higher throughput, which results in a ~3 times higher SNR. Firstly, we introduce the principle of the digital J-Stop method based on the negative effects of the beam divergence. Then the digital J-Stop method is validated using the FTIR transmission spectra of water vapor collected from a commercial spectrometer. Our results show that the digital J-Stop method can simply and effectively remove the width-broadening and peak-shifting effects caused by the larger physical J-Stop.

## 2. Theory and methods

Figure 1 shows the main structure of a FTIR spectrometer, augmented with the proposed digital J-Stop. A conventional FTIR spectrometer is basically composed of a light source, a physical J-Stop, a beam splitter (BS), a fixed mirror M_{1}, a moving mirror M_{2}, a photodiode (PD), a collimating lens L_{1} and a focusing lens L_{2}. The dotted M_{1}’ which is the image of M_{1} formed by BS is shown for intuitively explaining the effect of the divergence angle *α* on the OPD. After executing the fast Fourier transform (FFT) algorithm, FTIR spectra are extracted from the interferograms collected by the PD. While a larger physical J-Stop can provide higher throughput, the width-broadening and peak-shifting are more serious. To break the trade-off, a digital J-Stop is introduced to recover the FTIR spectra impaired by the larger physical J-Stop.

Figure 1(a) shows the spectral changes induced by the beam divergence. When the light source is an ideal monochromatic point one, the beam entering the interferometer can be perfectly collimated, as shown by the blue arrow lines in Fig. 1(a). In this case, the interferometer collected by the PD is a cosine signal as a function of the OPD which is equal to twice the distance *l* between M_{2} and M_{1}’ (M_{1}). After FFT transforming, a spectrum P_{0} is obtained, as shown by the blue curve in Fig. 1(a). The spectral width of P_{0} is determined by the maximum OPD and the apodization function used during the interferogram digital processing [3,12].

However, the real light source has a finite size, which directly produces the divergence angle of the beam entering the interferometer. For limiting the divergence angle, a physical J-Stop is set up, whose diameter is *d*. In this way, the divergence angle *α* is given by $\alpha \approx \mathrm{tan}\alpha =d/2{f}_{1}$, where *f*_{1} is the focal length of the collimating lens L_{1}. At the divergence angle *α*, the transmitted and reflected principle rays undergo different paths, as shown by the red arrow lines in Fig. 1(a). Owing to the change in the OPD, the spectrum P_{α} is different from P_{0}, as shown by the red curve in Fig. 1(a). The dotted curves in Fig. 1(a) denote the spectra corresponding to different divergence angles varying between 0 and *α*. When the OPD of the collimated rays is ${\delta}_{1}=2l$, the OPD of the divergent rays can be given by:

If the central wavenumber of the monochromatic source is *ν*_{0}, the peak of P_{0} will be at the same position. However, the peak of P_{α} will be shifted to *ν*_{α}, which can be given by [3]:

_{α}is shifted towards a lower wavenumber. The wavenumber shift is mainly related with the divergence angle

*α*.

The real spectrum P_{r} is superposed by the spectra resulting from different divergence angles, as shown by the orange curve in Fig. 1(a). For clearly showing the spectral changes, the intensity is normalized. The spectral width of P_{r} also becomes broadened in comparison with P_{0}, in addition to the obvious peak-shift towards the lower wavenumber.

To recover the width-broadened and peak-shifted spectra, the digital J-Stop method can be applied. Figure 1(b) shows the FTIR spectra coming from the physical and digital J-Stop. The olive curve denotes the spectrum collected with the 4 mm J-Stop, which is obviously width-broadened and peak-shifted. The 4 mm J-Stop provides higher throughput. The blue curve denotes the spectrum collected with the 1 mm J-Stop, which is chosen as a reference. The reference spectrum is multiplied by 10 for a clear display, owing to the big intensity difference. The red curve denotes the recovered spectrum with a digital J-Stop. The recovered spectrum is extracted from the higher-throughput spectrum by removing the width-broadening and peak-shifting. The SNR of the recovered spectrum is obviously better than the reference. Moreover, the recovered spectrum can coincide well with the reference, which implies that the recovery result is accurate and reliable.

The digital J-Stop method is a numeric recovery technique. In this method, the increase of the throughput can be achieved without any more physical components. The spectrum collected with a larger physical J-Stop gets recovered to generate a spectrum without width-broadening and peak-shifting. Let *p*_{0} denote the recovered spectrum, and let *p*_{r} denote the spectrum collected with a larger physical J-Stop. In theory, all the spectra coming from the divergence angles have the same spectral properties with the ideal spectrum P_{0}. The recovery problem can be solved according to the following formula:

*a*denotes the intensity distribution at the physical J-Stop, Δ

*ν*denotes the wavenumber shift step,

*n*denotes the number of wavenumber shift steps,

*e*denotes the noise during the measurement. The intensity distribution

*a*is the only parameter that needs to be measured simply in advance. The wavenumber shift step Δ

*ν*is always equal to the sampling wavenumber interval of FTIR spectrometers. The number

*n*is limited by the maximum divergence angle.

A matrix *H* is built as a projection operator, which can express the rules of the width-broadening and peak-shifting. The projection operator *H* is just a function of the intensity distribution *a* and the wavenumber shift step Δ*ν*, which is given by $H\left(a,\Delta \nu \right)$. When Δ*ν* is equal to the sampling interval, *H* is a *m* × *m* sparse matrix, where *m* is the length of *p*_{r}. The non-zero elements of *H* can be determined by ${H}_{x,y}={a}_{i}$, where $y=x+i$. There is no need to know the specific optical configurations in different FTIR spectrometers. In this way, the recovery problem can be transformed to solve the following formula:

In this paper, *a* is determined by the Gaussian distribution which is consistent with the characteristics of the source, in order to simplify the analysis. Δ*ν* is set to 0.03 cm^{−1} depending on the settings of the FTIR spectrometer. *n* is set to 10, which is not many for the following calculation.

To solve the Eq. (4), we apply the Landweber iterative algorithm to seek the maximum likelihood parameters until convergence, according to the following formula [13,14]:

where*β*denotes the adjustable parameter, ranging from 0.01 to 1.

In general, the digital J-Stop method can be applied easily, without additional physical components and knowledge of optical parameters of FTIR spectrometers, as we show in Code 1 [15]. A spectrum with higher SNR can be effectively obtained by recovering the higher-throughput spectrum using a digital J-Stop. The recovery result can be accurate and reliable enough to coincide with the reference.

## 3. Results and discussion

For validating the digital J-Stop method, we collected a set of FTIR transmission spectra of water vapor at a pressure of 5 kPa. A Bruker VERTEX 80v FTIR spectrometer was used, configured with different diameters of the physical J-Stop: 1, 2, 3, 4 and 6 mm, respectively. The limiting spectral resolution determined by the maximum OPD was set to 0.08 cm^{−1}, and the apodization function was set to the Norton-Beer Strong. For improving the SNR of the raw spectra, the average sampling number was set to 100.

Figure 2 shows the raw spectra collected for validating the digital J-Stop method. As the diameter of the physical J-Stop increases, the intensity of the raw spectrum increases obviously. When the diameter is larger than 4 mm, the change becomes small. It can be explained by the non-uniformity of the source intensity and the limited size of the interferometer and the PD. The result implies that the throughput of the 4 mm J-Stop is almost the highest, which is ~12 times higher than the 1 mm J-Stop. Therefore, the spectrum collected with the 4 mm J-Stop is chosen to be tested later.

Figure 3(a) shows four spectra collected with different diameters of the physical J-Stop: 1, 2, 3 and 4 mm, respectively. For clearly showing the spectral differences, the intensity is normalized. The spectral range is from 1471.25 to 1474.25 cm^{−1}, which is a part of the wide absorption band of water. As the intrinsic linewidth of water is narrow, the width-broadening and peak-shifting can be distinguished clearly. It can be clearly seen that the peak broadening and shifting are indeed more serious in the case of a larger physical J-Stop. The full width at half maximum (FWHM) of the peaks at ~1473.65 cm^{−1} is calculated by the Gaussian profile fit as follows: 0.118, 0.122, 0.141, 0.154 cm^{−1}, for the diameter of 1, 2, 3 and 4 mm, respectively. The maximum wavenumber difference among the peaks at ~1473.65 cm^{−1} is 0.088 cm^{−1}, larger than the limiting spectral resolution.

To demonstrate the digital J-Stop method, the spectrum collected with the 4 mm J-Stop is recovered, as shown in Fig. 3(b). The spectrum collected with the 1 mm J-Stop is chosen as a reference, because the diameter of 1 mm is recommended by the conventional choice for the trade-off. It can be clearly seen that the recovered spectrum by the digital J-Stop method can coincide well with the reference. The FWHM of the peak at ~1473.65 cm^{−1} in the recovered spectrum is 0.096 cm^{−1}, better than the reference. The noise level of the spectra are calculated by the root mean square (RMS) value as follows: 2.09 × 10^{−5} and 6.21 × 10^{−5}, for the recovered spectrum and the reference. As such, the SNR of the recovered spectra can be ~3 times higher than the spectra collected with the 1 mm J-Stop.

## 4. Conclusions

The proposed digital J-Stop method has been demonstrated to increase the throughput without impairment to the spectral resolution. The throughput is improved by ~12 times using the numeric recovery method, without additional physical components and knowledge of the optical configurations in FTIR spectrometers. The recovered spectrum is accurate and reliable to coincide well with the reference, which can reduce the difficulty of model transfer in the quantitative analysis. Moreover, the spectral resolution gets better. With the significant improvement in the throughput, the SNR has been increased by ~3 times, which is beneficial to the weak-signal measurement and rapid detection. As the result shows, it is potential to increase the SNR of various FTIR spectrometers simply and effectively by a digital J-Stop.

## Funding

National Natural Science Foundation of China (NSFC) (21273159, 21305101 and 61378048); Natural Science Foundation in Tianjin (13JCQNJC05100); Tianjin Research Program of Application Foundation and Advanced Technology (14JCZDJC34700); Open Funding Project of State Key Laboratory of Precision Measuring Technology and Instruments (PIL1605); Program for New Century Excellent Talents in University (NCET-11-0368).

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