1. INTRODUCTION

Scanning Fourier transform infrared (FTIR) spectrometers today represent the standard instrument for quantitative and qualitative laboratory analysis of gases, fluids, and solids in the mid-infrared spectral range [1], especially if moderate spectral resolution is sufficient for the application; however, static Fourier transform spectrometers (sFTS) can offer a viable alternative to these conventional systems, as they comprise no moving parts and therefore can be built in a cost-effective and, at the same time, rugged way. For the mid-infrared spectral range, so-called source-doubling interferometers creating a pair of divergent light sources in front of a Fourier optic emerged advantageous, as they work with light sources independent of their sizes and hence offer a comparatively high light throughput [2]. While various designs have been proposed, using birefringent [3], spatially modulated prism [4], or common-path interferometers [5–9], this paper presents a novel broadband static Fourier transform spectrometer (bsFTS) based on a single-mirror interferometer, as described in [10,11].

Dispersion effects can have a negative impact on the performance of these systems, especially when using a lens as a Fourier optic, as the focal length and therefore the contrast as well as the sampling frequency of the interferogram are then both wavelength- and temperature-dependent [12,13]. Although correction algorithms exist [13], the lens typically has to be temperature-controlled and still leads to a loss of spectral bandwidth. Setups comprising concave mirrors as Fourier optics have been proposed; however, these systems either work with narrowband light sources or large focal lengths leading to low spectral resolutions [5,7,9] or include customized and hence comparatively expensive mirror assemblies [4,8].

Therefore, in this paper, we present a bsFTS based on a single-mirror interferometer, where the Fourier lens is replaced by a spherical concave mirror to eliminate dispersion effects. The system works in the mid-infrared spectral range from 3.6 μm to 17 μm, respectively, from $2800{\mathrm{cm}}^{-1}$ to $600{\mathrm{cm}}^{-1}$at a spectral resolution of about $12{\mathrm{cm}}^{-1}$, while only off-the-shelf mirrors, a beam splitter, and an uncooled broadband microbolometer array are used. As no lenses are included in the spectrometer, its performance is potentially independent of temperature and wavenumber. In addition, high signal-to-noise ratios (SNR) can be achieved, as the system shows in principle no internal light losses, and the second detector dimension can be used for averaging. As for any sFTS, measurement speed is only limited by the detector readout frequency and thermal time constant [14]. With the currently used detector, measurement rates up to 25 Hz can be achieved.

At first, we introduce relevant fundamentals of static single-mirror Fourier spectrometers and the source-doubling principle. After a simulation of the wavenumber and temperature-dependent behavior of both lens- and mirror-based sFTS, we present our experimental setup and provide a proof of principle of the proposed concept by means of different measurement results.

2. SINGLE-MIRROR FOURIER TRANSFORM SPECTROMETERS

Before describing the proposed setup in detail, we give a brief overview of static single-mirror Fourier transform spectrometers (sSMFTS) as well as the source-doubling principle in general. Furthermore, we characterize the temperature and wavelength dependence of different Fourier optics.

A. Basic Principle of sSMFTS

Figure 1 shows the basic design principle of static single-mirror Fourier transform spectrometers. Light from an extended light source hits a beam splitter with thickness $T_{\mathrm{bs}}$ at point A, where it is split into one part being transmitted to point $B_1$ and another part being reflected twice by the beam splitter and a plane mirror to point $B_2$. For the sake of convenience, only the central rays of the light source are indicated here. In that way, the two central rays are separated by a distance $s_Q$ in front of a Fourier lens with focal length $f_F$. Because the refractive index $n_{\mathrm{bs}}$ of the beam splitter is considerably larger than that of the surrounding medium $n_s$, the optical path lengths from A to B1 and from A to B2 can be matched by a suitable choice of the distance $d_{\mathrm{bs}-\mathrm{pm}}$ between the beam splitter and plane mirror. Eventually, the Fourier lens collimates the wave fronts of the two interferometer arms onto its focal plane, where the resulting interference pattern is captured by a 2D microbolometer array.

 

Fig. 1. Design principle of a static single-mirror Fourier transform spectrometer [10].

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The interferogram $I(x,y)$ formed on the 2D detector array is then given by Eq. (1), where $S(\tilde{\nu })$ is the light source spectrum and $B(\tilde{\nu })$ is the wavenumber-dependent instrument response function, including optics and detector characteristics. The horizontal position along the detector array is denoted by $x$, the vertical position by $y$:

 

Fig. 2. Typical pattern of optical path differences generated by a single-mirror interferometer. The main path difference modulation occurs along the $x$ axis. As indicated, the signal can be averaged along the curved lines of equal optical path difference along the $y$ axis.

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B. Source-Doubling Principle and Characteristics of Fourier Lenses

The sSMFTS principle can also be illustrated, as shown in Fig. 3. In this virtual source model, the two interferometer arms are represented as two separate divergent light sources $V_1$ and $V_2$, emitting spherical wave fronts. A Fourier lens, typically made of germanium, then collimates and tilts the wave fronts onto a detector array and thus creates an interference pattern. This is often referred to as source-doubling principle. As indicated in this depiction, the horizontal separation of the virtual sources $s_Q(\lambda ,T)$ as well as the focal length of the Fourier lens $f_F(\lambda ,T)$ depend on both wavelength $\lambda$ and temperature $T$. Here, the former can be calculated using Eq. (4), with the variables being defined in Fig. 1. This approach assumes that the main effect on the distance between the virtual sources is the change in the refractive index of the beam splitter and that temperature-related expansions of the assembly and components can be neglected in a first approximation. Furthermore, it is assumed that the refractive index of the surrounding medium, in most cases air, exhibits minor changes with temperature and wavelength variations [10]:

 

Fig. 3. Temperature- and wavelength-dependent virtual source model of a static single-mirror Fourier transform spectrometer.

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In principle, two main effects can be observed in Fig. 4: different Michelson contrasts depending on wavenumber and temperature as well as a shift of the wavenumber axis in the spectrum. The former is mainly caused by the shift of the focal plane and may lead to quantitative measurement errors and a limitation of the spectral range. We illustrate this effect by simulating the ideal Michelson contrast over the relevant spectral and temperature range using the method from [12]. As can be seen in Fig. 4(a), the Michelson contrast decreases with the increasing wavenumber. Note that this effect becomes greater with shorter focal lengths, as indicated by dashed lines, as well as with larger light sources, and therefore somewhat limits the maximum spectral resolution and light throughput of the system [12]. Here, effects such as aberrations of the Fourier lens or detector characteristics are not considered, and an ideal linear interference pattern is assumed.

 

Fig. 4. (a) Simulation of the Michelson contrast for typical lens-based spectrometer configurations. Solid lines indicate a focal length of 75 mm, dashed lines a focal length of 40 mm. The radius of the light source is set to 1 mm. (b) Simulation of the wavenumber shift due to a changing distance between the virtual sources and a varying focal length of the Fourier lens.

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Another problem leading to a wrong assignment of the wavenumber axis in the spectrum is the varying linear sampling frequency ${\nu }_s$ of the interferogram. According to Eq. (7), it depends both on the distance of the virtual sources and on the focal length of the lens, leading to a shift of the wavenumber axis ${\tilde{\nu }}_{\mathrm{shift}}$, according to Eq. (8). The wavenumber shift for the spectrometer configuration described above is depicted in Fig. 4(b):

3. PROPOSED DESIGN

Due to the drawbacks of lenses as Fourier optics in static spectrometers described in the previous section, we propose a novel design of a broadband static Fourier transform spectrometer (bsFTS) using a single-mirror interferometer and a commercially available spherical mirror as Fourier optics. A schematic depiction of the setup can be found in Fig. 5.

 

Fig. 5. (a) Top view of the proposed bsFTS. (b) Side view of the proposed bsFTS.

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In analogy to Fig. 1, Fig. 5(a) shows a top view of the assembly. The convex lens is now replaced by a spherical mirror, which, like the lens, is on-axis in this dimension. In this way, the contrast along the $x$ axis or interferogram axis is not limited by a tilt of the spherical mirror. After being collimated by the spherical mirror, light is then directed upward and parallel to the $z$ axis by a plane fold mirror and finally interferes on the detector array. The side view in Fig. 5(b) shows that the spherical mirror is tilted slightly upward around the $x$ axis by an angle $\mathrm{\varphi }$, which should ideally be kept as small as possible to achieve a sufficient contrast along the $y$ axis or averaging axis and therefore a high SNR without time averaging. The fold mirror is eventually tilted by an angle of $\alpha =90°-\mathrm{\varphi }$ around the optical axis so that the detector array can be placed in parallel to the $x-y$ plane.

In theory, this configuration eliminates many of the previously mentioned disadvantages of the Fourier lens setup. First, the focal plane of the spherical mirror is constant over a wide temperature range; second, it is completely independent of the wavenumber. Therefore, ideally there is no loss of Michelson contrast. In addition, the bandwidth of the system is no longer limited by the transmittivity of the lens but only by the characteristics of the beam splitter and the used detector. Figure 6(a) shows the wavenumber shift of this system. The only effect to consider here is the varying distance between the virtual sources $s_Q(\lambda ,T)$, according to Eq. (4). The resulting wavenumber shift is small compared with Fig. 4(b) and can be corrected algorithmically without problems. In addition, when using a ZnSe beam splitter, there is no noticeable temperature dependence.

 

Fig. 6. (a) Simulation of the wavenumber shift due to a changing distance between the virtual sources and a fixed focal length of the spherical mirror. (b) Simulation of the standard deviation of optical path differences imaged onto the 2D detector by a tilted spherical mirror. The off-axis angle is 16° to the $y$ axis.

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In practice, problems may arise due to expansion of the components and temperature-dependent distances between them. This should be manageable by a proper mechanical design and will be investigated in future work. One of the main problems of using a tilted concave mirror, however, lies in the aberrations that have not been considered up to now. When using an off-axis spherical mirror as in this setup, imaging quality in the tilting direction deteriorates. This leads to an increasingly imprecise allocation of optical path differences on the 2D detector and thus to a loss of contrast in this direction. This effect increases with increasing the off-axis angle $\mathrm{\varphi }$. Figure 6(b) shows the results of a ray-tracing simulation of the proposed setup with an off-axis angle of 16°. The OPD standard deviation increases along the averaging axis with an optimum at around one-third of the detector height and is relatively constant along the interferogram axis. As a result, in practice it may not be possible to average over the entire $y$ dimension of the detector, but a certain region has to be selected. As in the experimental setup presented in Section 4, the detector dimensions amount to $13.6\mathrm{mm}\times 10.2\mathrm{mm}$, and the focal length of the spherical mirror is set to 75 mm. The light source radius is again set to 1 mm, which approximately corresponds to the dimension of the real light source used in the experimental setup. While a spherical mirror is inexpensive and still provides sufficient results in this configuration, as can be seen from the measurement results in Section 5, it may still be interesting for the future to replace it with an optimized free-form mirror to compensate for the aberrations described above. In particular, a mirror comprising spherical and parabolic components might prove beneficial.

4. EXPERIMENTAL SETUP

Based on the theoretical considerations in Section 3, we set up a prototype of the bsFTS for transmission measurements using a spherical mirror as a Fourier element. A ray-tracing model of the assembly is shown in Fig. 7. A light source is collimated by an off-axis parabolic mirror ${\mathrm{OAP}}_1$ with a focal length of 50.8 mm and an off-axis angle of 45°. After passing a long-pass filter and the sample, light is focused again by ${\mathrm{OAP}}_2$, having a focal length of 101.6 mm and an off-axis angle of 90°. ${\mathrm{OAP}}_2$ is positioned in such a way that, contrary to Figs. 1 and 5, the focal point and thus the extended light source are created behind the beam splitter and exactly below the lowest point of the fold mirror. In this way, we are able to place the fold mirror close above the optical axis and thus keep the off-axis angle of the spherical mirror relatively small at 16°. The beam splitter is made of zinc selenide with a thickness of 3.1 mm and a diameter of 25 mm at a parallelism of less than 10 arc seconds. The plane mirror has a diameter of 12.7 mm.

 

Fig. 7. Ray-tracing model of the experimental transmission measurement setup.

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After hitting the spherical concave mirror with a focal length of 75 mm and a diameter of 25 mm, light is eventually guided to the detector array by an additional fold mirror with a diameter of 20 mm. All mirrors are coated with protected gold. For the detector, we use the microbolometer array UV840A by GUIDE Infrared with $800\mathrm{pix}\times 600\mathrm{pix}$ and a customized broadband transparent germanium window. The pixel pitch amounts to 17 μm; therefore, the detection area is $13.6\mathrm{mm}\times 10.2\mathrm{mm}$. In combination with a long-pass filter having a cut-in wavenumber of about $2770{\mathrm{cm}}^{-1}$, the pixel pitch of the camera and the distance of the virtual sources of about 6.6 mm, the selected focal length of the spherical mirror satisfies the Nyquist criterion, according to Eq. (3). In principle, it would also be possible to set up the prototype without a fold mirror by using a beam splitter with a smaller diameter and then placing the detector array perpendicular to the optical axis and directly above the beam splitter. However, adjusting and setting up the system with a fold mirror is somewhat easier because the detector array can simply be placed in parallel to the $x-y$ plane.

Figure 8 shows the corresponding laboratory prototype used to perform the experiments in the course of this paper. Most mounts were fabricated using a 3D printer, with Thorlabs components used for adjustment. For a light source, we use a broadband silicon carbide thermal emitter (Hawkeye Technologies IR-Si207); further, the off-axis parabolic mirrors and the spherical mirror can be adjusted with appropriate mounts (Thorlabs K6XS, Thorlabs POLARIS-K1 and Thorlabs KMSS/M), while the long-pass filter, beam splitter, and fold mirror are fixed. The decisive component for calibration is the plane mirror, which can be tilted by a Thorlabs KM05FL/M mount to ensure that the two central beams are aligned parallel to each other, as well as moved with a LINOS translation stage to allow the distance $d_{\mathrm{bs}-\mathrm{pm}}$ to be varied. It should be noted that the height of the setup is only adapted to other laboratory components and could easily be reduced.

 

Fig. 8. Photo of the current laboratory prototype used to carry out the experiments in this paper.

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While this design is sufficient to verify the principle and test the system for bandwidth, spectral resolution, and signal-to-noise ratio in the course of this paper, the setup will be modified for future investigations. For example, the 3D printed parts should be replaced by materials with a low coefficient of thermal expansion to allow a verification of the long-term and temperature stability of the spectrometer.

5. MEASUREMENT RESULTS

For validation of the concept presented here, we use the above described transmission measurement setup to record background and transmission spectra. To verify the wavenumber accuracy of the setup, we measure a 3 ml polystyrene calibration foil with the broadband static Fourier spectrometer as well as with a classical laboratory FTIR instrument as a reference. First, we record the background spectrum of the system. The corresponding 2D detector image is shown in Fig. 9(a). In order to see the interference pattern more clearly, the image was high-pass filtered and zoomed to the midpeak area.

 

Fig. 9. (a) High-pass-filtered detector image of the background interference pattern recorded with the presented setup. Zoomed to the interferogram midpeak. (b) High-pass-filtered apodized 1D background interferogram averaged over the highlighted 350 pixels in the $y$ direction. The signal is normalized to the maximum peak.

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As expected from the simulation of the OPD standard deviation in Fig. 6(b), an area of about 350 pixels as highlighted can be identified where the contrast of the interferogram is quite high. Outside this region, the contrast decreases in both directions. This means that the achievable SNR is strongly dependent on the area of the detector image being evaluated. If an area too small or too large around the highest contrast is evaluated, either too few detector rows are available for averaging, or many detector rows with low contrast are considered, both resulting in low SNR values. The former can be compensated by averaging over time, leading to a trade-off between SNR and measurement speed. Therefore, as mentioned earlier, it could be interesting to replace the spherical mirror with a customized mirror to further improve the system performance.

For obtaining the 1D interferogram, as shown in Fig. 9(b), the detector image is averaged along the lines of equal optical path difference in the highlighted detector area, and a triangular window is applied for apodization. The signal is normalized to the maximum peak. It is apparent that the midpeak was shifted to one side of the detector array by adjusting the distance $d_{\mathrm{bs}-\mathrm{pm}}$ between the beam splitter and plane mirror accordingly. This leads to a greater maximum OPD in the interferogram and thus to a higher spectral resolution, according to Eq. (2). Here, ${\mathrm{OPD}}_{\max }$ amounts to about 0.9 mm, which corresponds to a spectral resolution of about $12{\mathrm{cm}}^{-1}$.

In analogy, Fig. 10 shows the detector image and the averaged interferogram of the polystyrene sample. For allowing a better comparison with the background measurement, the signal in Fig. 10(b) is again normalized to the maximum peak of the background interferogram. As expected, the midpeak of the sample interferogram is clearly attenuated and the envelope of the signal is slightly broadened because the polystyrene absorbs a significant part of the incoming radiation.

 

Fig. 10. (a) High-pass-filtered detector image of the polystyrene interference pattern recorded with the presented setup. Zoomed to the interferogram midpeak. (b) High-pass-filtered apodized 1D polystyrene interferogram averaged over the highlighted 350 pixels in the $y$ direction. The signal is normalized to the maximum peak of the background interferogram.

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As described in great detail in [10], the interferogram is sampled nonlinearly in a single-mirror interferometer. Therefore, we use a nonuniform Fourier transform to obtain the spectra. The resulting background spectrum is shown in Fig. 11(a). Its shape is mainly determined by three influence factors. The upper end of the wavenumber range is defined by the transmittivity of the long-pass filter, as shown in Fig. 11(a). This filter cuts in at $2780{\mathrm{cm}}^{-1}$ and also causes the signal drops around $1225{\mathrm{cm}}^{-1}$ and $900{\mathrm{cm}}^{-1}$ due to poor transmission properties. The significant signal drop between $2375{\mathrm{cm}}^{-1}$ and $1800{\mathrm{cm}}^{-1}$ is caused by the microbolometer transfer function. This is determined, on the one hand, by the typical behavior of a 2.5 μm optical cavity [18] and, on the other hand, by the integrated germanium window. At the lower end of the wavenumber range, all three components show poor transmittance. The main factor here is the beam splitter, which is not coated for wavenumbers below $650{\mathrm{cm}}^{-1}$. Apparently, the performance of the spectrometer could be significantly improved by using customized parts for this specific spectral range. Furthermore, characteristic absorption peaks caused by ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2\mathrm{O}$ molecules can be observed in the spectrum. Figure 11(b) contains the single-shot background signal-to-noise ratio of the broadband sFTS. Therefore, we record 500 consecutive background spectra and calculate the noise as the standard deviation of the signal. The SNR curve roughly follows the spectral intensity curve discussed above with maxima around $1350{\mathrm{cm}}^{-1}$ and $2560{\mathrm{cm}}^{-1}$.

 

Fig. 11. (a) Measured background spectrum using the presented prototype as well as transmittance curves of the long-pass filter, beam splitter, and microbolometer array influencing the background envelope. (b) Background signal-to-noise ratio of the proposed prototype averaging over 350 pixels in the $y$ direction without time averaging.

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Finally, as a verification of the wavenumber accuracy of the bsFTS, we calculate the transmission spectrum of the 3 ml polystyrene calibration foil and compare it with a classical FTIR laboratory device (Thermo Fisher—Avatar 300). The corresponding spectra are shown in Fig. 12. Here, the bsFTS spectrum was wavenumber-corrected with the results from Fig. 6(a). It can be clearly seen that the positions of the characteristic spectral bands of the calibration foil are correctly allocated by the bsFTS. Because the laboratory device is set to a higher spectral resolution of $8{\mathrm{cm}}^{-1}$, the absorption peaks of the bsFTS appear less pronounced. The slight offset of the transmittance in the area from $2100{\mathrm{cm}}^{-1}$ to $2800{\mathrm{cm}}^{-1}$ can be caused by different angles of penetration in the two measurement systems. The probe is placed in the focal point in the reference FTIR, whereas it is passed collimated in the bsFTS.

 

Fig. 12. Comparison of the proposed broadband static Fourier transform spectrometer (bsFTS) with a scanning FTIR by measuring a 3 mil polystyrene calibration standard.

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6. SUMMARY AND OUTLOOK

In this paper, we presented a novel broadband static Fourier transform spectrometer for the mid-infrared spectral range using a spherical concave mirror as Fourier element. The design is based on a single-mirror interferometer and therefore works independently of the light source size, allowing high light throughput and high signal-to-noise ratios. Because no lenses are implied in this spectrometer, it can be used over a wide spectral range and, with a suitable mechanical design, in principle shows no temperature or wavelength dependence. As only optical standard components and an uncooled microbolometer array are employed, the spectrometer can be designed at a reasonable cost. After giving an overview of the basic principle of static single-mirror Fourier transform spectrometers and the source-doubling principle in general, we carried out simulations to investigate the system behavior with different Fourier optics. Whereas a lens suffers from a varying refractive index and therefore causes different contrasts and sampling frequencies in the interferogram, a spherical mirror is comparably stable. Also, we investigated the effects of a tilted mirror, which cause a loss of contrast in the tilt direction. Based on these considerations, we devise an experimental setup using a spherical mirror tilted around the averaging axis, therefore maintaining the contrast along the interferogram axis. The setup was evaluated in a spectral range from $2800{\mathrm{cm}}^{-1}$ to $600{\mathrm{cm}}^{-1}$ at a spectral resolution of $12{\mathrm{cm}}^{-1}$. Measurements showed a high single-shot SNR as well as sufficient wavenumber accuracy compared with a laboratory FTIR spectrometer. Future work will include setting up and evaluating a temperature-stable prototype, using customized optical elements to improve the spectrometer efficiency as well as increase the spectral resolution. One possible future application of the system is the use for environmental gas sensing in combination with a scanning FTIR spectrometer, as described in [19].