## Abstract

Fourier transform spectroscopy has established itself as the standard method for spectral analysis of infrared light. Here we present a robust and compact novel static Fourier transform spectrometer design without any moving parts. The design is well suited for measurements in the infrared as it works with extended light sources independent of their size. The design is experimentally evaluated in the mid-infrared wavelength region between 7.2 μm and 16 μm. Due to its large etendue, its low internal light loss, and its static design it enables high speed spectral analysis in the mid-infrared.

© 2016 Optical Society of America

## 1. Introduction

Due to its high signal-to-noise ratio and its excellent spectral resolution, Fourier transform infrared (FTIR) spectroscopy has established itself as the standard method for spectral analysis of infrared light. According to this method, an interferogram is created by two beam interference that is then Fourier transformed. Most FTIR spectrometers used today are based on the Michelson interferometer design such that this interferogram is provided in the temporal domain.

Static Fourier transform spectrometers (sFTS) generate interferograms in the spatial domain, where they are detected by a detector array. As no moving parts are required, they are more robust and less complex than their counterparts working in the temporal domain. Their signal-to-noise ratio (SNR), spectral resolution and bandwidth, however, are inferior due to the use of detector arrays rather than single detectors. Various concepts of sFTS have been reported in the last years including double-mirror [1], modified Mach–Zehnder [2], spatially modulated prism [3], static Michelson, and static Mach–Zehnder interferometers [4]. The most relevant ones are the common-path [5, 6] and birefringent sFTS [7, 8].

In this paper, we present a new design for a static Fourier transform spectrometer adapted for an operation in the mid-infrared wavelength region. As it works with extended light sources independent of their size it features a large etendue. In addition it shows low internal light loss. Because it uses a novel single-mirror interferometer design, it is referred to as a static single-mirror Fourier transform spectrometer (sSMFTS).

## 2. Design principles

Figure 1 shows an overview of the proposed design in which only the focal rays are indicated.

The beam originating from the light source is divided at *A* by a beam splitter of thickness *T*_{bs}. The transmitted beam travels to *B*_{1}, whereas the reflected beam is reflected once more at the mirror to *B*_{2}. Thereby, both focal rays passing *B*_{1} and *B*_{2} must be aligned parallel to one another, while *B*_{1} and *B*_{2} are positioned at the same horizontal distance *d _{h}* from the convex lens. The length of the focal ray passing

*B*

_{2}between the light source and the convex lens is equal to the focal length

*f*of the convex lens. Since the light source emits divergent light, the convex lens collimates both beams onto its focal plane where the detector array detects their interference.

The key element in this design is a beam splitter with a high refractive index *n*_{bs} compared to the refractive index of the surrounding medium *n _{s}*. Hence, the beam transmitted by the beam splitter experiences a difference between its geometric and optical path. The distance

*d*

_{bs-m}between the mirror and the beam splitter has to be set such that the paths of the focal rays from

*A*to

*B*

_{1}and from

*A*to

*B*

_{2}have the same optical path length. Using Snells law with an incidence angle of 45 degrees, the distance

*d*

_{bs-m}between the beam splitter and mirror can be calculated with Eq. (1) and the separation

*s*between the two focal rays can be calculated using Eq. (2):

For *d*_{bs-m} in Eq. (1), the optical path differences of both interferometer arms are equal such that the interferogram is centered on the detector. By increasing *d*_{bs-m}, the central peak of the interferogram can be shifted to one side of the detector, resulting in a single-sided interferogram.

As all optical components including the beam splitter are passed just once by both beams, the sSMFTS design shows low internal light loss. In addition to that, adjustment is simple since the mirror just needs to be positioned correctly in order to accomplish the required parallel alignment of both interferometer arms.

An optical equivalent of the proposed design is shown in Fig. 2. Here, two wavefronts generated by the two virtual sources *S*_{1} and *S*_{2} represent the two beams split by the beam splitter. The detector array is placed in the focal plane of the lens as well as the virtual source *S*_{1}. The other virtual source *S*_{2} cannot be placed in the focal plane because both interferometer arms have the same optical path length but different geometric path lengths. This horizontal shift varies with the observed axis due to the astigmatism induced by the tilted beam splitter and is therefore called Δ*f*_{y/z}.

Due to the position of the detector array in the focal plane of the convex lens, the interference on the detector only depends on the angle of incidence of the rays onto the lens. This so-called source-doubling interferometer configuration is explained in detail by the example of the common-path sFTS in [2]. It guarantees a high visibility of the interferogram independent of the size and shape of the light source used and therefore a large etendue. Thus, even an imperfect collimation of the diverging light source by the convex lens does not degrade the interferogram.

As there are several other source-doubling interferometer designs reported, we now give a short comparison between them and the here proposed sSMFTS. Common-path and birefringent interferometers show a minimum of 50 percent internal light loss, whereas modified Mach–Zehnder, spatially modulated prism, and the here proposed static single-mirror interferometers experience no internal light loss at all. Of these source-doubling designs without light loss, the modified Mach–Zehnder interferometer is more difficult to align and to miniaturize than the other two. As the spatially modulated prism interferometer compensates for nonlinear optical effects, it has a more complex design than the sSMFTS.

The main modulation of the optical path difference (OPD) along the y-axis of the detector array is caused by the tilt between the wavefronts generated by the lens. This modulation Δ*x*(*y*, *z*)_{lin} at point (*y*, *z*) is linear and can be calculated by means of small-angle approximation using Eq. (3):

Due to the non-zero horizontal position shift Δ*f*_{y/z}, both wavefronts cannot be made ideal flat by the lens. Therefore, the linear modulation of the OPD Δ*x*(*y*, *z*)_{lin} is superposed by the nonlinear modulation Δ*x*(*y*, *z*)_{nonlin}. The resulting optical path difference Δ*x*(*y*, *z*) in the focal plane of the convex lens is given in Eq. (4):

The nonlinear OPD modulation Δ*x*(*y*, *z*)_{nonlin} is calculated by subtracting the wavefront of *S*_{1} from *S*_{2} while ignoring the linear tilt effect. Figure 3(a) and (b) shows the contours of Δ*x*(*y*, *z*)_{nonlin} and Δ*x*(*y*, *z*), respectively, as multiples of the corresponding constant sampling interval Δ*x _{s}* generated by the linear tilt effect. This constant sampling interval can be determined by Δ

*x*= Δ

_{s}*x*(

*p*,

*z*)

_{lin}with Eq. (3) and the detector pixel pitch

*p*. Both OPD data are hereby based on the characteristics of the prototype specified in Section 3.

As these nonlinear influences have an extent of ten times the constant sampling interval on both axes, they have to be addressed. The nonlinear distribution of the data points with the same OPD on the z-axis requires an averaging along the curves shown in Fig. 3(b). On the y-axis, the nonlinear effect induces non-uniformly distributed OPD intervals for the Fourier transformation. Hence, the spectrum is calculated by a non-uniform discrete Fourier transformation algorithm [9, 10].

Another effect that must be considered in the proposed sSMFTS design is the wavenumber-dependence of the refractive index, which influences all characteristics based on this index, including the separation *s* between the virtual sources and the focal length *f*. According to Eqs. (3) and (4), the OPD at the detector and thereby the sampling frequency *v _{s}* of the detector are also wavelength-dependent. Because the linear spacing in the wavenumber domain after the discrete Fourier transformation depends normally on the sampling frequency at the design wavenumber

*v*

_{s,νdesign}, every other wavenumber is incorrectly assigned.

Therefore, we create a lookup table to assign the correct wavenumber to every sample point in the wavenumber domain. We did this by calculating the sampling frequencies *v*_{s,ν} of the prototype specified in Section 3 for every wavenumber using the refractive index data for Zinc Selenide (ZnSe) [11, 12] and Germanium [13]. Equation (5) returns the factor *γ _{ν}* by which the sampling frequency at every wavenumber

*v*

_{s,ν}is stretched in respect to the sampling frequency at the design wavenumber

*v*

_{s,νdesign}:

The wavenumber shift Δ*ν*_{shift} between the incorrectly linearly distributed wavenumbers evaluated at the design wavenumber *ν*_{lin} and the correct wavenumbers *ν*_{correct} can then be calculated using Eq. (6):

The resulting nonlinear wavelength shift Δ*ν*_{shift} is shown in Fig. 4 with the design wavenumber of 943 cm^{−1}. This wavenumber corresponds to the design wavelength of 10.6 μm for which all infrared optics used here are designed.

Equation (7) gives the spectral resolution Δ*ν̃* of the proposed sSMFTS design as the inverse of the maximum OPD Δ*x _{max}* included in the interferogram [14]. The impact of different design choices like the thickness of the beam splitter on the OPD and therefore on the resolution can be determined using Eqs. (2), (3), and (4). Since the sSMFTS experiences a nonlinear wavenumber shift according to Eq. (6), the calculated resolution Δ

*ν̃*needs to be corrected for this shift.

## 3. System configuration

To evaluate the sSMFTS design proposed in this paper, we set up a prototype for spectral transmission measurements. An overview of the system configuration used is shown in Fig. 5.

The beam of an extended silicon nitride broadband infrared light source at a temperature of 1400 K is collimated by the convex ZnSe lens *L _{c}*. It is then directed through the optical measurement cell onto the convex lens

*L*

_{1}, which creates the divergent light source shown in Fig. 1 at its focal point. The position of the focal point is shifted beyond the beam splitter such that two focal points

*C*

_{1}and

*C*

_{2}are generated. With the optimal positions for

*C*

_{1}and

*C*

_{2}, the acceptance angle

*θ*and the diameter of the focal point

*D*

_{fp}are maximized for the highest etendue.

The ZnSe beam splitter used has a diameter of 25.4 mm and a thickness of 3.1 mm. The focal length and the diameter of the convex lens *L*_{1} are 100 mm and 25.4 mm, respectively; the correspondent values of the convex lens *L*_{2} tilting the wavefronts are 40 mm and 25.4 mm, respectively. The lens *L*_{1} is made of ZnSe, whereas *L*_{2} is made of Germanium. This material selection was based on market availability. The mirror is coated with protected gold and has a diameter of 12.7 mm.

As a detector we used the microbolometer array core FLIR Tau2 336 with 336px × 256px and a pixel pitch *p* of 17 μm, which leads to a detection area of 5.7mm × 4.4mm. The focal length of *L*_{1} is chosen such that a fully illuminated diameter of *L*_{1} corresponds to a spot diameter of 10.2 mm on the detector plane, which results in a full illumination of the detector array. The constant sampling interval of the detector at the design wavelength of 10.6 μm can be calculated as Δ*x _{s}* = 2.8μm. Because the lowest detectable wavelength of the detector is around 7.2 μm, the chosen combination of

*L*

_{2}and the detector satisfies the Nyquist criterion without the need for an optical long-pass filter. Due to export regulations, the maximum frame rate of the detector is cut to 8.3 Hz in contrast to a full frame rate of 60 Hz.

## 4. Experimental results

To demonstrate the performance of the proposed sSMFTS design, we use the above-described system configuration to measure the transmission spectrum of a 1.5 MIL polystyrene standard. Therefore, we first take the background spectrum of the light source; afterwards, we take the probe spectrum by inserting the polystyrene standard in the measurement cell.

By analyzing the background interference pattern in Fig. 6(a), its curved characteristic along the z-axis of the detector, and therefore the nonlinear OPD modulation, can clearly be seen. For a better view, the detector image is zoomed to the central peak of the interferogram, which is shifted to the side of the detector array. This is caused by different optical path lengths in both interferometer arms created by an intended increase of the distance between the beam splitter and mirror *d _{bs−m}*.

Taking the average value for each sampled OPD along the curves specified in Eq. (4) reduces the noise of the background interferogram shown in Fig. 6(b). The shift of the central peak results in a single-sided interferogram, which leads to higher maximum optical path differences and therefore according to Eq. (7) to higher wavenumber-resolution of the spectrum.

In contrast to other interferometer concepts, the beam splitter of the sSMFTS induces significant wavenumber dependent phase shifts to the interference pattern. This results in interferograms with broadened central peaks and therefore less modulation depth. The non-uniform direct component of the interferogram is caused by misalignment of the prototype since we are not able to control the position of all optical components in every direction with the current setup.

Before the non-uniform discrete Fourier transformation, we normalize the interferogram and apply a triangular window for apodization. For interpolation of the spectrum, we use zero-filling. The magnitude spectra of the background and the probe can be seen in Fig. 7(a), in which all the wavenumbers are corrected according to Eq. (6). Here, the boundaries of the spectral response curve for the evaluated sSMFTS prototype can be located around 1390 cm^{−1} and 625 cm^{−1}. The maximum wavenumber corresponds essentially to the lowest detectable wavelength of 7.2 μm specified by FLIR. As the above-mentioned non-uniform direct component of the interferogram is Fourier transformed outside the response curve of the detector, it has no effect on the shown spectrum.

According to Eq. (7), the spectral resolution of the evaluated prototype can be calculated using the maximum OPD included in the interferogram. An approximation of the expected maximum OPD gives the linear OPD model with Eq. (3). As we place the central peak of the interferogram at around 20 % of the detector length, the maximum linear OPD is expected to be 0.77 mm with the characteristics described in Section 3. The additional nonlinear effect increases the maximum OPD to 0.81 mm, as can be seen in Fig. 6(b). This corresponds to a spectral resolution of 12 cm^{−1} for the evaluated prototype.

The SNR levels of the background spectrum are calculated using 1000 consecutive spectral measurements without time averaging. As can be seen in Fig. 7(b), they approximately follow the spectral response curve of the detector. The measurement frequency is set hereby to the maximum of 8.3 Hz. The use of the full frame rate model instead of the regulated one in the evaluated prototype would show the same SNR levels, but with a 60 Hz frame rate, since the regulated model does not do any internal time averaging of the detector signal. Both models have the same thermal time constant and therefore the same noise characteristics.

For further characterization of the proposed sSMFTS, we measure the reference transmission spectrum of the polystyrene film with a traditional FTIR spectrometer, the Avatar 330 spectrometer by Thermo Fisher. Figure 8 shows the measured transmission spectrum of the polystyrene film by the sSMFTS against the reference FTIR spectrum.

In terms of wavenumber accuracy, it can be seen that the evaluated sSMFTS prototype provides the characteristic peaks of the polystyrene film at the same wavenumbers as the FTIR reference spectrometer. The sSMFTS prototype has, at 12 cm^{−1}, a lower spectral resolution than the reference FTIR spectrometer (0.5 cm^{−1}). This results in the broadened peaks of the sSMFTS spectrum.

## 5. Conclusion

In this publication we proposed a novel static Fourier transform spectrometer design which operates independently of the size and shape of the light source used. It shows low internal light loss and can be adjusted easily. The nonlinear interference effects on the detector plane induced by this design were addressed and algorithmically corrected.

Due to its high etendue, the proposed static single-mirror Fourier transform spectrometer design can be used in the mid-infrared. It was successfully evaluated with a broadband light source between 1390 cm^{−1} and 625 cm^{−1}. This spectral bandwidth is given by the spectral response curve of the used microbolometer detector array and covers the so-called fingerprint region in the infrared spectrum.

As the infrared fingerprint region of many lubricants contains information about their degradation, the proposed spectrometer could for example replace dispersive instruments now used for online oil condition monitoring [15]. With its robust design it is also well suited for condition monitoring in hazardous environments.

The evaluated prototype shows a spectral resolution of 12 cm^{−1} and, tested with a polystyrene standard, a good wavenumber accuracy. The spectral resolution is limited by the number of pixels of the used detector array, which can be increased by currently available detector arrays having higher numbers of pixels. With customized lenses and beam splitters, the spectral resolution, as well as the SNR, could be further improved.

Due to its high SNR, the proposed spectrometer design can be used for transmission, reflection, and fiber-coupled spectroscopy. In addition, the high SNR without averaging the detector images over time enables high speed spectroscopy up to 60 Hz using commercially available components.

## Acknowledgments

The authors gratefully acknowledge the funding by the Federal Ministry for Economic Affairs and Energy of Germany and the contributions from our project partner Comline Elektronik Elektrotechnik GmbH. This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

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