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.

 figure: Fig. 1

Fig. 1 Schematic illustration of the proposed static single-mirror Fourier transform spectrometer design.

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The beam originating from the light source is divided at A by a beam splitter of thickness Tbs. The transmitted beam travels to B1, whereas the reflected beam is reflected once more at the mirror to B2. Thereby, both focal rays passing B1 and B2 must be aligned parallel to one another, while B1 and B2 are positioned at the same horizontal distance dh from the convex lens. The length of the focal ray passing B2 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 nbs compared to the refractive index of the surrounding medium ns. Hence, the beam transmitted by the beam splitter experiences a difference between its geometric and optical path. The distance dbs-m between the mirror and the beam splitter has to be set such that the paths of the focal rays from A to B1 and from A to B2 have the same optical path length. Using Snells law with an incidence angle of 45 degrees, the distance dbs-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):

dbs-m=Tbs(nbs+nssin(π4arcsin(nsnbs2))1(nsnbs2)ns.2);
s=dbs-m+Tbs(nssin(π4arcsin(nsnbs2))1(nsnbs2)2).

For dbs-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 dbs-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 S1 and S2 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 S1. The other virtual source S2 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 Δfy/z.

 figure: Fig. 2

Fig. 2 Virtual sources model of the proposed static single-mirror Fourier transform spectrometer design. The ellipse in the schematic drawing of the virtual source S2 illustrates the astigmatism induced by the tilted beam splitter.

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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 Δx(y, z)lin at point (y, z) is linear and can be calculated by means of small-angle approximation using Eq. (3):

Δx(y,z)lin=sfy.

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

Δx(y,z)=Δx(y,z)lin+Δx(y,z)nonlin.

The nonlinear OPD modulation Δx(y, z)nonlin is calculated by subtracting the wavefront of S1 from S2 while ignoring the linear tilt effect. Figure 3(a) and (b) shows the contours of Δx(y, z)nonlin and Δx(y, z), respectively, as multiples of the corresponding constant sampling interval Δxs generated by the linear tilt effect. This constant sampling interval can be determined by Δxs = Δx(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.

 figure: Fig. 3

Fig. 3 Simulation of the optical path differences at the detector. The contours of the nonlinear influences and the resulting OPD are calculated with parameters matching the characteristics of the prototype specified in Section 3.

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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 vs 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 vs,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 vs,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 γν by which the sampling frequency at every wavenumber vs,ν is stretched in respect to the sampling frequency at the design wavenumber vs,νdesign:

γν=vs,νvs,νdesign

The wavenumber shift Δνshift between the incorrectly linearly distributed wavenumbers evaluated at the design wavenumber νlin and the correct wavenumbers νcorrect can then be calculated using Eq. (6):

Δνshift=νcorrectνlin=νlin(γν1).

The resulting nonlinear wavelength shift Δνshift 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.

 figure: Fig. 4

Fig. 4 Calculated nonlinear wavenumber shift Δνshift between the incorrectly linearly distributed wavenumbers and the correct wavenumbers due to the wavenumber-dependence of the refractive index.

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Equation (7) gives the spectral resolution Δν̃ of the proposed sSMFTS design as the inverse of the maximum OPD Δxmax 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 Δν̃ needs to be corrected for this shift.

Δν˜=1Δxmax.

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.

 figure: Fig. 5

Fig. 5 Overview of the system configuration used to evaluate the proposed static single-mirror Fourier transform spectrometer design. The use of an extended light source results in focal points with a non-zero diameter Dfp.

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The beam of an extended silicon nitride broadband infrared light source at a temperature of 1400 K is collimated by the convex ZnSe lens Lc. It is then directed through the optical measurement cell onto the convex lens L1, 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 C1 and C2 are generated. With the optimal positions for C1 and C2, the acceptance angle θ and the diameter of the focal point Dfp 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 L1 are 100 mm and 25.4 mm, respectively; the correspondent values of the convex lens L2 tilting the wavefronts are 40 mm and 25.4 mm, respectively. The lens L1 is made of ZnSe, whereas L2 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 L1 is chosen such that a fully illuminated diameter of L1 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 Δxs = 2.8μm. Because the lowest detectable wavelength of the detector is around 7.2 μm, the chosen combination of L2 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 dbs−m.

 figure: Fig. 6

Fig. 6 Detector image of the background interference pattern and the corresponding interferogram. The detector image is a close-up of the central peak of the interference pattern.

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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.

 figure: Fig. 7

Fig. 7 Background and probe spectra as well as the corresponding background SNR. The SNR was calculated at a measurement frequency of 8.3 Hz.

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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.

 figure: Fig. 8

Fig. 8 Comparison of the evaluated static single-mirror Fourier transform spectrometer prototype with a traditional FTIR spectrometer using a polystyrene standard.

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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.

References and links

1. D. G. Winters, P. Schlup, and R. A. Bartels, “Highly achromatic Fourier-transform spectrometer,” Opt. Express 15, 1361–1368 (2007). [CrossRef]   [PubMed]  

2. M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8, 1457–1462 (1991). [CrossRef]  

3. F. M. Reininger, “The application of large format, broadband quantum well infrared photodetector arrays to spatially modulated prism interferometers,” Infrared Physics & Technology 42, 345–362 (2001). [CrossRef]  

4. K. Möller, “Wave-front-dividing array interferometers without moving parts for real-time spectroscopy from the IR to the UV,” Appl. Opt. 34, 1493–1501 (1995). [CrossRef]   [PubMed]  

5. G. Zhan, “Static Fourier-transform spectrometer with spherical reflectors,” Appl. Opt. 41, 560–563 (2002). [CrossRef]   [PubMed]  

6. H. Mortimer, “Compact interferometer spectrometer,” US Patent US20120281223 (2012).

7. M. Hashimoto and S. Kawata, “Multichannel Fourier-transform infrared spectrometer,” Appl. Opt. 31, 6096–6101 (1992). [CrossRef]   [PubMed]  

8. M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and E. L. Dereniak, “Spatial heterodyne interferometry with polarization gratings,” Opt. Lett. 37, 4413–4415 (2012). [CrossRef]   [PubMed]  

9. L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004). [CrossRef]  

10. J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005). [CrossRef]  

11. J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979). [CrossRef]  

12. B. Tatian, “Fitting refractive-index data with the sellmeier dispersion formula,” Appl. Opt. 23, 4477–4485 (1984). [CrossRef]   [PubMed]  

13. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980). [CrossRef]  

14. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007). [CrossRef]  

15. B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

References

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  1. D. G. Winters, P. Schlup, and R. A. Bartels, “Highly achromatic Fourier-transform spectrometer,” Opt. Express 15, 1361–1368 (2007).
    [Crossref] [PubMed]
  2. M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8, 1457–1462 (1991).
    [Crossref]
  3. F. M. Reininger, “The application of large format, broadband quantum well infrared photodetector arrays to spatially modulated prism interferometers,” Infrared Physics & Technology 42, 345–362 (2001).
    [Crossref]
  4. K. Möller, “Wave-front-dividing array interferometers without moving parts for real-time spectroscopy from the IR to the UV,” Appl. Opt. 34, 1493–1501 (1995).
    [Crossref] [PubMed]
  5. G. Zhan, “Static Fourier-transform spectrometer with spherical reflectors,” Appl. Opt. 41, 560–563 (2002).
    [Crossref] [PubMed]
  6. H. Mortimer, “Compact interferometer spectrometer,” US Patent US20120281223 (2012).
  7. M. Hashimoto and S. Kawata, “Multichannel Fourier-transform infrared spectrometer,” Appl. Opt. 31, 6096–6101 (1992).
    [Crossref] [PubMed]
  8. M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and E. L. Dereniak, “Spatial heterodyne interferometry with polarization gratings,” Opt. Lett. 37, 4413–4415 (2012).
    [Crossref] [PubMed]
  9. L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004).
    [Crossref]
  10. J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005).
    [Crossref]
  11. J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
    [Crossref]
  12. B. Tatian, “Fitting refractive-index data with the sellmeier dispersion formula,” Appl. Opt. 23, 4477–4485 (1984).
    [Crossref] [PubMed]
  13. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980).
    [Crossref]
  14. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007).
    [Crossref]
  15. B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

2012 (1)

2007 (1)

2005 (1)

J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005).
[Crossref]

2004 (1)

L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004).
[Crossref]

2002 (1)

2001 (1)

F. M. Reininger, “The application of large format, broadband quantum well infrared photodetector arrays to spatially modulated prism interferometers,” Infrared Physics & Technology 42, 345–362 (2001).
[Crossref]

1995 (1)

1992 (1)

1991 (1)

1984 (1)

1980 (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980).
[Crossref]

1979 (1)

J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
[Crossref]

Bartels, R. A.

Connolly, J.

J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
[Crossref]

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007).
[Crossref]

Dereniak, E. L.

DiBenedetto, B.

J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
[Crossref]

Donadio, R.

J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
[Crossref]

Dorigo, D. G.

B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

Escuti, M. J.

Greengard, L.

J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005).
[Crossref]

L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004).
[Crossref]

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007).
[Crossref]

Hashimoto, M.

Ikonen, E.

Junttila, M.-L.

Kauppinen, J.

Kawata, S.

Koch, A. W.

B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

Kudenov, M. W.

Lee, J.-Y.

J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005).
[Crossref]

L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004).
[Crossref]

Li, H. H.

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980).
[Crossref]

Miskiewicz, M. N.

Möller, K.

Mortimer, H.

H. Mortimer, “Compact interferometer spectrometer,” US Patent US20120281223 (2012).

Reininger, F. M.

F. M. Reininger, “The application of large format, broadband quantum well infrared photodetector arrays to spatially modulated prism interferometers,” Infrared Physics & Technology 42, 345–362 (2001).
[Crossref]

Schardt, M.

B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

Schlup, P.

Tatian, B.

Wiesent, B. R.

B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

Winters, D. G.

Zhan, G.

Appl. Opt. (4)

Infrared Physics & Technology (1)

F. M. Reininger, “The application of large format, broadband quantum well infrared photodetector arrays to spatially modulated prism interferometers,” Infrared Physics & Technology 42, 345–362 (2001).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. Chem. Ref. Data (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980).
[Crossref]

Journal of Computational Physics (1)

J.-Y. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” Journal of Computational Physics 206, 1–5 (2005).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

J. Connolly, B. DiBenedetto, and R. Donadio, “Specifications of Raytran material,” Proc. SPIE 0181, 141–144 (1979).
[Crossref]

SIAM review (1)

L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM review 46, 443–454 (2004).
[Crossref]

Other (3)

H. Mortimer, “Compact interferometer spectrometer,” US Patent US20120281223 (2012).

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007).
[Crossref]

B. R. Wiesent, D. G. Dorigo, M. Schardt, and A. W. Koch, “Miniaturized IR-spectrometer for online oil condition monitoring tasks,” presented at the OilDoc Conference and Exhibition, Rosenheim, Germany, 22–24 Jan. 2013.

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Figures (8)

Fig. 1
Fig. 1 Schematic illustration of the proposed static single-mirror Fourier transform spectrometer design.
Fig. 2
Fig. 2 Virtual sources model of the proposed static single-mirror Fourier transform spectrometer design. The ellipse in the schematic drawing of the virtual source S2 illustrates the astigmatism induced by the tilted beam splitter.
Fig. 3
Fig. 3 Simulation of the optical path differences at the detector. The contours of the nonlinear influences and the resulting OPD are calculated with parameters matching the characteristics of the prototype specified in Section 3.
Fig. 4
Fig. 4 Calculated nonlinear wavenumber shift Δνshift between the incorrectly linearly distributed wavenumbers and the correct wavenumbers due to the wavenumber-dependence of the refractive index.
Fig. 5
Fig. 5 Overview of the system configuration used to evaluate the proposed static single-mirror Fourier transform spectrometer design. The use of an extended light source results in focal points with a non-zero diameter Dfp.
Fig. 6
Fig. 6 Detector image of the background interference pattern and the corresponding interferogram. The detector image is a close-up of the central peak of the interference pattern.
Fig. 7
Fig. 7 Background and probe spectra as well as the corresponding background SNR. The SNR was calculated at a measurement frequency of 8.3 Hz.
Fig. 8
Fig. 8 Comparison of the evaluated static single-mirror Fourier transform spectrometer prototype with a traditional FTIR spectrometer using a polystyrene standard.

Equations (7)

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d bs-m = T bs ( n bs + n s sin ( π 4 arcsin ( n s n bs 2 ) ) 1 ( n s n bs 2 ) n s . 2 ) ;
s = d bs-m + T bs ( n s sin ( π 4 arcsin ( n s n bs 2 ) ) 1 ( n s n bs 2 ) 2 ) .
Δ x ( y , z ) lin = s f y .
Δ x ( y , z ) = Δ x ( y , z ) lin + Δ x ( y , z ) nonlin .
γ ν = v s , ν v s , ν design
Δ ν shift = ν correct ν lin = ν lin ( γ ν 1 ) .
Δ ν ˜ = 1 Δ x max .

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