## Abstract

A coherent-dispersion spectrometer combining a solid Sagnac interferometer with a dispersing prism is presented, which reduces the multiplex disadvantage of Fourier transform spectroscopy used in the ultraviolet and visible regions while maintains the simultaneous wavelength detection. The spectrometer generates multiple interferograms simultaneously, each with a separate wavelength range and located in a separate row of the detector. The mathematical expressions are given for describing the coherent dispersion, the design calculations are illustrated by an example for the spectral range from 200 nm to 600 nm, and the numerical simulations are shown for the interferogram and spectrum. The unique design of the optics makes the spectrometer very stable, compact, relatively small-sized and, therefore, very suitable for broadband ultraviolet-visible space exploration instruments.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In 1949, astrophysicist Peter Fellgett applied the Michelson interferometer to measure the light from celestial bodies, and computed the first Fourier transform spectrum. Currently, Fourier transform spectrometer (FTS) is well accepted and much used for various kinds of scientific research and industrial applications. In a long period of time, the throughput (Jacquinot) [1, 2] and multiplex (Fellgett) [3, 4] advantages were considered to be the two basic benefits of Fourier transform over dispersive spectrometers. The effect of multiplex in FTS on the signal to noise ratio (SNR) results from two factors: (1) the simultaneous measurement of all wavelengths and (2) an *N*-fold longer measurement time of each individual wavelength for a spectra containing *N* wavelengths. An infrared FTS is used optimally when detector noise exceeds all other noise sources and is independent of the power of the incident radiation impinging on the detector [5]. This is the usual case for mid-infrared spectrometry, and the SNR advantage of the multiplex is then the square root of *N*.

In the visible and ultraviolet spectral region, where the photon noise is the limiting factor, the noise level is proportional to the square root of the incident power [5, 6]. The effect of shot noise is the main reason why FTS was not popular for measuring the ultraviolet and visible spectra. Source shot noise is uniformly distributed throughout the baseline in an FTS system, and there is no degradation of spectral resolution, but the overall SNR is reduced [7]. Source flicker noise degrades both spectral resolution and SNR, but not uniformly throughout the FTS baseline. In the photon noise limited cases, such as in the ultraviolet-visible region, the multiplex gain depends on the square root of the ratio of the intensity of a spectral line to the mean intensity of the whole spectra. The FTS performance is weakened for measuring a broadband source in the visible and ultraviolet spectral range [8, 9]. As a consequence, one greatest disadvantage of all forms of ultraviolet-visible multiplexed spectrometers is the multiplex disadvantage [5–12]. Unfortunately, it is difficult to solve the multiplex disadvantage, except for filtering the light before detection [13–15].

The common-path interferometer [16–18] has a number of characteristics. The lack of any moving parts makes the common-path interferometer very stable against a variety of disturbances. The interferogram is spatially resolved and simultaneously recorded by detector array, which makes the common-path interferometer suitable for the fast spectrometers. The common-path interferometer is a source-doubling interferometer, in which the virtual sources and detector array are, respectively, located in the front and back focal planes of the lens.

This paper presents a coherent-dispersion spectrometer (CDS), which is a unique method to reduce the multiplex disadvantage inherent in interferometry for the ultraviolet and visible spectral ranges while maintain the simultaneous wavelength detection. After a detailed description of the principle, the design calculations are given for the spectral range from 200 nm to 600 nm, and the numerical simulations are shown for the interferogram and spectrum. The conclusion is given in the end.

## 2. Principle

Figure 1 shows the optical layout of the coherent-dispersion spectrometer (CDS) combining interferometric and dispersive spectrometer for ultraviolet-visible spectroscopy. The heart of the CDS is a combination of a solid Sagnac interferometer and a dispersing prism. The solid Sagnac interferometer is made up of two prisms (i.e., prism 1 with reflective surface M1 and prism 2 with reflective surface M2 in Fig. 1) coupled with beam-splitting coating, and it is essentially a common-path interferometer. In Fig. 1, the light from the radiation source is divided into two beams by the beam-splitter *BS* in x-axis direction [17]. Each of these two beams travels the same path in the opposite directions and is recombined by *BS* after reflections at two reflective surfaces M1 and M2 (i.e., source→*BS*→*M1*→*M2* and source→*BS*→*M2*→*M1*) [17]. These two diverging beams are dispersed by a dispersing prism in y-axis direction, collimated by the lens in x-axis direction and in the meanwhile collected by the exact same lens in y-axis direction, and form multiple interferograms located in the different rows of a two-dimensional detector. The detector is located at the back focal plane of the lens.

For the CDS, the solid Sagnac interferometer is actually the wavelength selector in one direction (x-axis direction), and a dispersing prism is the wavelength disperser in the orthogonal direction (y-axis direction) [19,20]. The CDS produces multiple interferograms simultaneously, each interferogram with separate wavelength ranges controlled by the dispersing prism and detector geometry [19], each interferogram located in separate rows of the detector. These interferograms constitute a rectangular focal plane image. For example, when 64 × 1024 detector pixels are used to record interferograms, 64 separate interferograms are obtained simultaneously, each interferogram covering a different wavelength range and located in a separate row (in x-axis direction) of the detector. Throughout this paper, rows refer to interferograms in x-axis direction, and a column corresponds to the dispersion axis in y-axis direction.

Figure 2 shows the ray tracing in the meridian plane of the CDS. Since the solid Sagnac interferometer is a source-doubling interferometer, S1 and S2 are two virtual sources of the radiation source divided by beam-splitter. Suppose that $\gamma $ is the vertex angle of the dispersing prism with an isosceles triangle structure, and $n\left({\lambda}_{i}\right)$ is the refractive index of the dispersing prism for wavelength ${\lambda}_{i}$. A ray enters the prism at angle $\alpha $, is refracted at angle ${\theta}_{1}\left({\lambda}_{i}\right)$ on the first surface, reaches the second surface at angle ${\theta}_{2}\left({\lambda}_{i}\right)$, and leaves the prism at angle ${\theta}_{3}\left({\lambda}_{i}\right)$. According to the law of refraction and the geometry, it can be obtained that

Suppose that $f$is the focal length of the lens, ${\lambda}_{c}$is the central wavelength of the source spectrum, $y\left({\lambda}_{c}\right)$ is the y-axis coordinate for central wavelength ${\lambda}_{c}$ at the detector plane, and $y\left({\lambda}_{i}\right)$ is the y-axis coordinate for wavelength ${\lambda}_{i}$ at the detector plane. Let the optical axis of the lens overlap with the light ray exiting the prism of the central wavelength. Let $y\left({\lambda}_{c}\right)=0$, i.e., let the y-axis coordinate for central wavelength ${\lambda}_{c}$ at the detector plane be zero. According to the characteristics of the lens and the geometry, it can be obtained that

From Eqs. (1) and (2), the y-axis coordinate for wavelength ${\lambda}_{i}$ at the detector plane can be written as

Suppose that the source spectrum covers a wavelength range from ${\lambda}_{\mathrm{min}}$ to ${\lambda}_{\mathrm{max}}$ (equivalent to wavenumber range from ${\sigma}_{\mathrm{min}}=1/{\lambda}_{\mathrm{max}}$ to ${\sigma}_{\mathrm{max}}=1/{\lambda}_{\mathrm{min}}$). As a result, the size of each column (in y-axis direction) of the detector must be greater than $\left|y\left({\lambda}_{\mathrm{max}}\right)-y\left({\lambda}_{\mathrm{min}}\right)\right|$. Let $M$ denote the number of pixels in each column (in y-axis direction) of the detector illuminated simultaneously by multiple bilateral interferograms. That is, $M$ separate interferograms are simultaneously obtained by the CDS. It can be obtained that

Since each interferogram produced simultaneously by the CDS contains a different wavelength range, we can choose different sampling intervals for different interferograms. Nevertheless, for convenience, according to Nyquist theory, the sampling interval of each interferogram produced simultaneously by the CDS can also be

Figure 3 shows the equivalent light path diagram of the CDS in the sagittal plane. S1 and S2 are two virtual sources of the radiation source divided by beam-splitter [17]. The dashed line in Fig. 1(b) indicates the position for reflective surface M2 which produces the zero split of beams due to the symmetrical arrangement of the reflective surfaces M1 and M2 [17]. The distance, $a$, between the solid and dashed lines for reflective surface M2 along the optical axis produces the separation between the two virtual sources. For the case shown in Fig. 1(b), the distance, $d$, between two virtual sources S1 and S2 is $d=\sqrt{2}a$.

Since the dispersing prism of the CDS is equivalent to a parallel plate in the sagittal plane as shown in Fig. 3, the shear angle ${\theta}_{shear}$ (i.e., the angle between two emergent plane waves, the angle of the two virtual sources to the optical center of the lens) is given by

and the sampling interval $X$ for each interferogram is given bywhere $b$ is the pixel size of the detector. Since the shear angle ${\theta}_{shear}$ is usually very small, it can be calculated in radians by Eq. (9) in engineering.The maximum optical path difference (OPD) for each interferogram is given by

where $N$ is the number of pixels in each row (in x-axis direction) of the detector illuminated simultaneously by multiple bilateral interferograms. That is, $N$ is the number of pixels per row of the detector used to acquire multiple bilateral interferograms simultaneously.The spectral resolution $\delta \sigma $ for each interferogram is determined by

and therefore, from Eqs. (8)-(11), the spectral resolution for each interferogram is given byFrom Eqs. (7) and (10)-(12), the maximum wave number ${\sigma}_{\mathrm{max}}$ for all interferograms generated simultaneously by the CDS is determined by

The spectral resolution $\delta \sigma $ and maximum observable wavenumber ${\sigma}_{\mathrm{max}}$ can be adjusted by changing the distance $a$ between the actual position of the reflective surface M2 and the zero split position of beams in Fig. 1, where $d=\sqrt{2}a$. The resolving power $R$ for each interferogram (located in separate rows of the detector) is determined merely by the number of pixels in each row of the detector illuminated by bilateral interferograms, namely, $R={\sigma}_{\mathrm{max}}/\delta \sigma =N/2$.If $x$ is the coordinate along x-axis at the detector plane, the optical path difference (OPD) for each interferogram (row) is $\text{OPD}=xd/f$. The interferogram $I\left(x\right)$ formed on the detector plane is given by

Figure 4 shows two solid Sagnac interferometers. The first one is composed of two identical prisms as shown in Fig. 4(a), where $d=\sqrt{2}\cdot \Delta \cdot \mathrm{tan}\left(22.5\xb0\right)$. The second one is made up of two different prisms as shown in Fig. 4(b), in which the dashed line position is defined as the same as in Fig. 1(b). For the second solid Sagnac interferometer, $d=\sqrt{2}\cdot a$.

Considering the potential effect of dispersion inside prisms 1 and 2, if there is any slight change in the distance $d$ between the virtual sources S1 and S2, there will be a slight change in the shear angle ${\theta}_{shear}$ and the optical path difference (OPD), so the interference fringes will have a slight displacement and the corresponding spectral resolution will decrease. Referring to Fig. 3, we can get $\frac{d}{2f}=\mathrm{tan}\frac{{\theta}_{shear}}{2}$, $\text{OPD}=2x\mathrm{tan}\frac{{\theta}_{shear}}{2}=r\frac{1}{\sigma}$ ($r$ is a real number) and ${\text{OPD}}_{\mathrm{max}}=2{x}_{\mathrm{max}}\mathrm{tan}\frac{{\theta}_{shear}}{2}$. Differential to $\text{OPD}=2x\mathrm{tan}\frac{{\theta}_{shear}}{2}=r\frac{1}{\sigma}$, it can be obtained that ${\delta}_{\text{OPD}}=2x\cdot \frac{1}{{\mathrm{cos}}^{2}\left({\theta}_{shear}/2\right)}\cdot \frac{1}{2}\cdot \delta {\theta}_{shear}=r\frac{1}{{\sigma}^{2}}\delta \sigma $. When the spectral resolution is $\delta \sigma =\frac{\sigma}{N/2}$, based on the above several equations, we can get $\delta {\theta}_{shear}=\frac{\mathrm{sin}{\theta}_{shear}}{N/2}\approx \frac{{\theta}_{shear}}{N/2}$. Similarly, when the spectral resolution is $\delta \sigma =\frac{\sigma}{N/2}$, we can also get $\delta d=\frac{d}{N/2}$.

If the light source is spatially incoherent, the pairing points in two virtual sources are coherent. Moreover, for each specific narrow wavelength range, each pair of points in two virtual sources produces the same interferogram with the same position located in a separate row of the detector due to the use of the Fourier transform optical arrangement. Anyway, whether the light source is spatially coherent or spatially incoherent, the pairing points in two virtual sources are coherent.

The light from the pairing points in two virtual sources are coherent, and then these coherent lights are dispersed by a dispersing prism. That is, the light from the radiation source is first converted into coherent light by a solid Sagnac interferometer and then dispersed by a dispersing prism. Therefore, the name of the instrument is called the coherent-dispersion spectrometer (CDS).

If there is a spatial extension of the two interfering wave fronts perpendicular to the sagittal plane, a superposition of interferograms for different wavelength ranges appears on every detector row rather than the clean one-wavelength-range-only interferogram. Therefore, in order to solve the problem of superposition of interferograms for different wavelength ranges, the CDS needs a slit in practical applications, in which the slit dimension corresponds to the detector row (interferogram dimension).

The major advantage of the CDS, compared with the original Sagnac interferometer and other common-path designs, is that the multiplex disadvantage inherent in ultraviolet-visible interferometry is greatly reduced. Any noise present in an optical signal is constrained to the rows where this noisy signal falls, and these noise sources have no effect on the interferograms with different wavelength ranges [19,20].

Another advantage of the CDS, compared with the spatial interferometry, is that the dynamic range requirement of the detector is reduced. A limitation to the use of array detector for spatial interferometry is the large dynamic range requirement of the detector (similar requirements exist for other interferometric arrangements [21]). For the spatial interferometry, it is important to be able to measure the center fringe as well as the weak fringes at the edges of the interferogram [19]. The dynamic range of the detector can limit the sensitivity of the overall system [17]. For the CDS method, the center fringes are spread into multiple separate rows (interferograms), each with a separate wavelength range, and therefore the dynamic range requirement of the detector is reduced [19,20].

The CDS obtains complete interferograms simultaneously. Namely, every point in the interferograms is measured during the exact same integral period for the CDS. As opposed to the scanning nature of the Michelson interferometer system, the simultaneous nature of the CDS enables it to have great benefit for measuring the transient sources.

The unique optical design enables the CDS, compared with traditional multichannel dispersive spectrometers, to obtain higher spectral resolution for a broadband wavelength range, and to obtain higher signal-to-noise ratio.

Due to the lack of any moving parts, a significant benefit of the CDS is stability against a variety of disturbances. The combination of a solid Sagnac interferometer and a dispersing prism makes the CDS very stable, compact, relatively small-sized, and very strong adaptable to space environment.

## 3. Design calculation and numerical simulation

Suppose that the source spectrum covers a wavelength range from 200 nm to 600 nm, i.e., a wavenumber range from 16666.7 cm^{−1} to 50000 cm^{−1}. Fused silica is suitable for a wavelength range spreading from 200 nm and up into the mid infrared, which is a proper material for the dispersing prism in a CDS. The refractive index formula of fused silica can be written as [22]

Assume that the focal length of the lens is $f=450mm$, the vertex angle of the dispersing prism is $\gamma =48\xb0$, the incident angle to the dispersing prism is $\alpha =24\xb0$, and the central wavelength is ${\lambda}_{c}=300nm$. From Eqs. (3), (4) and (15), the y-axis coordinate for wavelength ${\lambda}_{i}$ at the detector plane is shown in Fig. 5. The y-axis coordinate for wavelength 600 nm is $y\left({\lambda}_{600}\right)\approx 0.36146mm$, and the y-axis coordinate for wavelength 200 nm is $y\left({\lambda}_{200}\right)\approx -0.60682mm$. Thus, the size of each column of the detector must be greater than $\left|y\left({\lambda}_{600}\right)-y\left({\lambda}_{200}\right)\right|=0.96828mm$. If the pixel size of the detector is $b=0.02mm$, the number of pixels in each column (in y-axis direction) of the detector must be greater than $\left|y\left({\lambda}_{600}\right)-y\left({\lambda}_{200}\right)\right|/b=\text{0}\text{.96828}/0.02=48.4$. Therefore, 49 separate interferograms are obtained simultaneously by the CDS.

Assume that the interferogram 1 contains wavelength 600 nm. According to Eqs. (3), (4) and (15), together with $b=0.02mm$, the wavelength (wavenumber) ranges for several separate interferograms are shown in Table 1. The interferogram 1 located in detector row 1 covers a wavelength range from 556 nm to 600 nm, the interferogram 39 located in detector row 39 covers a wavelength range from 220 nm to 222 nm, and so on.

The maximum wavenumber of a source spectra from 200 nm to 600 nm is ${\sigma}_{\mathrm{max}}=50000{\text{cm}}^{-1}$. For convenience, based on Eq. (7), the sampling interval for each interferogram can be $X\le 1/\left(2\times 50000c{m}^{-1}\right)=0.0001mm$. According to Eq. (9), the shear angle is ${\theta}_{shear}\approx X/b\le 0.1/20=0.005\text{radians}$. From Eq. (8), the distance between two virtual sources is $d=f\cdot {\theta}_{shear}=450mm\times 0.005=2.25mm$. It can also be obtained from Eq. (13), i.e., $d=f/\left(2b{\sigma}_{\mathrm{max}}\right)=450mm/\left(2\times 0.02mm\times 50000c{m}^{-1}\right)=2.25mm$. Accordingly, we can get $a=d/\sqrt{2}=2.25mm/\sqrt{2}=1.591mm$.

If the spectral resolution for each interferogram is $\delta \sigma =200{\text{cm}}^{-1}$, from Eq. (13), the number of pixels in each row (x-axis direction) of the detector illuminated simultaneously by multiple bilateral interferograms is $N=2{\sigma}_{\mathrm{max}}/\delta \sigma =2\times 50000c{m}^{-1}/200c{m}^{-1}=500$. Namely, the number of pixels per row of the detector used to acquire multiple bilateral interferograms simultaneously is $N=500$. Therefore, the x-axis coordinates of the detector are given by $-5mm\le x\le 5mm$. Based on Eqs. (3), (4), (14) and (15), together with the above given and calculated parameter values, several bilateral interferograms (i.e., interferograms 1, 5, 12, 28 and 39 shown in Table 1) obtained simultaneously by the CDS at the detector plane are shown in Fig. 6. The interferogram 1 located in detector row 1 contains wavelength 580 nm, the interferogram 5 located in detector row 5 contains wavelength 454 nm, the interferogram 12 located in detector row 12 contains wavelength 350 nm, the interferogram 28 located in detector row 28 contains wavelength 254 nm, and the interferogram 39 located in detector row 39 contains wavelength 220 nm。

Figure 7 shows the CDS spectrum obtained from Fourier transform of several unilateral interferograms that are, respectively, half of the several bilateral interferograms in Fig. 6. Spectral peaks in the range of 220 nm to 580 nm (i.e., 17241.4 cm^{−1} to 45454.5 cm^{−1}) are visible in the two-dimensional detector output.

Figure 8 shows the CDS interferogram 1 that contains only wavelength 556 nm, 580 nm, 600 nm with equal intensity and the spectrum obtained from Fourier transform of the interferogram 1 when the spectral resolution is 200 cm^{−1}. Three spectral peaks are visible. That is, the spectra of wavelengths 556 nm, 580 nm and 600 nm were obtained.

Figure 9 shows the CDS interferogram 39 that contains only wavelength 220 nm, 222 nm with equal intensity and the spectrum obtained from Fourier transform of the interferogram 39 when the spectral resolution is 200 cm^{−1}. Two spectral peaks are visible. Namely, the spectra of wavelengths 220 nm and 222 nm were obtained.

If the spectral resolution for each interferogram is $\delta \sigma =50{\text{cm}}^{-1}$, from Eq. (13), the number of pixels in each row of the detector used to acquire multiple bilateral interferograms simultaneously is $N=2{\sigma}_{\mathrm{max}}/\delta \sigma =2\times 50000c{m}^{-1}/50c{m}^{-1}=2000$. So the x-axis coordinates of the detector are given by $-20mm\le x\le 20mm$. In this case, the interferogram 1 containing only wavelength 556 nm, 580 nm, 600 nm with equal intensity and the Spectrum obtained from Fourier transform of the interferogram 1 are shown in Fig. 10, in which three spectral lines are completely separated. In addition, the interferogram 39 containing only wavelength 220 nm, 222 nm with equal intensity and the Spectrum obtained from Fourier transform of the interferogram 39 are shown in Fig. 11, in which two spectral lines are completely separated.

The sum of all columns of the CDS spectrum in Fig. 7 is shown in Fig. 12. Thus Fig. 12 is similar to the spectrum that would be obtained with the common-path instrument.

## 4. Conclusion

This work investigated a coherent-dispersion spectrometer (CDS) combining interferometric and dispersive spectrometry to offer significant advantages compared with conventional spectrometer configurations for ultraviolet-visible spectroscopy. Most importantly, the CDS greatly reduces the multiplex disadvantage of ultraviolet-visible multiplexed spectroscopy by confining the noise to the particular row of the CDS spectrum where the noise appears. The second important advantage of the CDS is simultaneous wavelength detection, i.e., simultaneous spectral acquisition. As opposed to the scanning nature of most other forms of interferometry such as Michelson interferometer, the simultaneous nature enables the **CDS** to have great potential for recording the spectra from transient sources. The third advantage of the CDS, compared with the spatial interferometry, is that the dynamic range requirement of the detector is reduced. The fourth advantage of the CDS, compared with traditional multichannel dispersive spectrometers, is higher spectral resolution for a broadband wavelength range and higher signal-to-noise ratio. The fifth advantage of the CDS is stability against a variety of disturbances. The unique design of the optics makes the CDS very stable, compact, relatively small-sized and very strong adaptability to space environment. The CDS will be very suitable for broadband space exploration instruments used in the ultraviolet and visible regions.

## Funding

National Natural Science Foundation of China (NSFC) (61605151).

## References and links

**1. **P. Jacquinot, “The luminosity of spectrometers with Prisms, Grating, or Fabry-Perot Etalons,” J. Opt. Soc. Am. **44**(10), 761–765 (1954). [CrossRef]

**2. **P. Jacquinot, “How the search for a throughput advantage led to Fourier transform spectroscopy,” Infrared Phys. **24**(2), 99–101 (1984). [CrossRef]

**3. **P. B. Fellgett, “Three concepts make a million points,” Infrared Phys. **24**(2-3), 95–98 (1984). [CrossRef]

**4. **P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. **60**(1), 91–93 (2006). [CrossRef]

**5. **P. R. Griffiths and J. A. de Haseth, *Fourier Transform Infrared Spectrometry* (Wiley-Interscience, 2007).

**6. **T. Hirschfeld, “Fellgett’s Advantage in UV-VIS Multiplex Spectroscopy,” Appl. Spectrosc. **30**(1), 68–69 (1976). [CrossRef]

**7. **E. Voigtman and J. D. Winefordner, “The multiplex disadvantage and excess low-frequency noise,” Appl. Spectrosc. **41**(7), 1182–1184 (1987). [CrossRef]

**8. **A. Barducci, D. Guzzi, C. Lastri, P. Marcoionni, V. Nardino, and I. Pippi, “Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers,” Opt. Express **18**(11), 11622–11649 (2010). [CrossRef] [PubMed]

**9. **A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio properties of multiplex dispersive spectrometry,” Appl. Opt. **49**(28), 5366–5373 (2010). [CrossRef] [PubMed]

**10. **P. Connes and G. Michel, “Astronomical Fourier spectrometer,” Appl. Opt. **14**(9), 2067–2084 (1975). [CrossRef] [PubMed]

**11. **P. Luc and S. Gerstenkorn, “Fourier transform spectroscopy in the visible and ultraviolet range,” Appl. Opt. **17**(9), 1327–1331 (1978). [CrossRef] [PubMed]

**12. **J. K. Kauppinen, I. K. Salomaa, J. O. Partanen, and M. R. Hollberg, “The Use of Carousel Interferometer in Fourier-Transform Ultraviolet Spectroscopy,” Opt. Rev. **4**(2), 293–296 (1997). [CrossRef]

**13. **E. A. Stubley and G. Horlick, “A Fourier transform spectrometer for UV and Visible measurements of Atomic emission sources,” Appl. Spectrosc. **39**(5), 800–804 (1985). [CrossRef]

**14. **T. Hirschfeld and D. B. Chase, “FT-Raman Spectroscopy: Development and Justification,” Appl. Spectrosc. **40**(2), 133–137 (1986). [CrossRef]

**15. **D. C. Tilotta, R. D. Freeman, and W. G. Fateley, “Hadamard transform visible Raman spectrometry,” Appl. Spectrosc. **41**(8), 1280–1287 (1987). [CrossRef]

**16. **K. Yoshihara and A. Kitada, “Holographic Spectra Using a Triangle Path Interferometer,” Jpn. J. Appl. Phys. **6**(1), 116 (1967). [CrossRef]

**17. **T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. **23**(2), 269–273 (1984). [CrossRef] [PubMed]

**18. **J. V. Sweedler and M. B. Denton, “Spatially Encoded Fourier Transform Spectroscopy in the Ultraviolet to Near-Infrared,” Appl. Spectrosc. **43**(8), 1378–1384 (1989). [CrossRef]

**19. **J. V. Sweedler, R. D. Jalkian, G. R. Sims, and M. B. Denton, “Crossed Interferometric Dispersive Spectroscopy,” Appl. Spectrosc. **44**(1), 14–20 (1990). [CrossRef]

**20. **J. V. Sweedler, “The use of charge transfer device detectors and spatial interferometry for analytical spectroscopy,” https://arizona.openrepository.com/handle/10150/184683.

**21. **T. H. Barnes, “Photodiode array Fourier transform spectrometer with improved dynamic range,” Appl. Opt. **24**(22), 3702–3706 (1985). [CrossRef] [PubMed]

**22. **M. Bass, G. Li, and E. V. Stryland, *Handbook of Optics***(McGraw-Hill, 2010)**.