Spectral resolution enhanced static Fourier transform spectrometer based on a birefringent retarder array

Jie Li; Chao Qu; Haiying Wu; Chun Qi

doi:10.1364/OE.27.015505

1. Introduction

Fourier transform spectrometer (FTS) is a vital tool for ultraviolet, visible and infrared spectra measurement in both scientific and industrial areas [1,2]. It offers significant advantages over the traditional dispersive spectrometers, namely the throughput and multiplex advantages [3]. There are many commercially available FTS, primarily used in the lab with high spectral resolution. Most of them are based on mechanically scanned mirrors. To achieve stability and proper sampling of the interferogram, a laser scanning servo-controller is always incorporated to maintain mirror alignment and position, making the instrument bulky and expensive [4]. Their size and weight make them ill-suited for on-site applications.

To reduce the size and cost, a number of compact FTS systems based on MEMS technique are developed [5–8]. Manzardo et al. presented a lamellar grating based FTS with a spectral resolution of 5.5 nm at 800 nm wavelength by using silicon micromachining [5]. Yu et al. developed a miniaturized Michelson interferometer based FTS on a silicon optical bench platform with a resolution of 45 nm near 1500 nm wavelength [6]. Haitham et al. proposed an integrated Mach-Zhender interferometer based FTS with a resolution of 25 nm at 1550 nm wavelength manufactured by deep reactive ion etching [7]. Very recently, an H-shaped electro-thermal MEMS mirror based micro FTS with no need of closed-loop control was presented by Wang et al [8]. Its maximum resolution achieves 36 cm⁻¹. Although even higher spectral resolution (better than 10 cm⁻¹) miniature MEMS FTS was reported [9], an extra laser reference mirror position detection system is required, which significantly increases both size and complexity. The designers have to make a balance between the spectral resolution, size and complexity of the system. These MEMS apparatuses generally suffer from vibration, heat generation, and electrical noise [10]. Additionally, the relatively small size of the mirrors and beamsplitters reduces the throughput [11].

To overcome these limitations, several different approaches were proposed to develop a static FTS [12–16]. They are best understood as shearing interferometers, in which the optical parts of the system are fixed in their position and the interferogram is encoded as a function of position on a detector array rather than as a function of time on a single channel detector. The most common forms are the tilted-mirror Michelson interferometer [12], Sagnac interferometer [13] and birefringent interferometer [14–16]. Compared with the mirror scanning FTS, they have the advantages of high stability in rigorous environments while preserving high throughput [17].

However, the conventional static FTS cannot easily reach very high spectral resolution since their resolving power is limited by the pixels numbers in a row of the detector array used to sample the interferogram [18]. A resolving power increasing method based on stepped mirrors and a 2D detector array has been presented [19,20]. In this method, one or two mirrors of a tilted-mirror Michelson interferometer were replaced by a set of mirrors organized in steps. Each mirror step produces a part of the interferogram to form a large optical path difference (OPD) interferogram. While this configuration could obtain extremely high spectral resolving power (~10⁴), it suffers from several disadvantages that make it difficult to implement in a portable sensor. These include (i) Sensitivity to light intensity variations over the beam aperture caused by defects and contamination of the optical surfaces; (ii) Sensitivity to the misalignment of the stepped mirrors; (iii) Vibration sensitivity due to the two non-common optical paths [21]. An alternative approach is to improve the spectral resolution by using a stacked Wollaston prism array as a multi-channel interferometer [22]. The spectral resolution is about 43.5 cm⁻¹ with a five-sub-Wollaston-prism array. Its drawback is that the small manufacture differences of the sub Wollaston prism wedge angles and the perpendicularity of the hypotenuse planes would introduce phase errors and highly affect the shapes of the interferogram. Ebizuka et al. [23] also proposed a birefringent spectrometer with increased spectral resolution and optical throughput based on a complex hybrid Savart plate and Wollaston prism structure. Its resolution and throughput is both increased by two times compared to those of a conventional static FTS. The three methods can boost the spectral resolving power of the static FTS, but all at cost of high complexity and manufacture difficulties.

In this paper, we propose a new type of spectral resolution enhanced compact static FTS (SESFTS) based on a Wollaston prism and a birefringent retarder array to overcome the aforementioned limitations of the conventional static FTS that offers much higher spectral resolution, while preserving their high throughput and wide free spectral range. Compared with the previous spectral resolution improving methods, the proposed structure is much more simple and robust. In Section 2, we present the optical structure, operation principle and system performance analysis of the developed SESFTS. Section 3 is experimental demonstration of the SESFTS prototype working in the visible and near-infrared band.

2. Theories

2.1 System structure

The schematic setup of the SESFTS system is depicted in Fig. 1. It comprises of a polarizer P, a high order birefringent retarder array BRA, a Wollaston prism WP, an analyzer A, a imaging lens L, and a digital camera. The BRA is formed with M cemented birefringent retarder plates with their fast axes oriented at 0° relative to the x-axis, which are also parallel to the optical axis of the first wedge of WP. Collimated light from the object is resolved by P into linearly polarized light at 45° to the fast axis of BRA. The BRA having a step profile segments the aperture into M + 1 parts. Where M is the number of the BRA steps. Each BRA step then splits the incoming light into two equal amplitude, orthogonally polarized components and introduces a phase delay between them. Transmission through the WP refracts the two polarized components so that they propagate with a small shearing angle. After passing through A, the two components rays are resolved into linearly polarized light in the same polarized orientation and focused by L on the photosensitive surface of the digital camera where they interfere. The digital camera then could record the spatial varying interferogram in a single integrated time. Note that since the segmentation of the BRA, the created interferogram consists of M + 1 sub-interferograms.

Fig. 1 (a) Schematic setup of the developed SESFTS. The optical axes of the polarization elements and fast axis of the retarder array are indicated by arrows and circles. (b) Side view of the developed SESFTS.

Download Full Size | PDF

2.2 Operation principle

Both of the WP and BRA could produce OPD in the developed system. The WP generates a varying OPD along x-axis between the two orthogonal polarization states. For a small prism wedge angle, the OPD of the WP can be given by [24]

\begin{matrix} Δ_{WP} = 2 (n_{o} - n_{e}) h \tan θ \\ = 2 (n_{o} - n_{e}) \frac{x}{M_{L}} \tan θ \end{matrix}

where

n_{o}

,

n_{e}

are the ordinary and extraordinary refractive indices of the Wollaston prism material and

θ

is the wedge angle of the prism. h is the displacement of the ray from the center of the prism, and has the relation

h = x / M_{L}

, where x is the displacement of the interference fringe line from the center of the photosensitive surface of the digital camera (zero-order fringe), and

M_{L}

is the magnification of the imaging lens L. The OPD introduced by the BRA is constant along x but variable among different steps. It can be expressed as

Δ_{BRA} = (n_{o} - n_{e}) i t

where

i = 1, 2, 3 \cdot \cdot \cdot M

is the number of the BRA step and t is the thickness for one step. For a specific sub-interferogram, the OPD is contributed by both of the two birefringent elements. Thus, the OPD of a sub-interferogram can be given by

\begin{matrix} Δ_{SI} = Δ_{WP} + Δ_{BRA} \\ = (n_{o} - n_{e}) (2 \frac{x}{M_{L}} \tan θ + i t) \end{matrix}

It should be mentioned that this expression is approximate for light incident with small incident angles. An example of the OPD versus spatial position, relative to each sub-interferogram, is illustrated in Fig. 2. Notable is the small slop in OPD along x, produced by the small wedge angle

θ

of the WP, and the relatively large steps in the OPD along y, which are realized through the steps thickness t of the BRA. The steps thickness t is given by

t = 4 (1 - Q) \frac{d}{M_{L}} \tan θ

where Q is the overlap ratio of each neighboring sub-interferograms, and d is the half size of the photosensitive surface of the digital camera along x-axis.

Fig. 2 OPD of each sub-interferogram as a function of camera sensor position in pixels.

Download Full Size | PDF

The BRA steps enable each sub-interferogram to get sequentially larger values of OPD. To emphasize this, the OPD of the sub-interferograms in Fig. 2 is numbered 0 and 1-9, representing no step and the BRA steps 1-9 in Fig. 1, respectively. Thus, an interferogram with increasing OPD can be obtained by connecting these sub-interferogram end to end. The increasing OPD can be calculated by

Δ_{t} = 2 (n_{o} - n_{e}) \frac{X}{M_{L}} \tan θ

where X is from

- d

to

d (1 + 2 M (1 - Q))

. This expression is quite similar to Eq. (1), and the only difference is that the displacement x (values from

- d

to d) is replicated by X. For no steps,

M = 0

, Eq. (5) is turned into Eq. (1). It shows that incorporating the BRA into the optics is equivalent to increasing the effective displacement of the WP. Notable is the overlap ratio Q. A small overlap could make the connection of sub-interferogram easy and keep the increasing of the OPD as large as possible. It also makes the connection robust to the manufacture errors of the BRA steps thickness, since the exact overlap ratio of each sub-interferogram could be obtained through the calibration procedure.

Using the OPD expression $Δ_{t}$ , the functional form of the interferogram can be calculated by integrating the total flux at displacement X for all frequencies [25]:

I (X) = \int_{0}^{\infty} S (σ) \cos [4 π σ (n_{o} - n_{e}) \frac{X}{M_{L}} \tan θ] d σ

where

σ

is the wavenumber, and is defined as the number of wavelengths per centimeters,

σ = 1 / λ

, and

S (σ)

is the spectral intensity at wavenumber

σ

. The interferogram

I (X)

is the Fourier cosine transform of the spectrum

S (σ)

with

2 (n_{o} - n_{e}) \tan θ / M_{L}

acting as the proportionality constant with respect to the variable X. Thus, the spectrum can be obtained by performing the inverse Fourier transform over the interferogram.

2.3 Spectral resolution

The spectral resolution of a FTS, depending on the maximum OPD within the interferogram and on the form of apodization used, can be quantified by using the full width at half maximum (FWHM) of the instrument line shape (ILS). For triangular apodization, the FWHM is defined as [25]:

\begin{matrix} FWHM = \frac{1 .79}{2 Δ_{tmax}} \\ = \frac{1. 79 M_{L}}{4 (n_{o} - n_{e}) d (1 + 2 M (1 - Q)) \tan θ} \end{matrix}

To show the capability of the spectral resolution enhancement of the proposed SESFTS, an example is depicted in Fig. 3. In the example, the WP and BRA are made from calcite, and the prism wedge angle $θ$ is 1.5°. Half of the sensor size is $d = 2.5 mm$ , and magnification of the imaging lens is $M_{L} = 0. 278$ , and the overlap ratio is $Q = 0. 2$ .

Fig. 3 Theoretical spectral resolution of a SESFTS example with different BRA step numbers: (a) M is from 0 to 4; (b) M is from 5 to 9.

Download Full Size | PDF

From Fig. 3, we can see that the spectral resolution increases rapidly with increasing step numbers M. With 9 BRA steps, the spectral resolution can reach about 7 cm⁻¹ which is nearly two orders of magnitude higher than that of a conventional static FTS with a similar CCD detector. It can be seen that the spectral resolution varies with wavelength since the dispersion of the birefringent index of calcite. In order to obtain high-precision spectral data, spectral calibration must be performed. Using Eq. (4), we can get the thicknesses of a single BRA step and the whole BRA which are 754 μm and 6.79 mm, respectively. It shows that the developed SESFTS has great potential to achieve extremely high spectral resolution with a small instrument size. It should be indicated that the spectral resolution is also limited by the effective number of pixels sampling the interferogram. According to the Nyquist criterion, we can deduce the detector-limited spectral resolution $Δ σ_{d}$ . For triangular apodization, it can be calculated by

Δ σ_{d} = σ_{c} \frac{1.79}{N}

where

σ_{c}

is the maximum unambiguously measurable wavenumber, and N is the effective number of sampling pixels from zero OPD to maximum OPD of the interferogram.

2.4 Relative signal to noise ratio (SNR)

Signal to noise ratio (SNR) is another key parameter of the spectrometer. For a FTS, the spectral SNR is also related to the maximum OPD $Δ_{\max}$ and can be expressed as [26]

SNR (λ) \propto \frac{S (λ)}{\bar{S}} \sqrt{\frac{1}{N_{λ}}} = \frac{S (λ)}{\bar{S}} \sqrt{\frac{1.79}{2 (σ_{\max} - σ_{\min}) Δ_{\max}}}

where

N_{λ}

is the number of spectral bands of the derived spectrum,

S (λ) / \bar{S}

is the ratio of the signal in a particular spectral band to the average signal over all the spectral bands, and

σ_{\max} - σ_{\min}

is the working spectral range.

Figure 4 is the theoretical relative SNR. As a rough estimate, we set $S (λ) / \bar{S} = 1$ and other parameters are the same as the calculation example in Section 2.3.

Fig. 4 Theoretical relative SNR of a SESFTS example with different BRA step numbers: (a) M is from 0 to 4; (b) M is from 5 to 9.

Download Full Size | PDF

It clearly shows that the spectral SNR of the SESFTS decreases as the number of the BRA steps increases. The trend is just the opposite of the spectral resolution. For a specific application, the balance between SNR and spectral resolution needs to be considered.

3. Experimental demonstration

A schematic of the prototype SESFTS is depicted in Fig. 5. Both of the birefringent elements are made from calcite. The BRA, formed with three sub-retarders, is 1.39 mm thick with a 1.0 mm cemented quartz substrate to strengthen it. The Wollaston prism is 20 × 20 × 3 mm³ with an internal wedge angle of 1.5 degrees. This structure could produce four sub-interferograms. The overlap ratio of each neighboring sub-interferograms is about 0.29. A 1388 × 1038 monochromatic digital CCD camera with a pixel size of 6.45 μm (AVT Manta G-145) is used to take the interferograms. An achromatic doublet with a focal length of 150 mm is used to collimate the input beam. A field stop with a 15 mm aperture is placed at the front focal plane of the doublet, yielding a 5.7° field angle.

Fig. 5 Photograph of the prototype of the SESFTS.

Download Full Size | PDF

3.1 Calibration

3.1.1 Optical path difference calibration

To connect the four sub-interferograms and reconstruct the corresponding spectrum, a highly precise knowledge of the OPD of each pixel is required and this could be done by measuring a polychromatic light and a laser source. Figure 6(a) and 6(b) show the interferograms which are taken with a broadband integrating sphere uniform light source and a He-Ne laser with a beam expender, respectively. From the polychromatic interferogram, the pixel location of the zero-OPD x₀ can be established. According to Eq. (3), the OPD of each sub-interferogram as a function of the FPA’s coordinates, $Δ_{SI} (x)$ , can be given by

Δ_{SI} (x) = (n_{o} - n_{e}) (2 \frac{x - x_{0}}{M_{L}} \tan θ + i t)

Thus, the theoretical interference patterns of the He-Ne laser can be described as

I (x) = \frac{1}{2} (1 + \cos [2 π σ_{HN} (n_{o} - n_{e}) (\frac{x - x_{0}}{M_{L}} \tan θ + i t + δ t_{i})])

where

n_{o} - n_{e} = 0.17

is the birefringence of calcite at the He-Ne laser wavenumber of

σ_{HN} = 1.58 \times 10^{4} {cm}^{- 1}

, and the magnification of the imaging lens L is

M_{L} = 0 .52

in the experiment.

δ t_{i}

(

i = 1, 2, 3

) is the small thickness error of the three sub-retarders.

Fig. 6 Inteferograms acquired by the SESFTS by viewing (a) an integrating sphere uniform light source and (b) a He-Ne laser with a beam expender. Both interferograms include four sub-interferograms.

Download Full Size | PDF

With Eq. (11), a theoretical interferogram is generated and then fit to the calibration interferogram of the He-Ne laser, Fig. 6(b), by using the iterative least squares fitting procedure. Known parameters are $σ_{HN}$ , $n_{o} - n_{e}$ , x₀, $M_{L}$ , $θ$ , i, and t. Meanwhile, $δ t_{i}$ is used as variable. Initial estimate of the connection points (overlap ratios) of sub-interferograms are obtained by using Eq. (4). And their exact locations are then obtained by comparing fringes of equal OPD at the boundaries of the overlap parts of each neighbor sub-interferograms.

3.1.2 Spectral calibration

For a standard FTS, spectral calibration usually begins with the calculation of the spectral wavenumber axis, $σ_{0}$ , in assuming that the OPD is achromatic. However, since the birefringence dispersion of the calcite material, used for the BRA and WP, compensation is required in the spectral calibration of the SESFTS. The birefringence dispersion compensation is performed by adjusting wavenumber axis [21]:

σ_{1} = σ_{0} \frac{n_{o} (σ_{HN}) - n_{e} (σ_{HN})}{n_{o} (σ_{1}) - n_{e} (σ_{1})}

where the birefringent indexes

n_{o}

and

n_{e}

in different wavenumbers are calculated by using the Sellmeier equation [16].

3.2 Results

The performance of the SESFTS was first evaluated by measuring a mercury-argon (Hg-Ar) light source. The obtained interferogram, depicted in Fig. 7, is formed by connecting the four sub-interferograms without averaging along the y-axis. The recovered spectrum from the interferogram was compared to spectrum acquired using an Ocean Optics Flame-S spectrometer with a FWHM spectral resolution of 1.2 nm. This comparison is shown in Fig. 8. Note that the SNR could be improved by averaging the sub-interferograms along the y-axis. One can see that the multiple emission lines of Hg and Ar. While the spectral data have not been corrected for the spectral response between the two spectrometers, the position accuracy of the emission peaks is better than 0.4 nm. The working spectral range extends from 400 to 1000 nm.

Fig. 7 Interferogram of the Hg-Ar light source acquired by the SESFTS. The interferogram is formed by connecting all four sub-interferograms.

Download Full Size | PDF

Fig. 8 Spectral data acquired from the SESFTS (bottom, blue line) and Ocean Optics Flame-S spectrometer (top, red line). The red numbers are the standards.

Download Full Size | PDF

The spectral resolution of the SESFTS prototype was tested by measuring three diode laser sources at wavelength 450 nm, 532 nm and 650 nm. Figure 9 is the experimental interferogram formed by connecting the four sub-interferograms acquired from the SESFTS.

Fig. 9 Interferogram of the three diode laser sources (450 nm, 532 nm, and 650 nm) acquired by the SESFTS. The interferogram is formed by connecting the four sub-interferograms.

Download Full Size | PDF

Figures 10(a)-10(d) show the reconstructed spectra from the SESFTS with 1, 1 + 2, 1 + 2 + 3 and 1 + 2 + 3 + 4 sub-interferograms, respectively. It can be seen that the spectral resolution increases with the number of the connected sub-interferograms. With the total four sub-interferograms, the spectral resolution achieves 24.2 cm⁻¹, 25.8 cm⁻¹ and 26.8 cm⁻¹ at 450 nm, 532 nm and 650 nm wavelength, respectively. The experimental and theoretical values of the spectral resolution are depicted in Fig. 11. They exhibit favorable accordance and the feasibility of the proposed idea is validated.

Fig. 10 Reconstructed spectra from the SESFTS with (a) 1 sub-interferogram; (b) 1 + 2 sub-interferograms; (c) 1 + 2 + 3 sub-interferograms; (d) 1 + 2 + 3 + 4 sub-interferograms.

Download Full Size | PDF

Fig. 11 Spectral resolution of the SESFTS prototype with 1, 1 + 2, 1 + 2 + 3 and 1 + 2 + 3 + 4 sub-interferograms. Experiment and theoretical data are indicated with triangle and solid lines, respectively.

Download Full Size | PDF

4. Conclusion

We have presented a new instrumental concept for a spectral resolution enhanced static Fourier transform spectrometer (SESFTS). By using an M-step birefringent step retarder array (BRA) and a Wollaston prism, the interferogram with a large optical path difference could be achieved. This allows high spectral resolution measurements over a wide spectral range. Operation principle is given with a design example with a 9-step BRA and 7 cm⁻¹ spectral resolution. Based on the theoretical concept, a SESFTS prototype with a 3-step BRA and approximate 25 cm⁻¹ spectral resolution was built and tested. Experiment tests of a mercury-argon light source and three laser sources have demonstrated the spectral capability of the instrument.

Funding

National Natural Science Foundation of China (61675161 and 61205187); Fundamental Research Funds for the Central Universities (zdyf2017003).

References

1. M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. 66(10), 4763–4797 (1995). [CrossRef]

2. J. L. Harley, B. A. Rankin, D. L. Blunck, J. P. Gore, and K. C. Gross, “Imaging Fourier-transform spectrometer measurements of a turbulent nonpremixed jet flame,” Opt. Lett. 39(8), 2350–2353 (2014). [CrossRef] [PubMed]

3. P. Griffiths and J. D. Haseth, Fourier Transform Infrared Spectrometry (John Wiley & Sons, Inc., 1986).

4. J. W. Brault, “New approach to high-precision Fourier transform spectrometer design,” Appl. Opt. 35(16), 2891–2896 (1996). [CrossRef] [PubMed]

5. O. Manzardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29(13), 1437–1439 (2004). [CrossRef] [PubMed]

6. K. Yu, D. Lee, U. Krishnamoorthy, N. Park, and O. Solgaard, “Micromachined Fourier transform spectrometer on silicon optical bench platform,” Sens. Actuators A Phys. 130-131, 523–530 (2006). [CrossRef]

7. H. Omran, M. Medhat, B. Mortada, B. Saadany, and D. Khalil, “Fully integrated Mach-Zhender MEMS interferometer with two complementary outputs,” IEEE J. Quantum Electron. 48(2), 244–251 (2012). [CrossRef]

8. D. Wang, H. Liu, J. Zhang, Q. Chen, W. Wang, X. Zhang, and H. Xie, “Fourier transform infrared spectrometer based on an electrothermal MEMS mirror,” Appl. Opt. 57(21), 5956–5961 (2018). [CrossRef] [PubMed]

9. A. Kenda, S. Lüttjohann, T. Sandner, M. Kraft, A. Tortschanoff, and A. Simon, “A compact and portable IR analyzer: progress of a MOEMS FT-IR system for mid-IR sensing,” Proc. SPIE 8032, 80320O (2011). [CrossRef]

10. W. Wang, J. Chen, A. S. Zivkovic, and H. Xie, “A Fourier transform spectrometer based on an electrothermal MEMS mirror with improved linear scan range,” Sensors (Basel) 16(10), 1611 (2016). [CrossRef] [PubMed]

11. T. Sandner, E. Gaumont, T. Grasshoff, G. Auböck, A. Kenda, T. Gisler, S. Langa, A. Herrmann, and J. Grahmann, “Translatory MEMS actuator with wafer level vacuum package for miniaturized NIR Fourier transform spectrometers,” Proc. SPIE 10545, 105450W (2018). [CrossRef]

12. J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998). [CrossRef] [PubMed]

13. 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]

14. C. Y. Huang and W. C. Wang, “Birefringent prism based Fourier transform spectrometer,” Opt. Lett. 37(9), 1559–1561 (2012). [CrossRef] [PubMed]

15. J. Li, J. Zhu, C. Qi, C. Zheng, B. Gao, Y. Zhang, and X. Hou, “Compact static imaging spectrometer combining spectral zooming capability with a birefringent interferometer,” Opt. Express 21(8), 10182–10187 (2013). [CrossRef] [PubMed]

16. J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284(5), 1127–1131 (2011). [CrossRef]

17. J. Courtial, B. A. Patterson, A. R. Harvey, W. Sibbett, and M. J. Padgett, “Design of a static Fourier-transform spectrometer with increased field of view,” Appl. Opt. 35(34), 6698–6702 (1996). [CrossRef] [PubMed]

18. R. Glenn Sellar and B. Rafert, “Effects of aberrations on spatially modulated Fourier transform spectrometers,” Opt. Eng. 33(9), 3087–3093 (1994). [CrossRef]

19. A. Lacan, F. M. Bréon, A. Rosak, F. Brachet, L. Roucayrol, P. Etcheto, C. Casteras, and Y. Salaün, “A static Fourier transform spectrometer for atmospheric sounding: concept and experimental implementation,” Opt. Express 18(8), 8311–8331 (2010). [CrossRef] [PubMed]

20. E. V. Ivanov, “Static Fourier transform spectroscopy with enhanced resolving power,” J. Opt. A, Pure Appl. Opt. 2(6), 519–528 (2000). [CrossRef]

21. M. W. Kudenov and E. L. Dereniak, “Compact real-time birefringent imaging spectrometer,” Opt. Express 20(16), 17973–17986 (2012). [CrossRef] [PubMed]

22. D. Komisarek, K. Reichard, D. Merdes, D. Lysak, P. Lam, S. Wu, and S. Yin, “High-performance nonscanning Fourier-transform spectrometer that uses a Wollaston prism array,” Appl. Opt. 43(20), 3983–3988 (2004). [CrossRef] [PubMed]

23. N. Ebizuka, M. Wakaki, Y. Kobayashi, and S. Sato, “Development of a multichannel Fourier transform spectrometer,” Appl. Opt. 34(34), 7899–7906 (1995). [CrossRef] [PubMed]

24. M. Françon and S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971), Chap. 2.

25. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, 1972), Chap. 3 and 5.

26. R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005). [CrossRef] [PubMed]

Spectral resolution enhanced static Fourier transform spectrometer based on a birefringent retarder array

Abstract

1. Introduction

2. Theories

2.1 System structure

2.2 Operation principle

2.3 Spectral resolution

2.4 Relative signal to noise ratio (SNR)

3. Experimental demonstration

3.1 Calibration

3.1.1 Optical path difference calibration

3.1.2 Spectral calibration

3.2 Results

4. Conclusion

Funding

References

Cited By

Figures (11)

Equations (12)

Optics Express