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Abstract
A compact ultrahigh-spectral-resolution imaging spectrometer (CUSRIS) is presented, which combines an entrance slit, a scanning Fabry-Perot interferometer (FPI), a static grating interferometer (SGI) and a cylindrical lens. The SGI consists of a beam splitter, a fixed reflection grating in Littrow configuration, and a fixed plane mirror. For each point of the entrance slit, one spectral image is obtained at each FPI spacing position, and multiple spectral images are obtained to synthesize an ultrahigh-spectral-resolution spectral image. First-order approximations of system performance are given. The CUSRIS is a unique concept that not only obtains spatial information and ultrahigh-resolution spectral information (e.g., resolving power higher than 1,000,000) in the near-infrared, short-wave infrared or mid-wave infrared region, but also has the advantages of compact size and short measurement time compared with the existing ultrahigh-spectral-resolution infrared imaging spectrometers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Imaging spectrometry was first defined by Goetz et al. in 1985 [1]. Imaging spectrometers can be roughly divided into four categories: color filter imaging spectrometer, dispersive imaging spectrometer, interferometric imaging spectrometer and coherent-dispersion imaging spectrometer [2–15]. Push-broom scanning is frequently used in hyperspectral imaging for remote sensing applications [16]. So far, there are only two types of ultrahigh-spectral-resolution infrared imaging spectrometers with resolving power higher than 1,000,000.
For the ultrahigh-spectral-resolution infrared imaging spectrometer that uses only one scanning Michelson-type interferometer [17–20] to obtain infrared hyperfine spectra (e.g., resolving power higher than 1,000,000), there are two disadvantages: (1) the physical size is very large, since a very large maximum optical path difference is needed and so a very large maximum displacement of the moving element is needed; (2) the measurement time is very long, since the number of sampling points for each interferogram is very large.
For the ultrahigh-spectral-resolution infrared imaging spectrometer combining a scanning Fabry-Perot interferometer (FPI) [21–27] with a scanning Michelson-type interferometer [28,29], there are three disadvantages: (1) the measurement time is long, since at each spacing position of the FPI, one scanning period of the moving element of the scanning Michelson-type interferometer is needed; (2) the system is not very compact because of the existence of two scanning systems (one for the scanning FPI, the other for the scanning Michelson-type interferometer); (3) the stability against various disturbances is not high due to the existence of the above two scanning systems.
The combination of a scanning FPI either with a fixed narrowband filter [30], a fixed prism [31], a fixed grating [32–35], or with a fixed FPI [36–42] does not have the above disadvantages, and a virtually imaged phased array (VIPA) used in combination with a grating also does not have the above disadvantages [43–48], however, they cannot provide resolving power higher than 1,000,000 in the infrared region.
Using only a compact instrument to quickly acquire both spatial information and hyperfine spectra with resolving power higher than 1,000,000 in the infrared region remains a big challenge. To solve this challenge, this paper proposes a compact ultrahigh-spectral-resolution infrared imaging spectrometer. Section 2 first describes the principle and then gives the comparisons between the new instrument and the existing ultrahigh-spectral-resolution infrared imaging spectrometers. Section 3 shows the preliminary numerical simulation of two examples with resolving power higher than 1,000,000 in the near-infrared and mid-wave infrared region, respectively. The last section is the conclusion.
2. Principle
Figure 1 shows the optics of the compact ultrahigh-spectral-resolution imaging spectrometer (CUSRIS), which combines an entrance slit, a scanning Fabry-Perot interferometer (FPI) and a static grating interferometer (SGI). The FPI needs to scan N steps in order to cover the full spectral range of interest, and this scanning is usually carried out by using a piezoelectric device. The SGI consists of a beam splitter, a fixed reflection grating in Littrow configuration, and a fixed plane mirror. The SGI obtains a one-dimensional spatial sampling of the interferogram, namely, the SGI obtains a one-dimensional spatial distribution of the optical path difference. The spectral resolution of the SGI only needs to be enough to separate the overlapping orders of the FPI. The entrance slit is located at the front focal plane of the collimating lens, and the front focal plane of the collimating lens is coincident with the back focal plane of the objective lens. The area-array detector is located at the back focal plane of the cylindrical lens.
Fig. 1. Optics of the compact ultrahigh-spectral-resolution imaging spectrometer (CUSRIS): (a) Equivalent Top view of the CUSRIS and (b) Equivalent Side view of the static grating interferometer (SGI) from the FPI side. ZPD: zero path difference.
On the one hand (imaging), each point of the entrance slit is imaged on a separate column of the area-array detector. Because of the existence of an entrance slit, the CUSRIS obtains one-dimensional spatial information in one scanning period of the FPI. Nevertheless, the CUSRIS can obtain two-dimensional spatial information when it is spatially scanned in a direction perpendicular to the plane formed by the entrance slit and the optical axis (i.e., push-broom scanning). On the other hand (spectrometry), for each point of the entrance slit, one interferogram is obtained at each FPI spacing position, and a total of N interferograms are sequentially obtained in one scanning period of the FPI. The Fourier transform of each interferogram yields a separate spectrum. Therefore, for each point of the entrance slit, one spectrum containing the wavelengths that satisfy the maximum transmission condition of the FPI is obtained at each FPI spacing position, and a total of N spectra are obtained to constitute an ultrahigh-resolution spectrum. The theoretical approximations of system performance based on the first-order properties of components are analyzed in detail as follows.
In this paper, d is the plate spacing of the FPI, b is the pixel size of the area-array detector, the rows of the area-array detector are parallel to the x-axis of the detector plane, the columns of the area-array detector are parallel to the y-axis of the detector plane, ${M_X}$ is the number of pixels in each row of the area-array detector used to record the spatial image, ${M_Y}$ is the number of pixels in each column of the area-array detector used to record the spectral image, f is the focal length of the objective lens, ${f_1}$ is the focal length of the collimating lens, and ${f_2}$ is the focal length of the cylindrical lens.
In Fig. 1, typical ray emerging from one representative point of the entrance slit is drawn, point A is imaged on a column (represented by line segment $A^{\prime}$) of the detector. At each FPI spacing position: for point A, one interferogram is obtained and its one-dimensional spatial sampling is recorded by the column $A^{\prime}$ of the detector, namely, each pixel in the column $A^{\prime}$ of the detector is used to record one sampling point with different optical path difference for the interferogram of point A. The number of sampling points for each interferogram is equal to the number of pixels in each column of the detector used to record the spectral image.
Figure 2 shows the equivalent light path diagram of the CUSRIS. $\phi$ is the field angle of the CUSRIS along the slit direction. Point $A(0 )$ of the entrance slit is located on the optical axis of the collimating lens, and point $B({x^{\prime}} )$ of the entrance slit is displaced by a distance $x^{\prime}$ from the optical axis. The spectral image of point $A(0 )$ is recorded by the column $A^{\prime}$ (line segment $A^{\prime}$) of the detector, the spectral image of point $B({x^{\prime}} )$ is recorded by the column $B^{\prime}$ (line segment $B^{\prime}$) of the detector, and x is the x-axis coordinate of the column $B^{\prime}$ (line segment $B^{\prime}$) of the detector. The line segments $A^{\prime}$ and $B^{\prime}$ have the same y-axis coordinates on the detector plane (see Fig. 2(b)). The length of line segment $A^{\prime}$ and line segment $B^{\prime}$ is the same as the aperture size of the FPI in the y-axis direction, which is given by
Fig. 2. Equivalent light path diagram: (a) for the CUSRIS in the sagittal plane and (b) for the Side view of the static grating interferometer (SGI) from the FPI side. ZPD: zero path difference.
According to the geometry and the first-order characteristics of the lens, it can be obtained that $x = {f_2}\tan \theta = {{({{f_2}f\tan \phi } )} \mathord{\left/ {\vphantom {{({{f_2}f\tan \phi } )} {{f_1}}}} \right.} {{f_1}}}$, $\theta = \arctan [{({{f \mathord{\left/ {\vphantom {f {{f_1}}}} \right.} {{f_1}}}} )\tan \phi } ]$, the length of the entrance slit is ${L_S} = {{{M_X}b{f_1}} \mathord{\left/ {\vphantom {{{M_X}b{f_1}} {{f_2}}}} \right.} {{f_2}}}$, the spatial resolution of the CUSRIS to the entrance slit is $\delta x^{\prime} = {{b{f_1}} \mathord{\left/ {\vphantom {{b{f_1}} {{f_2}}}} \right.} {{f_2}}}$, the angular resolution of the CUSRIS along the slit direction is given by
If there is a vacuum medium between the two plates of the FPI, the maximum transmission equation of the FPI is given by
where d is the plate spacing of the FPI, $\theta$ is the incident angle within the FPI, $\lambda$ is the wavelength of light, and m is the order that is an integer.The free spectral range (FSR) in wavenumber of the FPI is $FS{R_\sigma } = {1 \mathord{\left/ {\vphantom {1 {({2d\cos \theta } )}}} \right.} {({2d\cos \theta } )}}$, the FSR in wavelength of the FPI is $FS{R_\lambda } = {{{\lambda ^2}} \mathord{\left/ {\vphantom {{{\lambda^2}} {({2d\cos \theta } )}}} \right.} {({2d\cos \theta } )}}$, and the reflective finesse of the FPI is ${F_r} = {{\pi \sqrt R } \mathord{\left/ {\vphantom {{\pi \sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$, where R is the reflectance of the inner surfaces of the FPI plates.
Suppose that ${\sigma _{\min }}$ and ${\sigma _{\max }}$ are the minimum and the maximum wavenumbers in the spectral range of interest, respectively, and $\delta {\sigma _{CUSRIS(0 )}}$ is the desired spectral resolution (in wavenumber) of the CUSRIS at zero field angle $\phi = 0$.
Let ${d_0}$ denote a central value of the FPI plate spacing, the free spectral range $FS{R_{\sigma 0}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}} )}}} \right.} {({2{d_0}} )}}$ should be equal to P times the spectral resolution $\delta {\sigma _{SGI}}$ of the static grating interferometer (SGI) at a central wavenumber ${\sigma _C}$, i.e., $FS{R_{\sigma 0}} = P \cdot \delta {\sigma _{SGI}}$, and P is determined by making the spectral broadening in the SGI negligible as compared with $FS{R_{\sigma 0}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}} )}}} \right.} {({2{d_0}} )}}$. The spectral resolution of the SGI at zero field angle $\phi = 0$ is given by
In general $P \ge 3$ and P does not need to be an integer.To cover the full spectral range of ${\sigma _{\min }} \le \sigma \le {\sigma _{\max }}$, the FPI needs to scan N steps. The $FS{R_{\sigma 0}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}} )}}} \right.} {({2{d_0}} )}}$ needs to be sampled N times with a sampling interval $\delta {\sigma _{CUSRIS(0 )}}$, and one wavelength satisfying the maximum transmission condition of the FPI is obtained from the SGI at each sampling point (at each FPI spacing position). For $FS{R_{\sigma 0}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}} )}}} \right.} {({2{d_0}} )}}$, the wavenumber position of central spectral peak, i.e., the wavenumber satisfying the maximum transmission condition (see Eq. (4)) of the FPI, moves from ${\sigma _C} - {{FS{R_{\sigma 0}}} \mathord{\left/ {\vphantom {{FS{R_{\sigma 0}}} 2}} \right.} 2}$ to ${\sigma _C} + {{FS{R_{\sigma 0}}} \mathord{\left/ {\vphantom {{FS{R_{\sigma 0}}} 2}} \right.} 2}$, i.e., from ${\sigma _C} - ({{N \mathord{\left/ {\vphantom {N 2}} \right.} 2}} )\cdot \delta {\sigma _{CUSRIS(0 )}}$ to ${\sigma _C} + ({{N \mathord{\left/ {\vphantom {N 2}} \right.} 2} - 1} )\cdot \delta {\sigma _{CUSRIS(0 )}}$. Thus, the spectral resolution (in wavenumber) of the CUSRIS at zero field angle $\phi = 0$ can be calculated by
where $P < N < {{{\pi }\sqrt R } \mathord{\left/ {\vphantom {{{\pi }\sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$. For convenience, let N be an even number. It can be obtained that $\delta {\sigma _{CUSRIS(0 )}} = ({{P \mathord{\left/ {\vphantom {P N}} \right.} N}} )\cdot \delta {\sigma _{SGI(0 )}}$.At the $k\textrm{ - th}$ FPI spacing position ${d_k}$ (where $- {N \mathord{\left/ {\vphantom {N 2}} \right.} 2} \le k \le {N \mathord{\left/ {\vphantom {N 2}} \right.} 2} - 1$ and k is the integer): (1) for point $A(0 )$ of the entrance slit, the $k\textrm{ - th}$ interferogram is obtained and its Fourier transform yields the $k\textrm{ - th}$ spectrum containing the wavenumbers ${\sigma _{\min }} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}} )}}} \right.} {({2{d_k}} )}} \le {\sigma _{\max }}$ (all values of m are integers); (2) for point $B({x^{\prime}} )$ of the entrance slit, the $k\textrm{ - th}$ interferogram is obtained and its Fourier transform yields the $k\textrm{ - th}$ spectrum containing the wavenumbers ${\sigma _{\min }} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}\cos \theta } )}}} \right.} {({2{d_k}\cos \theta } )}} \le {\sigma _{\max }}$ (all values of m are integers).
The displacement of the $k\textrm{ - th}$ FPI plate spacing from the spacing ${d_0}$ is given by
where $- {N \mathord{\left/ {\vphantom {N 2}} \right.} 2} \le k \le {N \mathord{\left/ {\vphantom {N 2}} \right.} 2} - 1$, k is the integer, and $N < {{{\pi }\sqrt R } \mathord{\left/ {\vphantom {{{\pi }\sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$.The spectral resolution (in wavenumber) of the CUSRIS at field angle $\phi$ can be written as
Figure 3 shows the equivalent side view of the static grating interferometer (SGI) from the FPI side. A fixed reflection grating in Littrow configuration is used to realize the stepped mirror (i.e. staircase mirror) [49–52] and, therefore, to solve the fabrication issues of the micro stepped mirrors [52]. Assuming that g is the groove spacing of grating, $\gamma$ is the groove angle of grating, S is the groove depth of grating, h is the equivalent stepped mirror depth, $S = g\sin \gamma$, and a total of Q grooves match a pixel width of the detector (i.e., $b = Q \cdot g\cos \gamma$, $h = b \cdot \tan \gamma$, where Q is an integer), so the equivalent stepped mirror depth is also given by $h = Q \cdot S = Qg\sin \gamma$.
Fig. 3. The fixed reflection grating in Littrow configuration in the equivalent Side view of the static grating interferometer (SGI) from the FPI side.
For a single wavenumber $\sigma$ with input spectral intensity $B(\sigma )$, at field angle $\phi$, the spectral intensity received by each grating groove is ${{B(\sigma )} \mathord{\left/ {\vphantom {{B(\sigma )} {Q{M_Y}}}} \right.} {Q{M_Y}}}$, the optical path difference recorded by the $\kappa \textrm{ - th}$ pixel of the detector includes ${{[{2h({\kappa - 1} )+ 2iS} ]} \mathord{\left/ {\vphantom {{[{2h({\kappa - 1} )+ 2iS} ]} {\cos \theta }}} \right.} {\cos \theta }}$ (where $i = 1,\textrm{ }\ldots ,\textrm{ }Q$), so the recorded intensity on the $\kappa \textrm{ - th}$ pixel of the detector is calculated by
The maximum optical path difference of the SGI at field angle $\phi$ is calculated by ${x_{\max (\phi )}} = {M_Y} \cdot {{2h} \mathord{\left/ {\vphantom {{2h} {\cos \theta }}} \right.} {\cos \theta }} = 2{M_Y}Qg\sin \gamma \cdot {{\sqrt {f_1^2 + {f^2}{{\tan }^2}\phi } } \mathord{\left/ {\vphantom {{\sqrt {f_1^2 + {f^2}{{\tan }^2}\phi } } {{f_1}}}} \right.} {{f_1}}}$, so the spectral resolution (in wavenumber) of the SGI at field angle $\phi$ is determined by
In the CUSRIS, the entrance slit only needs to meet the requirements of spatial resolution, and it has nothing to do with the spectral resolution. The spectral resolution of the CUSRIS is determined by both the scanning FPI and the static grating interferometer (SGI). The use of entrance slit in the CUSRIS will not decrease the optical throughput and therefore will not reduce the signal-to-noise ratio (SNR).
Table 1 shows the comparisons of the CUSRIS with the existing ultrahigh-spectral-resolution infrared imaging spectrometers with resolving power higher than 1,000,000 in the near-infrared, short-wave infrared or mid-wave infrared region.
Table 1. Comparisons of the CUSRIS with the existing ultrahigh-spectral-resolution infrared imaging spectrometer with resolving power higher than 1,000,000 in the near-infrared, short-wave infrared or mid-wave infrared region
3. Preliminary numerical simulation with two examples
3.1. First example in the near-infrared region
Focusing on the CO2 absorption lines at 1.6 µm and 2.0 µm [52], for a source spectra with a central wavenumber ${\sigma _0} = 6250\textrm{ c}{\textrm{m}^{ - 1}}$ (wavelength 1.6 µm) and a spectral bandwidth $\Delta \sigma = 100\textrm{ c}{\textrm{m}^{ - 1}}$ (i.e., a wavenumber range of 6200 cm−1 to 6300 cm−1), the desired spectral resolution is $\delta {\sigma _{CUSRIS(0 )}} = 0.005\textrm{ c}{\textrm{m}^{ - 1}}$ to obtain resolving power higher than 1,000,000. When the reflectance of the inner surfaces of the FPI plates is $R = 0.95$, the number of scanning steps of the FPI should be $N < {{{\pi }\sqrt {0.95} } \mathord{\left/ {\vphantom {{{\pi }\sqrt {0.95} } {({1 - 0.95} )}}} \right.} {({1 - 0.95} )}} \approx 61$. According to Eqs. (5) and (6), some key parameters of the CUSRIS are shown in Table 2. Referring to Reference [52], the spectral resolution of the SGI at zero field angle is $\delta {\sigma _{SGI(0 )}} = 0.05\textrm{ c}{\textrm{m}^{ - 1}}$.
Table 2. Some key parameters of the CUSRIS for the first example
Based on Eq. (16), the equivalent stepped mirror depth should be $h \le {1 \mathord{\left/ {\vphantom {1 {({4 \times 100\textrm{c}{\textrm{m}^{ - 1}}} )}}} \right.} {({4 \times 100\textrm{c}{\textrm{m}^{ - 1}}} )}} = 25\mu m$. If the pixel size of the detector is $b = 20\mu m$ and the grating groove angle is $\gamma = 45^\circ$, the equivalent stepped mirror depth is $h = 20\mu m \times \tan 45^\circ{=} 20\mu m$. Let $Q = 3$, the grating groove spacing is $g = {h \mathord{\left/ {\vphantom {h {({Q\sin \gamma } )}}} \right.} {({Q\sin \gamma } )}} = {{20\mu m} \mathord{\left/ {\vphantom {{20\mu m} {({3 \times \sin 45^\circ } )}}} \right.} {({3 \times \sin 45^\circ } )}} \approx 9.4281\mu m$, so we can choose a reflection grating with 106 grooves/mm. Table 3 shows the main parameters of the SGI in the CUSRIS.
Table 3. Main parameters of the SGI in the CUSRIS for the first example
To cover the full wavenumber range from 6200 cm−1 to 6300 cm−1, from Eq. (7), the plate spacing of the FPI is approximately assigned from 2.4999620 cm to 2.5000400 cm. That is, the FPI should scan N=40 steps with approximate 20 nm in step interval.
Assuming that the focal length of the collimating lens is equal to the focal length of the objective lens (i.e., ${f_1} = f$), the focal length of the cylindrical lens is ${f_2} = 100\textrm{ mm}$, and the number of pixels in each row of the area-array detector used to record the spectral image is ${M_X} = 512$, from Eq. (3), the maximum half field angle of the CUSRIS along the slit direction is ${\phi _{\max }} = \arctan [{{{512 \times 0.02mm} \mathord{\left/ {\vphantom {{512 \times 0.02mm} {({2 \times 100mm} )}}} \right.} {({2 \times 100mm} )}}} ]\approx \textrm{2}\textrm{.931}^\circ$. Based on Eq. (9), the resolving power of the CUSRIS for the spectral range from 1 µm to 2 µm is shown in Fig. 4. For the maximum half field angle ${\phi _{\max }} = \textrm{2}\textrm{.931}^\circ$, the influence of the decrease in resolving power is negligible, so the two diagrams in Fig. 4 are almost identical.
Fig. 4. Resolving power of the CUSRIS at zero field angle $\phi = 0$ and the field angle ${\phi _{\max }} = \textrm{2}\textrm{.931}^\circ$ for the spectral range from 1 µm to 2 µm when $N = 40$ and ${d_0} = 2.5\textrm{ }cm$ (for the first example).
According to Eq. (11), the interferogram produced by the CUSRIS with a spectral resolution of 0.005 cm−1 at zero field angle is shown in Fig. 5, which consists of three interferograms. The first interferogram obtained at $d = \textrm{2}\textrm{.4999980 cm}$ includes only wavenumber 6249.805 cm−1, 6250.005 cm−1 and 6250.205 cm−1. The second interferogram obtained at $d = 2.5\textrm{ cm}$ includes only wavenumber 6249.8 cm−1, 6250 cm−1 and 6250.2 cm−1. The third interferogram obtained at $d = \textrm{2}\textrm{.5000020 cm}$ includes only wavenumber 6249.795 cm−1, 6249.995 cm−1 and 6250.195 cm−1.
Fig. 5. Interferogram produced by the CUSRIS with a spectral resolution of 0.005 cm−1 at zero field angle.
Figure 6 shows the spectrum obtained from the Fourier transform of the three interferograms shown in Fig. 5. We now effectively have a three-dimensional spectrum: one dimension being the wavenumber recovered from the SGI, one being the intensity from the transformed interferograms, and the last being the plate spacing of the FPI. More specifically, the three-dimensional spectrum shown in Fig. 6 is synthesized from three spectra: the first spectrum obtained from the SGI at $d = \textrm{2}\textrm{.4999980 cm}$ contains only wavenumber 6249.805 cm−1, 6250.005 cm−1 and 6250.205 cm−1; the second spectrum obtained from the SGI at $d = 2.5\textrm{ cm}$ contains only wavenumber 6249.8 cm−1, 6250 cm−1 and 6250.2 cm−1; the third spectrum obtained from the SGI at $d = \textrm{2}\textrm{.5000020 cm}$ contains only wavenumber 6249.795 cm−1, 6249.995 cm−1 and 6250.195 cm−1; any one of the three spectra has the same wavenumber interval of 0.2 cm−1, which can be very clearly resolved by the SGI with a spectral resolution of 0.05 cm−1. Spectral peaks of the three wavenumbers 6249.995 cm−1, 6250 cm−1 and 6250.005 cm−1 are clearly distinguished. Spectral peaks of the three wavenumbers 6249.795 cm−1, 6249.8 cm−1 and 6249.805 cm−1 are clearly distinguished. Spectral peaks of the three wavenumbers 6250.195 cm−1, 6250.2 cm−1 and 6250.205 cm−1 are clearly distinguished. Therefore, it can be easily obtained that the spectral resolution of the synthesized three-dimensional spectrum is 0.005 cm−1.
3.2. Second example in the mid-wave infrared region
For the second example, a source spectra with a central wavenumber ${\sigma _0} = 2500\textrm{ c}{\textrm{m}^{ - 1}}$ (wavelength 4 µm) and a spectral bandwidth $\Delta \sigma = 100\textrm{ c}{\textrm{m}^{ - 1}}$ (i.e., a wavenumber range of 2450 cm−1 to 2550 cm−1), the desired spectral resolution is $\delta {\sigma _{CUSRIS(0 )}} = 0.0025\textrm{ c}{\textrm{m}^{ - 1}}$ to obtain resolving power higher than 1,000,000. According to Eqs. (5) and (6), some key parameters of the CUSRIS for the second example are shown in Table 4. The spectral resolution of the SGI at zero field angle is $\delta {\sigma _{SGI(0 )}} = 0.04\textrm{ c}{\textrm{m}^{ - 1}}$. The plate spacing of the FPI is approximately assigned from 3.9999040 cm to 4.0001 cm, i.e., the FPI should scan N=50 steps with approximate 40 nm in step interval. Suppose that the pixel size of the area-array detector is $b = 20\mu m$. The main parameters of the SGI in the CUSRIS for the second example are shown in Table 5.
Table 4. Some key parameters of the CUSRIS for the second example
Table 5. Main parameters of the SGI in the CUSRIS for the second example
Suppose that ${f_1} = f$, ${f_2} = 100\textrm{ mm}$ and ${M_X} = 512$, so ${\phi _{\max }} \approx \textrm{2}\textrm{.931}^\circ$. The resolving power of the CUSRIS for the spectral range from 1 µm to 4 µm is shown in Fig. 7. For ${\phi _{\max }} = \textrm{2}\textrm{.931}^\circ$, the influence of the decrease in resolving power is negligible, so the two diagrams in Fig. 7 are almost identical.
Fig. 7. Resolving power of the CUSRIS at zero field angle $\phi = 0$ and the field angle ${\phi _{\max }} = \textrm{2}\textrm{.931}^\circ$ for the spectral range from 1 µm to 4 µm when $N = 50$ and ${d_0} = 4\textrm{ }cm$ (for the second example).
Figure 8 shows the interferogram produced by the CUSRIS for the second example, which consists of three interferograms. The first interferogram obtained at $d = \textrm{3}\textrm{.9999960 cm}$ includes only wavenumber 2499.8775 cm−1, 2500.0025 cm−1 and 2500.1275 cm−1. The second interferogram obtained at $d = 4\textrm{ cm}$ includes only wavenumber 2499.875 cm−1, 2500 cm−1 and 2500.125 cm−1. The third interferogram obtained at $d = \textrm{4}\textrm{.0000040 cm}$ includes only wavenumber 2499.8725 cm−1, 2499.9975 cm−1 and 2500.1225 cm−1. Figure 9 shows the spectrum obtained from the Fourier transform of the three interferograms in Fig. 8. This spectrum is synthesized from three spectra, each with the same wavenumber interval of 0.125 cm−1, each can be very clearly resolved by the SGI with a spectral resolution of 0.04 cm−1. Spectral peaks of the three wavenumbers 2499.8725 cm−1, 2499.875 cm−1 and 2499.8775 cm−1 are clearly distinguished. Spectral peaks of the three wavenumbers 2499.9975 cm−1, 2500 cm−1 and 2500.0025 cm−1 are clearly distinguished. Spectral peaks of the three wavenumbers 2500.1225 cm−1, 2500.125 cm−1 and 2500.1275 cm−1 are clearly distinguished. So it can be readily obtained that the spectral resolution of this synthesized three-dimensional spectrum is 0.0025 cm−1.
Fig. 8. Interferogram produced by the CUSRIS with a spectral resolution of 0.0025 cm−1 at zero field angle (for the second example).
Table 6 shows some parameters of both the CUSRIS and a standard Michelson interferometer for the same spectral resolution of 0.005 cm−1 in the near-infrared region. Table 7 shows some parameters of both the CUSRIS and a standard Michelson interferometer for the same spectral resolution of 0.0025 cm−1 in the mid-wave infrared region. The CUSRIS can provide resolving power higher than 1,000,000 in the near-infrared, short-wave infrared or mid-wave infrared region, but its physical size will be much smaller than a standard Michelson interferometer with the same spectral resolution.
Table 6. Some parameters of both the CUSRIS and a standard Michelson interferometer for the same spectral resolution 0.005 cm-1 in the near-infrared region (for the first example)
Table 7. Some parameters of both the CUSRIS and a standard Michelson interferometer for the same spectral resolution 0.0025 cm-1 in the mid-wave infrared region (for the second example)
The coherent-dispersion imaging spectrometer (CDIS) in [13] is the first to achieve ultra-high / high spectral resolution broadband spectral imaging in the ultraviolet-visible spectral region. The CDIS in [13] and the coherent-dispersion stereo-imaging spectrometer (CDSIS) in [14] can provide ultrahigh spectral resolution in the infrared region. However, when the CDIS and the CDSIS are used to acquire hyperfine spectra (e.g., resolving power higher than 1,000,000) in the infrared region, they have the following two disadvantages: (1) the physical size is very large and (2) the measurement time is very long. The compact coherent-dispersion imaging spectrometer (CCDIS) in [15] does not have the above disadvantages. However, due to the limitation of the resolving power of a single grating to the overlapping orders of the FPI, the CCDIS in [15] cannot provide resolving power higher than 1,000,000 in the infrared region. Table 8 shows the comparisons of the CUSRIS, the CDIS in [13], the CDSIS in [14], and the CCDIS in [15].
Table 8. Comparisons of the CUSRIS, the CDISa, the CDSISb, and the CCDISc
4. Conclusion
The principle of the CUSRIS is described in detail, the theoretical approximations of system performance based on the first-order properties of components are given, and the results of preliminary numerical simulation with two examples are shown. Compared with the ultrahigh-spectral-resolution infrared imaging spectrometer that uses only one scanning Michelson-type interferometer to obtain hyperfine spectra with resolving power higher than 1,000,000 in the infrared region, the CUSRIS has much smaller physical size and much fewer scanning steps (see Tables 1, 6 and 7). Compared with the ultrahigh-spectral-resolution infrared imaging spectrometer combining a scanning Fabry-Perot interferometer with a scanning Michelson-type interferometer, the CUSRIS has much fewer scanning steps (much shorter measurement time), higher stability and higher compactness. However, compared with the above two types of ultrahigh-spectral-resolution infrared imaging spectrometers, the CUSRIS also has a trade-off, that is, the spectral range is narrow. In summary, the CUSRIS provides a unique concept that uses only a very compact instrument to quickly obtain both spatial information and ultrahigh-resolution spectral information (e.g., resolving power higher than 1,000,000) in the near-infrared, short-wave infrared or mid-wave infrared region.
Funding
National Natural Science Foundation of China (61605151).
Disclosures
The authors declare no conflicts of interest.
References
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