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Abstract
An imaging spectrometer combining an entrance slit, a Fabry-Perot interferometer (FPI) and a plane transmission grating is presented. Each unit of the entrance slit is imaged on a separate column of the detector and different wavelengths are dispersed across different rows of that column. To cover the full spectral range, the FPI needs to scan N steps. For each unit of the entrance slit, one spectrum is obtained at each FPI spacing position and a total of N spectra are sequentially obtained to constitute a high resolution spectrum. The combination of imaging, interferometry and dispersive spectrometry enables the instrument to obtain spatial information and high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region. First-order approximations of system performance are given. The unique design of the optics will make the instrument compact and suitable for high-spectral-resolution broadband ultraviolet-visible spectral imaging.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the ultraviolet-visible spectral region, the photon noise is the limiting factor, and the noise level is proportional to the square root of the incident power [1–4]. Furthermore, with the rapid development of photodetectors, the influence of photon noise has become more significant compared with detector noise. The grating spectrometers can obtain high-resolution spectral information of a narrow-band source [5–8]. The gratings normally provide resolving powers from 1000 to 100,000 [9]. The resolving power of a single grating is limited by the product of the number of grating lines and the diffraction order utilized [9]. Together with the constraints of the minimum grating groove spacing, to use the gratings at higher spectral resolutions, requires the use of increasingly larger gratings. To illuminate large gratings, optics with large aperture and long focal length are required to minimize aberrations. Owing to imaging considerations, a large entrance focal length combined with a narrow entrance slit is also required in order to obtain high resolution. Therefore, the grating spectrometers with resolving power higher than 50,000 are commonly large and expensive. On the other hand, in order to obtain higher spectral resolution, the echelle grating is normally used in high diffraction orders (the overlapping high diffraction orders are usually separated by a prism or another grating), which makes the spectral range of high-resolution grating spectrometers generally not wide. In short, dispersive spectrometry cannot obtain high-resolution broadband spectral information. Interferometric spectrometry can acquire high-resolution spectral information of a narrow-band source in the ultraviolet-visible spectral region, however, interferometric spectrometry is only suitable for acquiring low or medium resolution spectral information of a broadband source in the ultraviolet-visible spectral region [1,10–15].
A very good method to obtain high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region is the coherent-dispersion spectrometry, which combines interferometric spectrometry with dispersive spectrometry to obtain spectral information. The coherent-dispersion spectrometry integrates an interferometer (a static Sagnac interferometer, a static Fabry-Perot etalon, or a Michelson-type interferometer) and a dispersive element (a prism or a grating) [16–24]. The coherent-dispersion spectrometry has the advantage of providing high spectral resolution of interferometry for a broadband source, while avoiding the multiplex disadvantage of interferometry in the ultraviolet-visible spectral region, and relaxing the dynamic range requirement of the detector [25].
Fabry-Perot interferometers (FPI) offer a compact method to separate closely-spaced wavelengths [26–40]. The instruments reported in [41–43] combine a FPI and a reflection grating, the instrument reported in [44] integrates a FPI and a prism, and the instrument reported in [45] integrates a fiber FPI and a fiber Bragg grating. However, all the instruments reported in [41–45] can only obtain spectral information (just spectrometers), but cannot obtain spatial information (not imaging spectrometers).
A very effective method to use only one instrument to obtain both spatial information and high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region is the coherent-dispersion imaging spectrometer (CDIS) [46,47], which combines imaging with coherent-dispersion spectrometry. The CDIS integrates an entrance slit, a moving corner-cube-mirror interferometer and a dispersing prism (or a transmission grating), however, this CDIS may still not be the best solution for some applications where size, weight and power are very precious. The main reasons are as follows: (1) in order to obtain high spectral resolution, a large change in optical path difference is needed, so a large displacement of the moving corner-cube mirror is needed, and so a large precision driving system for the moving corner-cube mirror is needed, together with the use of a moving corner-cube-mirror interferometer with two corner-cube mirrors, the physical size is large and the weight is heavy [1,48]; (2) in order to perform high-spectral-resolution measurement of a broadband source, not only a large displacement of moving corner-cube mirror is needed, but also the number of sampling points of each interferogram is very large, so a long scanning time is needed, and therefore a long spectral measurement time is needed [48–50].
How to use only a compact, lightweight instrument to obtain both spatial information and high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region is still a big challenge. The aim of this paper is to solve this challenge by proposing a new ultraviolet-visible imaging spectrometer, which combines an entrance slit, a Fabry-Perot interferometer (FPI) and a plane transmission grating. After a detailed description of the principle, the comparison between the new instrument and the previous CDIS is given, preliminary numerical simulations are illustrated by an example within the spectral range from 270 nm to 500 nm. Finally, the conclusion is given.
2. Principle
Figure 1 shows the optical layout of the compact coherent-dispersion imaging spectrometer (CCDIS), which includes an objective lens, an entrance slit, a collimating lens, a Fabry-Perot interferometer (FPI), a plane transmission grating, a collecting lens, and an area-array detector. The heart of the CCDIS is a combination of an entrance slit, a FPI and a plane transmission grating. The entrance slit is located at the front focal plane of the collimating lens, so the light from each unit of the entrance slit is converted into a parallel beam by the collimating lens. The back focal plane of the objective lens is coincident with the front focal plane of the collimating lens. The area-array detector is located at the back focal plane of the collecting lens. On the one hand (imaging), each unit of the entrance slit is imaged on a separate column of the detector and different wavelengths are dispersed across different rows of that column. Due to the use of an entrance slit, the CCDIS gets one-dimensional spatial information, however, the CCDIS can obtain two-dimensional spatial information when it is spatially scanned perpendicular to the entrance slit. On the other hand (spectrum), in order to cover the full spectral range, the FPI needs to scan N steps, and this scanning is generally carried out by using a piezoelectric device. For each unit of the entrance slit, one low-resolution spectrum including the wavelengths that satisfy the maximum transmission condition of the FPI is obtained at each FPI spacing position, and a total of N low-resolution spectra are sequentially obtained to constitute a high-resolution spectrum. For a given unit of the entrance slit, these N low-resolution spectra are sequentially recorded by a given column of the detector. The spectral resolution of each low-resolution spectrum only needs to be sufficient to separate the overlapping orders of the FPI, and the spectral resolution of each low-resolution spectrum is controlled by the transmission grating and detector geometry. The combination of imaging with coherent-dispersion spectrometry enables the CCDIS to obtain both spatial information and high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region. The unique optical layout will make the CCDIS compact. The theoretical approximations of system performance based on the first-order properties of components are analyzed in detail as follows.
Fig. 1. Optical Layout of the compact coherent-dispersion imaging spectrometer (CCDIS): (a) Equivalent Top view and (b) Equivalent front view. The heart of the CCDIS is a combination of an entrance slit, a Fabry-Perot interferometer (FPI) and a plane transmission grating.
Throughout this paper, 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 y-axis of the detector plane, b is the pixel size of the area-array detector, ${L_S}$ is the length of the entrance slit, d is the plate spacing of the FPI, f is the focal length of the objective lens, ${f_1}$ is the focal length of the collimating lens, ${f_2}$ is the focal length of the collecting lens, and ${\lambda _C}$ is the central wavelength of a source spectra covering a wavelength range from ${\lambda _1}$ to ${\lambda _K}$ (where ${\lambda _1}$ is the minimum wavelength, ${\lambda _K}$ is the maximum wavelength).
Figure 2 shows the equivalent light path diagram of the CCDIS. Typical rays emerging from two representative points of the entrance slit are drawn. Point $A(0 )$ is located on the optical axis of the collimating lens, and point $B({x^{\prime}} )$ is displaced by a distance $x^{\prime}\;({x^{\prime} \le {{{L_S}} \mathord{\left/ {\vphantom {{{L_S}} 2}} \right.} 2}} )$ from the optical axis. Four representative image points are shown: ${A_1}({0,{y_1}} )$ is the image point on the detector of point $A(0 )$ formed by wavelength ${\lambda _1}$, ${A_K}({0,{y_K}} )$ is the image point on the detector of point $A(0 )$ formed by wavelength ${\lambda _K}$, ${B_1}({x,{y_1}} )$ is the image point on the detector of point $B({x^{\prime}} )$ formed by wavelength ${\lambda _1}$, ${B_K}({x,{y_K}} )$ is the image point on the detector of point $B({x^{\prime}} )$ formed by wavelength ${\lambda _K}$. Let the optical axis of the collecting lens overlap with the first-order diffracted light ray from the transmission grating of the central wavelength ${\lambda _C}$, so the y-axis coordinate on the detector plane of the central wavelength ${\lambda _C}$ is zero (i.e., ${y_C} = 0$). According to the characteristics of the lens and the geometry, it can be obtained that $\tan \phi = {{x^{\prime}} \mathord{\left/ {\vphantom {{x^{\prime}} f}} \right.} f}$ and $\tan \theta = {{x^{\prime}} \mathord{\left/ {\vphantom {{x^{\prime}} {{f_1}}}} \right.} {{f_1}}} = {x \mathord{\left/ {\vphantom {x {{f_2}}}} \right.} {{f_2}}}$. Thus, ${L_S} = {{Qb{f_1}} \mathord{\left/ {\vphantom {{Qb{f_1}} {{f_2}}}} \right.} {{f_2}}}$, the maximum value of the angle $\theta$ is given by
Fig. 2. Equivalent light path diagram of the CCDIS: (a) in the sagittal plane and (b) in the meridian plane.
The spatial resolution of the CCDIS to the entrance slit is given by
The grating equation of the plane transmission grating is given by [9] where G is the diffraction order that is an integer, ${\lambda _i}$ is the wavelength of light, g is the groove spacing of grating, $n({{\lambda_i}} )$ is the refractive index of the plane transmission grating for wavelength ${\lambda _i}$, $\alpha$ is the incidence angle measured from the grating normal, ${\beta _m}({{\lambda_i}} )$ is the m-order diffraction angle measured from the grating normal for wavelength ${\lambda _i}$.According to the grating equation, the characteristics of the lens and the geometry, it can be obtained that
The free spectral range (FSR) in wavenumber of the FPI is given by
The FSR in wavelength of the FPI is given by The reflective finesse of the FPI is given byThe transmittance function of the FPI is ${T_{FPI}}({\sigma, d} )= {{{{({1 - R} )}^2}} \mathord{\left/ {\vphantom {{{{({1 - R} )}^2}} {[{1 + {R^2} - 2R\cos ({4{\pi}\sigma d\cos \theta } )} ]}}} \right.} {[{1 + {R^2} - 2R\cos ({4{\pi }\sigma d\cos \theta } )} ]}}$ [51]. Therefore, the transmitted spectrum through the CCDIS can be expressed as
Suppose that the desired spectral resolution of the CCDIS at angle $\theta = 0$ is $\delta {\sigma _{CCDIS(0 )}}$. According to the desired spectral resolution $\delta {\sigma _{CCDIS(0 )}}$ and wavenumber range ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$, the central value of the FPI plate spacing and the number of the FPI scanning steps can be given: ${d_0}$ is a central value of the FPI plate spacing, and N is the number of scanning steps of the FPI. On the one hand, the free spectral range $FS{R_{\sigma (0 )}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}\cos \theta } )}}} \right.} {({2{d_0}\cos \theta } )}}$ must be able to be resolved by the grating system consisting of a plane transmission grating, a collecting lens and an area-array detector. On the other hand, to cover the full wavenumber range ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$, the FPI needs to scan N steps, where $N < {{{\pi }\sqrt R } \mathord{\left/ {\vphantom {{{\pi }\sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$. Accordingly, when $\theta = 0$, the free spectral range $FS{R_{\sigma 0}} = {1 \mathord{\left/ {\vphantom {1 {({2{d_0}} )}}} \right.} {({2{d_0}} )}}$ needs to be scanned N steps and the scanning interval is $\delta {\sigma _{CCDIS(0 )}}$, and so the wavenumber position of central spectral peak 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 ${\sigma _C} - ({{N \mathord{\left/ {\vphantom {N 2}} \right.} 2}} )\cdot \delta {\sigma _{CCDIS(0 )}}$ to ${\sigma _C} + ({{N \mathord{\left/ {\vphantom {N 2}} \right.} 2} - 1} )\cdot \delta {\sigma _{CCDIS(0 )}}$, where ${\sigma _C} = {1 \mathord{\left/ {\vphantom {1 {{\lambda_C}}}} \right.} {{\lambda _C}}}$ is the central wavenumber of the wavenumber range ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$. Therefore, the spectral resolution (in wavenumber) of the CCDIS at $\theta = 0$ can be calculated by
where $N < {{{\pi }\sqrt R } \mathord{\left/ {\vphantom {{{\pi }\sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$ and, for convenience, let N be an even number.The spectral resolution (in wavenumber) of the CCDIS at angle $\theta$ can be expressed as
For the point $A(0 )$ of the entrance slit: (1) 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), the maximum transmission wavenumbers of the FPI are ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}} )}}} \right.} {({2{d_k}} )}} \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$ (all values of m are integers), the $k \textrm{- th}$ spectrum including the wavenumbers ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}} )}}} \right.} {({2{d_k}} )}} \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$ (all values of m are integers) is obtained by the grating system; (2) a total of N spectra are sequentially obtained by the grating system in one scan period of the FPI, and these N spectra constitute a high-resolution continuous spectrum; (3) the resolving power of the CCDIS can be calculated by ${R_{CCDIS(0 )}} = {{2Nd} \mathord{\left/ {\vphantom {{2Nd} \lambda }} \right.} \lambda }$.
For the point $B({x^{\prime}} )$ of the entrance slit: (1) 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), the maximum transmission wavenumbers of the FPI are ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}\cos \theta } )}}} \right.} {({2{d_k}\cos \theta } )}} \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$ (all values of m are integers), the $k \textrm{- th}$ spectrum including the wavenumbers ${1 \mathord{\left/ {\vphantom {1 {{\lambda_K}}}} \right.} {{\lambda _K}}} \le \sigma = {m \mathord{\left/ {\vphantom {m {({2{d_k}\cos \theta } )}}} \right.} {({2{d_k}\cos \theta } )}} \le {1 \mathord{\left/ {\vphantom {1 {{\lambda_1}}}} \right.} {{\lambda _1}}}$ (all values of m are integers) is obtained by the grating system; (2) a total of N spectra are sequentially obtained by the grating system in one scan period of the FPI, and these N spectra constitute a high-resolution continuous spectrum.
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 maximum value of the FPI plate spacing is given byAccording to Eq. (22), the smaller the value of N, the smaller the value of wavelength ${\lambda _2}$. According to Eq. (7), the smaller the value of wavelength ${\lambda _2}$, the larger the value of $y({{\lambda_2}} )$, and therefore the smaller the value of $|{{y_1} - y({{\lambda_2}} )} |$. In other word, the smaller the value of N, the smaller the value of $|{{y_1} - y({{\lambda_2}} )} |$. Therefore, the value of N is determined by $N < {{{\pi }\sqrt R } \mathord{\left/ {\vphantom {{{\pi }\sqrt R } {({1 - R} )}}} \right.} {({1 - R} )}}$, $\delta {\sigma _{CCDIS(0 )}} = {1 \mathord{\left/ {\vphantom {1 {({2N{d_0}} )}}} \right.} {({2N{d_0}} )}}$ and $|{{y_1} - y({{\lambda_2}} )} |> b$.
In order to distinguish the maximum transmission wavenumbers of the FPI at each spacing position for each unit of the entrance slit, the resolution of the collecting lens, ${R_{CL}}$, should satisfy both ${R_{CL}} \le {{|{{y_1} - y({{\lambda_2}} )} |} \mathord{\left/ {\vphantom {{|{{y_1} - y({{\lambda_2}} )} |} 2}} \right.} 2}$ and ${R_{CL}} \le b$. We can choose $|{{y_1} - y({{\lambda_2}} )} |= 2b$ and ${R_{CL}} = b$.
The spectral resolution (in wavenumber) of the CCDIS at field angle $\phi$ can be given by
Some considerations for the implementation of the FPI are as follows. Assuming that there is a vacuum cavity between two substrates transparent to ultraviolet-visible spectrum, the outer surfaces of the substrates are wedged in order to prevent parasitic reflections, and the inner surfaces of the substrates are coated with high reflectivity films. Fused silica is a typical substrate material used in ultraviolet-visible spectral region. The higher the reflectance of the inner surfaces of the FPI plates, the higher the reflective finesse of the FPI, but the lower the transmission efficiency. Therefore, the choice of the reflectance of the inner surfaces of the FPI plates is a balance between the reflective finesse and the transmission efficiency.
The comparisons between the CCDIS and the CDIS reported in [46] are described in the following three aspects. Firstly, the role of the dispersive element (a dispersing prism or a transmission grating) in the CCDIS and the CDIS is not exactly the same. In the CDIS reported in [46]: the role of a dispersive element (a prism or a grating) is to divide a broad-band spectrum into a number of separate narrow-band spectra, so the wavelength range of each narrow-band spectrum is controlled by the dispersive element and detector geometry, but the spectral resolution of each narrow-band spectrum is determined by Michelson-type interferometer. In the CCDIS: the role of a transmission grating is to separate the overlapping orders of the FPI, so the spectral composition of each spectrum obtained at each FPI spacing position is controlled by the FPI, but the spectral resolution of each spectrum obtained at each FPI spacing position is determined by the transmission grating and detector geometry, which in turn affects the spectral resolution of the full spectral range. According to Eq. (4), the resolving power of a transmission grating is limited by the product of the number of grating lines and the diffraction order utilized (e.g. first order diffraction order of transmission grating is utilized in the CCDIS). Due to the limitation of the resolving power of a single grating to the overlapping orders of the FPI, the CDIS reported in [46] can achieve higher spectral resolution than the CCDIS.
Secondly, the method and sequence for acquiring the spectrum of each unit of the entrance slit in the CCDIS and the CDIS are not exactly the same. In the CDIS reported in [46]: for each unit of the entrance slit, the CDIS simultaneously produces multiple interferograms in one scan period of moving corner-cube mirror, each interferogram with a separate narrow-band spectrum, the Fourier transform of each interferogram yields a separate high-resolution narrow-band spectrum, and all of these high-resolution narrow-band spectra constitute a high-resolution broad-band spectrum. Furthermore, for a given unit of the entrance slit, each interferogram is located in a separate pixel of a given column of area-array detector. In the CCDIS: for each unit of the entrance slit, the CCDIS sequentially produces N low-resolution spectra in one scan period of the FPI (i.e., one low-resolution spectrum is obtained at each FPI spacing position), each low-resolution spectrum containing different wavelengths that satisfy the maximum transmission condition of the FPI at each spacing position, and these N low-resolution spectra constitute a high-resolution spectrum.
Thirdly, due to the advantages of the FPI over Michelson-type interferometer, compared with the CDIS reported in [46], the CCDIS has the following advantages: (1) the unique optical layout will make the CCDIS more compact; (2) when performing high-spectral-resolution measurement of a broadband source, the FPI only needs to scan N steps (e.g. $N < 61$ when the reflectance is 0.95) and the maximum displacement of the FPI plate spacing from the central value ${d_0}$ is very small, so the number of scanning steps of the CCDIS is much smaller than that of the CDIS.
Fourthly, compared with the CDIS reported in [46], the disadvantages of the CCDIS include two points: (1) the CCDIS is sensitive to mechanical and optical tolerances of the FPI plates; (2) the CDIS will be able to obtain higher spectral resolution than the CCDIS.
Table 1 shows the comparisons of the CCDIS and the CDIS reported in [46] for high-spectral-resolution broadband spectral imaging in ultraviolet-visible spectral region.
Table 1. Comparisons of the CCDIS and the CDIS reported in [46] for high-spectral-resolution broadband ultraviolet-visible spectral imaging
3. Preliminary numerical simulation with an example
Suppose that a source spectra covers a wavelength range from 270 nm to 500 nm (i.e., a wavenumber range from 20000 cm−1 to 37037.04 cm−1), where ${\lambda _1} = 270 \;\textrm{nm}$, ${\lambda _K} = 500 \;\textrm{nm}$, ${\lambda _C} = 380 \;\textrm{nm}$ (the central wavelength), and the desired spectral resolution of the CCDIS at the angle $\theta = 0$ is $\delta {\sigma _{CCDIS(0 )}} = 0.5 \;\textrm{cm}^{ - 1}$.
Fused silica is a proper material for a plane transmission grating used for the spectral range from 270 nm to 500 nm, and its refractive index formula is given by [52]
Fig. 3. The relationship between the central value of the FPI plate spacing and the number of the FPI scanning steps when the spectral resolution of the CCDIS at $\theta = 0$ is 0.5 cm−1.
The Andor Newton 940 Series 2048 × 512 pixel Spectroscopy CCD with 13.5 × 13.5 µm pixel size is suitable for the CCDIS, so let $b = 0.0135 \;\textrm{mm}$. According to Eqs. (7), (8), (21), (22) and (26), the value of $|{{y_1} - y({{\lambda_2}} )} |- b$ will increase with the increase of the focal length ${f_2}$, the number of grating grooves per millimeter (i.e., ${1 \mathord{\left/ {\vphantom {1 g}} \right.} g}$), the incident angle $\alpha$, or all three. Assuming that the focal length of the collecting lens is ${f_2} = 100 \;\textrm{mm}$, the transmission grating with 900 grooves/mm, and the incident angle of the plane transmission grating is $\alpha = 30^\circ$. The relationship between $|{{y_1} - y({{\lambda_2}} )} |- b$ and N is shown in Fig. 4. Together with the requirement of $|{{y_1} - y({{\lambda_2}} )} |- b > 0$ [see Eq. (21)], we can choose $N = 32$. From Eqs. (10) and (15), we can choose ${d_0} = 0.31236 \;\textrm{mm}$.
Fig. 4. The relationship between $|{{y_1} - y({{\lambda_2}} )} |- b$ and N (i.e., the relationship between ${y_1} - {y_2} - b$ and $N$) when the spectral resolution of the CCDIS at $\theta = 0$ is 0.5 cm−1.
In order to cover the full wavelength range from 270 nm to 500 nm, based on Eq. (19), the FPI plate spacing is approximately assigned from 0.312271 mm to 0.312455 mm. That is, the FPI should scan $N = 32$ steps of approximately 6 nm in step interval.
The y-axis coordinates on the detector plane of the wavelengths from ${\lambda _1} = 270 \;\textrm{nm}$ to ${\lambda _K} = 500 \;\textrm{nm}$ are shown in Fig. 5. The size of each column of the area-array detector must be greater than $|{{y_1} - {y_K}} |= 24.4956 \;\textrm{mm}$. Since ${{|{{y_1} - {y_K}} |} \mathord{\left/ {\vphantom {{|{{y_1} - {y_K}} |} b}} \right.} b} = {{24.4956mm} \mathord{\left/ {\vphantom {{24.4956mm} {0.0135mm}}} \right.} 0.0135{mm}} = 1814.5$, the number of pixels in each column of the area-array detector used to record the spectral image is $M = 1815$.
Fig. 5. The y-axis coordinates of different wavelengths at the detector plane when ${f_2} = 100 \;\textrm{mm}$, $\alpha = 30^\circ$ and the transmission grating with 900 grooves/mm.
From Eq. (18), the resolving power of the CCDIS at the angle $\theta = 0$ is shown in Fig. 6. When the number of pixels in each row of the area-array detector used to record the spectral image is $Q = 480$, based on Eq. (1), the maximum value of the incident angle within the FPI is ${\theta _{\max }} = \arctan ({{{480 \times 0.0135mm} \mathord{\left/ {\vphantom {{480 \times 0.0135mm} {({2 \times 100mm} )}}} \right.} {({2 \times 100mm} )}}} )\approx 1.856^\circ$. The resolving power of the CCDIS at the angle ${\theta _{\max }} = 1.856^\circ$ is shown in Fig. 7. Based on Eq. (17), the spectral resolution in wavelength of the CCDIS is shown in Fig. 8.
Fig. 6. Resolving power of the CCDIS at the angle $\theta = 0$ for the wavelength range from 270 nm to 500 nm when $N = 32$ and ${d_0} = 0.31236\;mm$.
Fig. 7. Resolving power of the CCDIS at the angle ${\theta _{\max }} = 1.856^\circ$ for the wavelength range from 270 nm to 500 nm when $N = 32$ and ${d_0} = 0.31236\;mm$.
Fig. 8. Spectral resolution in wavelength of the CCDIS at the angle $\theta = 0$ and the angle ${\theta _{\max }} = 1.856^\circ$ for the wavelength range from 270 nm to 500 nm when $N = 32$ and ${d_0} = 0.31236\;mm$.
When the spectral resolution in wavenumber of the CCDIS is $\delta {\sigma _{CCDIS(0 )}} = 0.5 \;\textrm{cm}^{ - 1}$, according to Eq. (14), a spectrum with a spectral resolution in wavelength of approximately 0.007 nm is shown in Fig. 9, which is synthesized from three low-resolution spectra. The first low-resolution spectrum obtained at $d = 0.312354\;mm$ contains only wavelength 379.762 nm, 379.993 nm and 380.224 nm, and its spectral resolution in wavelength is approximately 0.231 nm. The second low-resolution spectrum obtained at $d = 0.31236\;mm$ contains only wavelength 379.769 nm, 380 nm and 380.231 nm, and its spectral resolution in wavelength is approximately 0.231 nm. The third low-resolution spectrum obtained at $d = 0.312366 \;mm$ contains only wavelength 379.776 nm, 380.007 nm and 380.238 nm, and its spectral resolution in wavelength is approximately 0.231 nm. These three low-resolution spectra (with a spectral resolution in wavelength of approximately 0.231 nm) constitute a high-resolution spectrum with a spectral resolution in wavelength of approximately 0.007 nm.
Fig. 9. Spectrum with a spectral resolution in wavelength of approximately 0.007 nm when the spectral resolution in wavenumber of the CCDIS is $\delta {\sigma _{CCDIS(0 )}} = 0.5 \;\textrm{cm}^{ - 1}$.
4. Conclusion
The principle of the CCDIS is described, the comparison between the CCDIS and the CDIS reported in [46] is given, the results of preliminary numerical simulation are shown, and the theoretical approximations of system performance based on the first-order properties of components are given. The combination of imaging, interferometric spectrometry and dispersive spectrometry enables the CCDIS to obtain both spatial information and high-resolution spectral information of an ultraviolet-visible broadband source. Compared with traditional dispersive imaging spectrometers or interferometric imaging spectrometers, the CCDIS can obtain high-resolution broadband spectral information in the ultraviolet-visible spectral region. Compared with the CDIS integrating an entrance slit, a moving corner-cube-mirror interferometer and a dispersing prism (or a transmission grating), the advantages of the CCDIS mainly include two aspects: (1) compactness; (2) the number of scanning steps is very small. However, the disadvantages of the CCDIS mainly focus on two points: (1) the CCDIS is sensitive to mechanical and optical tolerances of the FPI plates; (2) because the resolving power of a single grating will limit the spectral resolution of the CCDIS, the CDIS will be able to obtain higher spectral resolution than the CCDIS. In summary, the CCDIS provides a good method that uses only a compact, lightweight instrument to obtain both spatial information and high-resolution spectral information of a broadband source in the ultraviolet-visible spectral region.
Funding
National Natural Science Foundation of China (61605151).
Disclosures
The authors declare no conflicts of interest.
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