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Abstract
The classic Offner spectrometer has the advantages of having a simple structure, light weight, and high imaging quality. It is easily to achieve a fixed spectral resolution but cannot meet usage requirements. Therefore, we present a practical method for designing a spectrometer with variable spectral resolution. Multiple off-axis convex (OAC) gratings are used to replace the convex grating in the classic Offner spectrometer. We derive the principle through ray tracing and establish an optimization process for the basic parameters of multiple OAC gratings. To demonstrate this method, a corresponding system is designed. The results show that a variable spectral resolution, with a variation ratio close to 4, of 0.45–1.91 nm is achieved over a wide bandwidth of 460–900 nm. Additionally, the smile and keystone of the system are well corrected.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, spectral imaging technology has been widely used in various fields such as food safety [1,2], environmental monitoring [3,4], archaeological and art preservation [5,6], agriculture [7], and biomedicine [8]. While spectrometers are used to identify and classify a wide range of complex objects, there are higher demands on the speed of spectral data acquisition and processing. In addition, spectrometers with a large field of view and high resolution are desired. However, such spectrometer systems will generate huge image data cubes, which will reduce the efficiency of information processing and transmission.
In practical applications, in the observation area of the full field, different targets require different spectral resolutions to be identified. If the spectrum data cube of all objects in the full field is acquired only with the highest resolution, the unnecessary data volume and transmission pressure are increased. If the spectrum data cube of all objects in the full field is acquired only at the lowest resolution, targets that require hyperspectral cannot be recognized. More specifically, we need a spectrometer that can simultaneously obtain high spectral resolution imaging of the target of interest and low spectral resolution imaging of the full field. When we only need the monitoring information of the full field, the low-resolution output data can meet the requirements. If we want to accurately analyze the spectral characteristics of the interested target, the high-resolution output data can meet the requirements. Therefore, a spectrometer with variable spectral resolution is urgently required to realize the above conditions.
Currently, spectrometers with variable spectral resolution can be broadly classified into three categories: First, spatial light modulation devices such as DMD [9,10] are used to modulate light at the slit to achieve changes in spatial and spectral resolutions. This approach can only obtain the spectral information of one modulation mode per imaging scan. Specifically, spectral information at different spectral resolutions cannot be obtained using a single scan. In the second category, different spectral resolutions are obtained using multiple channel imaging systems, which require multiple detectors [11] or multiple slits, gratings, and reflectors stitched together [12]. This approach is structurally complex and does not meet the compactness requirements. In the third category, a spectrometer with variable spectral resolution is achieved by designing and machining varied-line-spacing gratings [13,14], which involves a complex and costly machining process.
Classic Offner spectrometers have a high imaging quality, simple structure, and smaller volume and weight than those of plane grating imaging spectrometers [15]. In this study, we took advantage of the simplicity and compactness of the traditional Offner spectrometer and made improvements accordingly. Herein, we propose a design concept for a spectrometer with variable spectral resolution based on multiple off-axis convex (OAC) gratings, which are used to replace the convex spherical grating in the classical Offner structure.
The remainder of this paper is organized as follows: Section 2 presents the structural principle of the spectrometer with variable spectral resolution and multiple OAC gratings. According to this principle, each beam of light diffracted from an OAC grating generates a sub-spectral image at the image plane. However, each sub-spectral image overlaps, so a method is required to avoid this. Section 3 presents the automatic optimization process established to prevent overlapping of sub-spectral images, and calculations of the initial parameters of a well example. Section 4 describes further optimization of the design example by considering the distortion and increasing the optimizable parameters of the system. The spectral resolution and distortion are also analyzed. As a result, the spectral resolution variation ratio is found to be approximately 4. To prove the practicability of the system, we analyzed the SNR, as shown in Section 5. Finally, conclusions are presented in Section 6.
This novel structure provides different spectral resolution imaging through a single scan and has the advantage of being both simple and compact. This method does not cause a large number of problems while obtaining hyperspectral data. In addition, compared with the flexibility of digital spectrometers, the proposed spectrometer does not require additional digital components, and has strong environmental adaptability and compact structure, and can be applied to the infrared band. Thus, it can be used for agricultural monitoring, crop identification, and classification.
2. Structural principle
We substituted multiple OAC gratings for each subregion of the convex grating in the classic Offner spectrometer. The parameters of each OAC grating include the curvature, grating density, decenter, diameter, angle of incidence, and diffraction angle. The algorithms used to solve these parameters are described in this section. Moreover, the spectrometer is uniformly defined in the global coordinate system through coordinate transformations to improve the convenience of construction.
2.1 Parameter definitions for the spectrometer
A classic Offner spectrometer in the YZ plane is displayed in Fig. 1. The beam emitted from the slit is reflected by the primary mirror and then diffracted at the convex grating. The tertiary mirror focuses the dispersive beam onto the image plane. The primary and tertiary mirrors are on the same spherical surface. Figure 2 presents a diagram of the ray tracing of the Offner structure in the XYZ three-dimensional space. The origin of the coordinate system O is at the center of the surface where the primary and tertiary mirrors are located. The coordinate system is a Cartesian three-dimensional coordinate system. The line, i.e., the Z-axis, between the center of the grating surface and the origin of the coordinate system is the optical axis of the Offner spectrometer, which is positive from the grating toward the origin. The XOY plane is perpendicular to the Z-axis.
Fig. 2. Schematic of light tracing *Coordinate system XYZ is the global coordinate and O is the origin of the global coordinate system. Coordinate system X'Y'Z’ represented by the dotted line at the slit is the local coordinates. The center of the slit is the origin of the local coordinate system, the direction of which is the same as that of the global coordinate system. (0, y0, z0) are the coordinates of the slit center in the global coordinate system.
At the slit, the X-axis indicates the slit length and the Y-axis indicates the slit width. As shown in Fig. 2, one of the rays originating from the slit is traced. To facilitate the corresponding calculation, we take the center point at the slit x0 = 0 as the starting point of the ray tracing. (0, y0, z0), (xp, yp, zp), (xg, yg, zg), $({{x_{\textrm{t},{\lambda_n}}},{y_{\textrm{t},{\lambda_n}}},{z_{\textrm{t},{\lambda_n}}}} )$, and $({{x_{\textrm{i},{\lambda_n}}},{y_{\textrm{i},{\lambda_n}}},{z_{\textrm{i},{\lambda_n}}}} )$ denote the coordinates of the ray at the center of the slit, the primary mirror, the grating, the tertiary mirror, and the image plane, respectively. $\overrightarrow {{l_{\textrm{sp}}}} $, $\overrightarrow {{l_{\textrm{pg}}}} $, $\overrightarrow {{l_{\textrm{gt},{\lambda _n}}}} $, and $\overrightarrow {{l_{\textrm{ti},{\lambda _n}}}} $ denote the light vector from the slit to the primary mirror, the primary mirror to convex grating, the convex grating to the tertiary mirror, and the tertiary mirror to the image plane, respectively. In the image plane, the Y-coordinate represents the coordinates of the spectral information at a certain spatial position, and the X-coordinate represents the coordinates of the spectral information at different adjacent spatial positions. , $\overrightarrow {{n_g}} $, and $\overrightarrow {{n_{t,{\lambda _n}}}} $ denote the unit normal vector of the primary mirror in (xp, yp, zp), convex grating in (xg, yg, zg), and tertiary mirror in $({{x_{\textrm{t},{\lambda_n}}},{y_{\textrm{t},{\lambda_n}}},{z_{\textrm{t},{\lambda_n}}}} )$, respectively.
θ is the angle between $\overrightarrow {{l_{\textrm{sp}}}} $ and the Z-axis, and φ is the angle between the projection of $\overrightarrow {{l_{\textrm{sp}}}} $ in the XY plane and the X-axis. Based on the relationship of the space geometry, θ and φ can be expressed by some parameters of the global coordinate system, as follows:
The primary mirror is spherical with curvature cp. The substrate of the convex grating is a spherical surface with curvature cg and D is the distance from the primary mirror to the convex grating. Thus, zp and zg are given by:
Equations (5) and (6) are obtained according to the law of reflection.
Similarly, $\overrightarrow {{n_\textrm{g}}} $ is given by:
According to Eqs. (1)–(10), the angle of incidence θi on the convex grating can be obtained from Eq. (11).
For a convex grating, ${\theta _{e,{\lambda _n}}}$ is the diffraction angle. The relationship between θi and ${\theta _{e,{\lambda _n}}}$ is determined by the grating density T in lines per millimeter (g/mm), given by:
where M is the diffraction order, equal to 1(the diffraction order other than the utilized first positive from one grating will not affect the neighboring configuration), and λn is the wavelength of the dispersed beam. Based on the relationship of space geometry, ${\theta _{e,{\lambda _n}}}$ can also be expressed as:When a single ray is reflected by the grating, the relationship between the incident light, reflected light, and the unit normal vector of the grating at the incident point can be expressed as:
The tertiary mirror is spherical with curvature ct. Then, ${z_{\textrm{t},{\lambda _n}}}$ is given by:
Equations (16 and (17) were obtained according to the law of reflection.
According to Eq. (12), when T is determined, it follows that ${\theta _{\textrm{e},{\lambda _n}}}$. Further, according to Eqs. (13)–(20), $({{x_{\textrm{i},{\lambda_n}}},{y_{\textrm{i},{\lambda_n}}},{z_{\textrm{i},{\lambda_n}}}} )$ can be obtained. More specifically, using the model comprising Eqs. (1)–(20), the initial structure of interest can be finalized once the relevant limitation factor is input into the model.
2.2 Substitution of multiple OAC gratings
There are m OAC gratings: g1, …, gm−1, gm. Figure 3(a) shows a diagram of the OAC gratings in the YZ plane. An OAC grating is simply a side section of a parent convex grating with a decenter d (Y offset in the tangential plane). g1, …, gm−1, gm were fitted to a sphere of curvature cg. The diameter of the OAC gratings is ϕy. The Y-axis coordinates of the edge ray and the center ray through the multiple OAC gratings are yp,max, yp,min, and yp,O. Figure 3(b) presents a diagram of the multiple OAC gratings in the XY plane. Each OAC grating has a different grating density, labeled T1, …, Tm−1, Tm. Therefore, m OAC gratings diffract m sub-spectral images with different dispersion widths.
Figure 4 shows the replacement of the convex grating in the classic Offner structure with the proposed multiple OAC gratings. The beam is diffracted at multiple OAC gratings with different grating densities Tm. According to Eq. (12), the diffraction angle $\theta_{e,{\lambda_n},{g_m}}$ of each OAC grating is influenced by the wavelength $\lambda_{n}$ and Tm. Moreover, according to Eqs. (13)–(20), the coordinates of the beam on the image plane also depend on λn and Tm. Figure 5 shows the image plane in the XY plane. The beam in the λ1∼λn bandwidth (central wavelength λc) is diffracted at OAC grating gm and focused on the image plane by the tertiary mirror; its dispersion width Δpm is expressed as:
where ${y_{\textrm{i},{\lambda _n},{g_m}}}$ and ${y_{\textrm{i},{\lambda _1},{g_m}}}$ are the Y-coordinates of the spectral lines with λn and λ1 generated by gm in the image plane. Similarly, the dispersion width Δpm−1 generated by gm−1 is expressed as:Δpm, Δpm−1, and Δhm,m−1 can be expressed by Y-coordinates on the image plane, which depend on λn and Tm. Therefore, the occurrence of sub-spectral image overlap is mainly determined by the wavelength λ1∼λn and the grating density T1, …, Tm−1, Tm of the OAC gratings.
2.3 Distortion definition
Owing to the dispersive behavior, two types of distortion require special attention when evaluating the spectrometer performance. These are spectral distortions (smiles), which are bent monochromatic line images along the X-axis, and spatial distortion (keystones), which refer to wavelength-dependent bent slit images along the Y-axis [16,17]. In this study, we used Eqs. (25) and (26) to define these two distortions [18]. Smile distortion is defined in Eq. (25), where ${y_{\textrm{max},{\lambda _n},{g_m}}}$ and ${y_{\textrm{min},{\lambda _n},{g_m}}}$ represent the maximum and minimum Y-coordinates of the bent monochromatic line in the sub-spectral image, respectively. Keystone distortion is defined in Eq. (26), where ${x_{\textrm{max},{\lambda _n},{g_m}}}$ and ${x_{\textrm{min},{\lambda _n},{g_m}}}$ represent the maximum and minimum X-coordinates in the same slit imaging in the sub-spectral image, respectively, both of which are wavelength dependent.
2.4 Definition of spectral resolution
The spectral resolution is determined by the detector pixel size aD, RMS spot radius aR,λ, dispersion width Δpm, and bandwidth λ1∼λn. When aR,λ is smaller than aD, the spectral resolution is limited by aD and is expressed as Eq. (27). The variation ratio βD of the spectral resolution of the spectrometer with multiple OAC gratings is given by Eq. (28). When aD and λ1∼λn of the system are determined, βD can be obtained by the maximum dispersion widths Δpmax and minimum dispersion widths Δpmin of the image plane. When aR,λ is larger than aD, the spectral resolution is limited by aR,λ, as expressed in Eq. (29), and βD can be realized using Eq. (30). The RMS spot radius varies with wavelength, so the spectral resolution variation ratio is determined by not only Δpmax and Δpmin, but also aR,λ at different wavelengths. In Section 3, for facilitation, Eq. (28) is used as the basis for the initial calculation in the optimization process of the spectrometer with multiple OAC gratings.
3. Spectrometer construction
3.1 Design specification
The following example details the methodology of a spectrometer with variable spectral resolution. A classic Offner spectrometer is designed as the initial system for the OAC grating spectrometer. A wide bandwidth of 460–900 nm was considered. The spectrum in this range was divided into thirteen sampling wavelengths, labeled as λ1∼λ13, and the central wavelength is 680 nm, labeled as λ7. The numerical aperture for the divergent beam from the entrance slit was set to 0.145, which matched the optical system with an F value of 3.4. A detector with 2,048 px × 2,048 px was chosen, with the pixel size of 6.5 µm × 6.5 µm. A convex grating with the grating density T of 150 g/mm, and the first positive diffraction order were utilized. The basic parameters of the initial system are listed in Table 1.
Table 1. Basic parameters of initial system
3.2 Optimization process of the spectrometer with multiple OAC gratings
We replaced the convex grating with multiple OAC gratings based on the initial system. In this example, the convex grating was split into three OAC gratings with grating densities T1, T2, and T3, and decenters ${d_{{g_1}}}$, ${d_{{g_2}}}$, and ${d_{{g_3}}}$. The optimization process is depicted in Fig. 6.
First, the parameters of the initial system were fed into the optimization process. For easy manufacturing, the grating density should be greater than 40 lp/mm and less than 1,000 lp/mm. The throughput aperture of each OAC grating should be approximately the same to ensure that the energy of each sub-spectral image is balanced. Therefore, we define the decenters of the OAC grating to satisfy ${d_{{g_2}}} = 0$ and $|{{d_{{g_1}}}} |= |{{d_{{g_3}}}} |= \frac{1}{3}{\phi _y}$. The Y-coordinates ${y_{{\textrm{p}_\textrm{c}},{g_1}}}$, ${y_{{\textrm{p}_\textrm{c}},{g_2}}}$, and ${y_{{\textrm{p}_\textrm{c}},{g_3}}}$ of the central rays of the three OAC gratings can be expressed as Eqs. (31)–(33).
Here, ${\phi _y} = |{{y_{\textrm{pmax}}} - {y_{\textrm{pmin}}}} |$. According to Eqs. (1)–(11), the incidence angles of the central ray in the OAC grating can be obtained. Second, the bandwidth and central wavelength are input into the optimization process. According to Eqs. (21)–(23), the dispersion widths Δp1, Δp2, and Δp3, and offsets Δh12 and Δh23 of the three sub-spectral images can be obtained. Then, the optimization process determines if there are overlaps between the sub-spectral images according to Eq. (24). In addition, according to the effective detector size L, the sum of the three dispersion widths must satisfy Eq. (34). If the optimization result does not satisfy Eqs. (24) and (34), the process returns to reset the grating density. If it is satisfied, the process outputs results for T1, T2, T3, ${d_{{g_1}}}$, ${d_{{g_2}}}$, ${d_{{g_3}}}$, βD, and ${y_{\textrm{i},{\lambda _n},{g_1}}} - {y_{\textrm{i},{\lambda _1},{g_3}}}$. The spectral resolution variation ratio βD can be derived using Eq. (28). ${y_{\textrm{i},{\lambda _n},{g_1}}} - {y_{\textrm{i},{\lambda _1},{g_3}}}$ denotes the Y-axis space occupied by all the dispersive spectral lines in the image plane.
Through the above optimization process, the five sets of data shown in Table 2 satisfying Eqs. (24) and (34) are output. In the first and fourth sets of data, βD is higher than 4. However, the ${y_{\textrm{i},{\lambda _n},{g_1}}} - {y_{\textrm{i},{\lambda _1},{g_3}}}$ value of 8.45 mm in the first dataset is much smaller than the effective detector size L of 13 mm, which reduces the detector utilization. Meanwhile, the fifth dataset shows a higher detector utilization but a lower spectral resolution variation ratio. In this study, we chose the fourth set of data for the initial parameters of the multiple OAC gratings.
Table 2. Five sets of optimization results
4. Performance analysis
Further optimization was performed using the constructed spectrometer. To increase the optimization parameters of the system, the primary and tertiary mirrors of the initial system were separated so that they are not in one mirror and have different curvatures. Meanwhile, the smile and keystone are controlled according to Eqs. (25) and (26). The results are presented in Fig. 7. The colors of the rays in Fig. 7(a) indicate different wavelengths. The colors of the light in Fig. 7(b) indicate the dispersive spectral lines diffracted from each OAC grating. The parameters of the spectrometer with multiple OAC gratings are listed in Table 3, and those of the multiple OAC gratings are provided in Table 4. The grating comprised three OAC gratings with grating densities of 300, 150, and 70 g/mm, respectively. The decenters of the three OAC gratings were adjusted slightly to improve the detector utilization, to 5.5, 0.45, and −4.5 mm, respectively. The grating aperture is 15.4 mm.
Table 3. Parameters of spectrometer with multiple OAC gratings
Table 4. Parameters of multiple OAC gratings
For our optimization, we reduced the distortion by optimizing the curvature and distances of the optical elements. The spectral smiles and keystones are shown in Fig. 8. The maximum smile, produced by OAC grating 1 at the end of the slit at the wavelength of 900 nm, is less than 3.5 µm, which is approximately half of the pixel size. In addition, the maximum keystone produced by the OAC grating 1 at the end of the slit at 900 nm is less than 3.2 µm, which is also approximately half of the pixel size.
Fig. 8. Distortion of spectrometer with multiple OAC gratings. (a)–(c) Spectral smiles for different wavelengths; (d)–(f) Spectral keystones for 0, 0.2, 0.4, 0.6, 0.8, and 1 normalized fields of view
Figure 9 displays the full-field spot diagram in the image plane. The X-axis indicates the imaging of the slit, and the coordinates are represented by the normalized field of view. The Y-axis indicates the dispersion spectral lines of the system, and the colors represent different dispersion wavelengths. Figure 9 shows that the system generates three sub-spectral-images with dispersion widths of Δp1, Δp2, and Δp3 in the image plane.
Figure 10 presents a comparison of the RMS spot radius for the three sub-spectral images. The RMS spot radius diffracted by OAC grating 3 from 460 to 900 nm is smaller than the detector pixel size of 6.5 µm. The RMS spot radius diffracted by OAC gratings 1 and 2 are smaller than the detector pixel size of 6.5 µm from 695 to 900 nm and from 460 to 812 nm, respectively.
The spectral resolution of the system, according to the definition provided in Section 2.4, is shown in Fig. 11(a). When the wavelength is 460–636 nm and 900 nm, the spectral resolution diffracted by OAC grating 1 is between 0.46 and 0.49 nm, and is limited by the RMS spot radius. When the wavelength lies between 665 and 871 nm, the spectral resolution, to which the detector pixel size is the main limitation factor, has the value of 0.45 nm. Moreover, when diffracted by OAC grating 2, the spectral resolution varies between 0.90 and 0.93 nm with the wavelength of 841–900 nm. The spectral resolution is 0.89 nm with wavelength of 460–812 nm. For these two spectral resolutions, the former is limited by the RMS spot radius, while the latter is limited by the detector pixel size. When OAC grating 3 is working, the spectral resolution is 1.91 nm at the full wavelength range of 460–900 nm and is limited by the detector pixel size.
Fig. 11. (a) Spectral resolution diffracted by three OAC gratings; (b) Spectral resolution variation ratio
According to Eqs. (28) and (30), the spectral resolution variation ratio of the system was calculated, as shown in Fig. 11(b). β is in the range of 3.86–3.94 at the band of 489–607 nm, and greater than the spectral resolution variation ratio of 4.02 for the initial parameters presented in Section 3.2, at the rest of the bands. The highest β value is 4.27.
5 Prediction of the signal-to-noise ratio
The two main factors affecting the SNR of each sub-image with different spectral resolutions are the diffraction efficiency of the OAC grating and the relative aperture of the optical system. According to the processing level of the convex blazed gratings [19,20], the average diffraction efficiencies of the three OAC gratings can reach 60%, 50%, and 40%. Because the convex grating is divided into three OAC gratings, the F value corresponding to each OAC grating is approximately 10. To reduce the etalon effect in the hyperspectral system, we used a detector from HORIBA, model 4.2MP-U-6.5-BI. In the same integration of 0.3 s, the SNR shown in Fig. 12, which was determined while considering the atmospheric effects, is not less than 80 for the three different spectral resolution images. The SNRs of the sub-images generated by the three OAC gratings all meet the detection requirements. Therefore, the proposed system can obtain images with different spectral resolutions at the same integration time. Moreover, the SNR will be higher in the case of overlaying the frame frequency or spatial merging. In addition, we can choose a larger detector and design an optical system with a smaller F number according to the design method in this study. The detailed process of SNR analysis is provided in Supplement 1.
6 Conclusions
We introduced a construction method for designing a spectrometer with variable spectral resolution and multiple OAC gratings. We derived the relationship between the image plane coordinates, the basic parameters of the multiple OAC gratings, and the system wavebands via ray tracing. Based on the theory presented in Section 2, we established an optimization process to calculate the optimal initial parameters of the multiple OAC gratings and designed an example system. We also controlled the system smile and keystone. The results demonstrate that the system distortion was well corrected. Finally, we obtained a spectral resolution variation ratio close to 4. Specifically, this system can obtain spectral imaging with the spectral resolution variable from 0.45 to 1.91 nm. Users can choose spectral imaging with different resolutions according to their own needs.
In the example provided in Section 3, the grating comprised three OAC gratings. In future applications, the system can obtain more different resolutions of spectral imaging using more than three OAC gratings. To ensure a high SNR, larger detector sizes will be used when the grating consists of more OAC gratings. Alternatively, if the basic parameters in the example are used, the optimization process can output a higher spectral resolution variation ratio when the detector size is larger. In this paper, we provide an example of a 4-fold spectral resolution variation ratio, but our proposed construction method also enables design of a spectrometer with variable spectral resolution having different specifications according to the requirements. Thus, the proposed system can be widely used in various fields.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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