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Abstract
Snapshot hyperspectral imaging is superior to scanning spectrometers due to its advantage in dimensionality, allowing longer pixel dwell time and higher data cube acquisition efficiency. Due to the trade-off between spatial and spectral resolution in snapshot spectral imaging technologies, further improvements in the performance of snapshot imaging spectrometers are limited. Therefore, we propose a cemented-curved-prism-based integral field spectrometer (CIFS), which achieves high spatial and high spectral resolution imaging with a high numerical aperture. It consists of a hemispherical lens, a cemented-curved-prism and a concave spherical mirror. The design idea of aplanatic imaging and sharing-optical-path lays the foundation for CIFS to exhibit high-resolution imaging in a compact structure. The numerical model between the parameters of optical elements and the spectral resolution of the system is established, and we analyze the system resolution influenced by the hemispherical lens and the cemented-curved-prism. Thus, the refractive index requirements of the hemispherical lens and the cemented-curved-prism for the optimal spatial and spectral resolution imaging of the system are obtained, providing guidance for the construction of CIFS. The designed CIFS achieves pupil matching with a 1.8 f-number lenslet array, sampling 268 × 76 spatial points with 403 spectral channels in the wavelength band of 400 to 760 nm. The spectral and spatial resolution are further evaluated through a simulation experiment of spectral imaging based on Zemax. It paves the way for developing integral field spectrometers exhibiting high spatial and high spectral resolution imaging with high numerical aperture.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As an effective product for hyperspectral imaging application, the imaging spectrometer obtains not only the spatial information of the target directly, but also its material structure according to the characteristic spectrum, realizing the combination of spatial and spectral dimensions [1–5]. With recent advances in hyperspectral imaging technologies, a series of spectral imagers called snapshot hyperspectral imaging spectrometer have emerged, which refer to the instantaneous acquisition of the data cube (x, y, λ) in a single shot [6–9]. The exposure time per voxel in snapshot imaging techniques is the same as the time required to obtain the data cube, resulting in higher light collection efficiency and better consistency of the data cube compared to scanning techniques [10–13]. The inherent advantages of snapshot imaging have driven this technology to become widely used in remote sensing, biomedical research, agricultural detection and many other fields [14–16].
Integral field spectroscopy (IFS) is a kind of snapshot hyperspectral imaging and one of the most efficient means of performing 3-D imaging spectroscopy [17,18]. IFS adopts either fiber arrays, mirror arrays or lenslet arrays to segment the input image [19,20], leaving empty pixel space on the detector to provide a spectrum for each spatial sample of the field of view (FOV), and one-to-one correspondence between voxels in the data cube and pixels on the detector array is established. Among them, lenslet array IFS has obvious advantages due to the mature performance and low processing cost of the core element lenslet array. Typically, the IFS technology divides full FOV with lenslet array as a carrier and remaps a 3-D data cube to a 2-D detector array, enabling parallel measurements of each voxel in the data cube. Although some new structural designs of integral field spectrometers based on lenslet arrays have advanced spatial and spectral resolution, further improvements are still needed.
In lenslet array IFS, high-density lenslet arrays provide more sufficient spatial sampling of the input image, enabling the imaging spectrometer to obtain more spatial details of the target. Low f-number lenslet arrays allow the pupil images occupying fewer pixels on the detector, leaving more vacant pixel space for further enhancement of the dispersion corresponding to each spectrum. Dwight and Tkaczyk proposed a snapshot imaging spectrometer for hyperspectral fluorescence microscopy, enabling the acquisition of a 200 × 200 × 27 data cube from 515 to 635 nm in its fixed mode [21]. The system captures more spatial sampled points by introducing a high-density lenslet array, but the pupil mismatch between optical elements results in a great loss of light throughput, reducing its potential to obtain more spatial and spectral information.
Imaging spectrometers using dispersion as their spectroscopic system are often more compact in structure and stable in performance, which are adopted by many designers. A coronagraphic high angular resolution imaging spectrograph was proposed by Kasdin et al [22], which provides a 138 × 138 × 72 data cube in the wavelength band of 1.15 to 2.4 µm. The selection of a single prism as its spectroscopic system introduces complex aberrations, which is not conducive to the acquisition of multiple spectral channels, and the system sacrifices the light throughput from the foreoptics to ensure the expected number of spectral channels. Due to the mutual opposition of the dispersion spectra of prisms and gratings, their combination is conducive to correcting spectral distortion and obtaining high spectral resolution. Spectrometers based on prism-grating dispersion have been adopted by some systems [23,24]. In 2018, a lenslet-array-type hyperspectral microscope system based on prism-grating dispersion with double Gaussian symmetry structure was proposed by Liu et al [25]. The operating band of the system is 400 to 800 nm with spectral resolution better than 1 nm. The system was improved and evaluated by Yu et al [26,27], which captures 28 × 14 spatial points with 180 spectral channels in the wavelength band of 500 to 600 nm. In addition to introducing up to 15 optical elements, the system also suffers from the crosstalk between adjacent diffraction orders, resulting in a trade-off between FOV and spectral resolution.
To meet different application requirements, systems with tunable spatial and spectral resolution are proposed [28,29], which are realized by replacing optical elements, introducing variable focal length lenses and other means. Such systems exhibit excellent flexibility and interactivity, though, they fail to extend the three available dimensions of the data cube to increase the total number of voxels. Therefore, it is still a challenge to achieve both high spatial and high spectral resolution imaging.
In this paper, we propose a cemented-curved-prism based integral field spectrometer (CIFS), which avoids the above problems. The CIFS consists of a hemispherical lens, a cemented-curved-prism and a concave spherical mirror, and the design idea of aplanatic imaging and sharing-optical-path lays the foundation for CIFS to achieve high resolution imaging in compact structure. The system resolution influenced by the hemispherical lens and the cemented-curved-prism is analyzed, providing guidance for establishing the CIFS that enables high spatial and high spectral resolution imaging with high numerical aperture. Simulation results show that CIFS achieves pupil matching with a 1.8 f-number lenslet array and provides 268 × 76 spatial sampled points with 403 spectral channels in the wavelength band of 400 to 760 nm.
2. Design idea
According to the theory of first-order optics, we determine the initial structural parameters of the CIFS, which satisfies aplanatic imaging. Furthermore, the numerical model between the parameters of optical elements and the spectral resolution of the system is established, and we analyze the system resolution influenced by the hemispherical lens and the cemented-curved-prism. Thus, the refractive index requirements of the hemispherical lens and the cemented-curved-prism for high spatial and high spectral resolution imaging of CIFS are obtained, providing guidance for the construction of CIFS.
2.1 Determination of initial structural parameters
The imaging spectrometer consists of a hemispherical lens, a cemented-curved-prism and a concave spherical mirror, as shown in Fig. 1. The hemispherical lens and the concave spherical mirror share a curvature center C1, which is located at the intersection of the Y-axis and the Z-axis. C2, C3 and C4 correspond to the curvature centers of the three spherical surfaces of the cemented-curved-prism, which are all distributed on the Y-axis.
Fig. 1. Schematic diagram of the chief ray path of a single wavelength in the imaging spectrometer. (a) The light path of the incident ray before passing through the spherical mirror. (b) The light path of the incident ray after passing through the spherical mirror.
Through paraxial ray tracing, we obtain:
The height of the final image point h'9 can be obtained by iterative calculation of Eq. (3) and simplified by Eq. (5), and when Δh3=-Δh2, the dispersion width w of the polychromatic light passing through the imaging spectrometer can be obtained as:
Based on the analysis of the system structure, the numerical model between the parameters of optical elements and the spectral resolution of the system is established. Furthermore, the system resolution influenced by the hemispherical lens and the cemented-curved-prism is analyzed as follows.
2.2 System resolution influenced by the hemispherical lens
The refractive index n1 of the hemispherical lens existing in Eq. (6) as the denominator terms is directly related to the dispersion width w. The change of the refractive index n1 is realized by replacing the hemispherical lens with different materials, without changing the structural parameter. The relationship between refractive index n1 and dispersion width w is shown in Fig. 3(a). The purple circles correspond to the values of the dispersion width w at different refractive indices n1, which are distributed in the shaded area. The blue curve reflecting the trend of dispersion width w is obtained by polynomial fitting of sampled data. In general, the dispersion width w decreases with the increase of n1. The color bar on the right reflects the distribution of Δn1 (Δn1 = n1,s-n1,l) through the gradient, as shown in Fig. 3(b). When n1 remains constant, the dispersion width w has a certain floating range within ±0.035 mm due to the difference of Δn1. The dispersion width w decreases slightly with the increase of Δn1, i.e., both refractive index n1 and refractive index difference Δn1 contribute to the spectral resolution, and the former has much more influence on the system resolution than the latter. Figure 3(c) and Fig. 3(d) show the distribution of root mean square (RMS) radii corresponding to short wavelength (400 nm) and long wavelength (760 nm) in the shaded area, respectively. The RMS radius corresponds to the outermost FOV with the largest diffuse spot, which obviously reflects the aberration change of the coaxial catadioptric system. The color bar on the right reflects the distribution of RMS radii through the gradient. The areas with RMS radii within 4.0 µm are marked by dotted lines and defined as reasonable imaging areas with high spatial resolution. A lower refractive index n1 is conducive to high spatial resolution imaging at short wavelength. With the increase of corresponding wavelength, the refractive index n1 for high spatial resolution imaging increases gradually. In order for the system to be expected to achieve a spectral resolution comparable to that shown in the model, the marked areas corresponding to the two extreme spectral bands are intersected to obtain the overlap area where high spatial and high spectral resolution imaging can be achieved in the full working band. Therefore, the selection range of refractive index n1 is limited to 1.60-1.63.
The relationship between the structural parameter of the hemispherical lens and the system resolution is discussed below. Without changing other parameters, the curvature radius r1 of the hemispherical lens is taken as the variable. The relationship between curvature radius r1 and system resolution is shown in Fig. 4. The dispersion width w remains constant when r1 changes, i.e., curvature radius r1 of the hemispherical lens has no effect on spectral resolution, only contributing to aberration correction. When the curvature center C1 of the spherical surface is located on the plane of the hemispherical lens, the corresponding curvature radius r1 is -281.5 mm, and the RMS radii in the full working band are within 1.0 µm. The RMS radii gradually increase with the continuous deviation of r1, i.e., the spatial resolution is significantly reduced with the further destruction of the concentric structure of the hemispherical lens.
It is preferable to select the material with refractive index n1 between 1.60 and 1.63 and relatively low refractive index difference Δn1 for the hemispherical lens, which is beneficial to achieve both high spatial and high spectral resolution imaging. The curvature center C1 is set on the plane of the hemispherical lens to achieve optimal spatial resolution imaging.
2.3 System resolution influenced by the cemented-curved-prism
The material parameters related to the cemented-curved-prism include n3,s, n3,l, n4,s and n4,l, which exist as numerator terms in Eq. (6). Different materials are replaced into the cemented-curved-prism to realize the change of refractive indices n3 and n4. The difference between changes in refractive index Δn4 and Δn3 (Δn4–Δn3) is introduced as the abscissa of the coordinate system to effectively reflect the relationship between material parameters of the cemented-curved-prism and dispersion width w. As shown in Fig. 5(a), the purple circles correspond to the values of the dispersion width w at different (Δn4–Δn3). The blue curve generated by polynomial fitting reflects the trend of dispersion width w, and the sampled points are distributed along the blue curve with the floating range within ±0.025 mm. In general, the dispersion width w increases steadily as the difference between Δn4 and Δn3 increases. The distribution of RMS radii corresponding to the short wavelength (400 nm) and long wavelength (760 nm) are shown in Fig. 5(b) and Fig. 5(c), and the areas with RMS radii within 4.0 µm are marked by dotted lines. The introduction of the material with higher refractive index n4 into the rear curved prism is conducive to achieving high spatial resolution imaging. However, with the decrease of wavelength, lower refractive index n3 is not conducive to achieving high spatial resolution imaging. We intersect the marked areas corresponding to the two extreme spectral bands to obtain the overlap area that achieves high spatial resolution in the full working band. Therefore, we select material combinations with refractive index n3 higher than 1.58 and n4 higher than 1.74 for the cemented-curved-prism.
According to Eq. (6) and Eq. (7), the dispersion width w is directly related to the Δh2 term, which is affected by the curvature radii r2, r3 and tilt angles φ2, φ3. We first take the tilt angles as variables to realize the change of Δh2 term. It is noted that we obtain Eq. (6) based on Δh3=-Δh2, which should be satisfied when structural parameters are taken as variables to avoid the influence of other parameters on the system resolution. Combined with Fig. 2, the condition to be met is r2·sin(φ2)=r4·sin(φ4). As shown in Fig. 6(a), the dispersion width w increases linearly with the increase of the vertex angle (φ2+φ3). The RMS radii are within 2.0 µm when the vertex angle of the front curved prism is controlled within 4.5 degrees. As the vertex angles continue to increase, the RMS radii will change from a steady state to a sharp increase, i.e., the excessive increase in vertex angles seriously reduces the spatial resolution. Then, we keep the tilt angles constant and take the curvature radii as variables to realize the change of Δh2 term. As shown in Fig. 6(b), the dispersion width w is hardly influenced by the change of the curvature radius r2. When the curvature center C2 of the spherical surface of the front curved prism is distributed on the Y-axis, the corresponding curvature radius r2 is -324 mm, and the RMS radii are within 2.0 µm. The RMS radii increase sharply with the continuous deviation of r2, r3, r4, which reflects that the destruction of the aplanatic imaging condition seriously reduces the spatial resolution.
Under a specific working waveband, dispersion width can be used to show the spectral resolution. The dispersion width w increases steadily as the difference between Δn4 and Δn3 increases as Fig. 5(a) shows. Meanwhile, a high refractive index of the cemented-curved-prism are favorable to obtain spatial samples with small root mean square (RMS) less than 4.0 µm as Fig. 5(b) and Fig. 5(c) show. Accordingly, refractive index of the front curved prism n3 is higher than 1.58 and refractive index of the rear prism n4 is higher than 1.74.
Under the same working waveband, how the vertex angle (φ2+φ3) of the front curved prism and the curvature radius r2 affect the spectral and spatial resolution are further studied and shown as Fig. 6(a) and Fig. 6(b), respectively. As Fig. 6(a) shows, the spectral resolution of the CIFS increases linearly with the vertex angle from 0° to 6°, the spatial resolution is less than 2.0 µm when the vertex angle is less than 4.5°. As Fig. 6(b) shows, curvature radius r2 hardly makes any affect on the spectral resolution, while the RMS radii of the spatial samplings over the entire FOV are within 2.0 µm when it is from about -331 mm to -317 mm, with the curvature center C2 distributed along the Y-axis.
Accordingly, a vertex angle of the front curved prism less than 4.5° is selected and the curvature center of r2 is designed on the Y-axis.
2.4 Comparison of the system resolution influenced by hemispherical lens and cemented-curved-prism
The coefficient K is introduced to indicate the degree to which the corresponding elements sacrifice the spatial resolution when the same spectral resolution is improved by replacing materials, and K can be expressed as:
In Eq. (9), ΔR represents the degree of change in RMS radius size within the sampled range, and Δw represents the degree of change in dispersion width w. As shown in Fig. 7, The K value of hemispherical lens is more than 14 times that of cemented-curved-prism. The spatial resolution sacrificed to improve spectral resolution by changing the refractive index of the hemispherical lens is enormous. Instead, it is reasonable to increase the spectral resolution without degrading the excessive aberration correction ability by changing the refractive index of the cemented-curved-prism. In other words, the hemispherical lens exhibits excellent aberration correction ability, acting as an effective means to realize high spatial resolution imaging with high numerical aperture, and the cemented-curved-prism provides outstanding dispersion ability for the system without introducing complex aberrations.Based on the analysis above, we obtain the refractive index requirements of the hemispherical lens and the cemented-curved-prism for high spatial and high spectral resolution imaging of CIFS. We also demonstrate the validity of the idea of aplanatic imaging in CIFS, which enables the system to achieve the optimal spatial resolution, providing guidance for the establishment of CIFS with both high spatial and high spectral resolution imaging potential with high numerical aperture.
3. Optical design
3.1 Design of the imaging spectrometer of the CIFS
During the optimization, two meniscus lenses are added to further improve the aberration correction ability of the system, enabling the system to image with high numerical aperture in compact structure, and allowing the cemented-curved-prism to have larger vertex angles, thus improving the dispersion ability. We list the specifications of imaging spectrometer in Table 1 and the lens data in Table 2. In detail, n1 is 1.618, Δn1 is 0.023, n3 is 1.804, n4 is 1.923, (Δn4–Δn3) is 0.072. Figure 8 present a schematic diagram of the optical path of the imaging spectrometer, Fig. 9 present the modulation transfer function (MTF) of imaging spectrometer, MTFs of different spectra at Nyquist frequency are higher than 0.8. Figure 10 present the spot diagram. The performance evaluation of spectral smile and keystone on the image surface is shown in Fig. 11 and Fig. 12, respectively. The maximum smile is controlled within 0.8µm and the maximum keystone is controlled within 0.5µm in the full working band.
Fig. 9. Modulation transfer function of the imaging spectrometer corresponding to typical wavelength: (a) 400 nm, (b) 580 nm, (c) 760 nm.
Fig. 10. Spot diagrams of different spectra of the imaging spectrometer: (a) 400 nm, (b) 580 nm, (c) 760 nm.
Fig. 11. Spectral smile of the imaging spectrometer on the image surface: (a) Field-Y = 5 mm, (b) Field-Y = 0, (c) Field-Y = -5 mm.
Fig. 12. Spectral keystone of the imaging spectrometer on the image surface: (a) Field-Y = 5 mm, (b) Field-Y = 0, (c) Field-Y = -5 mm.
Table 1. Specifications of the imaging spectrometer
Table 2. Lens data of the imaging spectrometer
As shown in Fig. 13, the purple, orange and green lines correspond to the spectral resolution of the system when the FOV is Field-Y = -5 mm, Field-Y = 0, Field-Y = 5 mm, respectively. The corresponding spectral resolution with every 12 nm step in the wavelength range of 400-760 nm is recorded, Field-Y = -5mm: the spectral resolution is within 0.2 nm at wavelength 400 nm, within 0.8 nm at wavelength 580 nm and within 1.8 nm at wavelength 760 nm.
3.2 Design of the CIFS system
The foreoptics is optimized with a focal length of 100 mm and an image space f-number of 45. The image space f-number of the lenslet array is set to 1.8, which matches that of imaging spectrometer. A hexagonal-arranged lenslet array with a pixelate scale of 268 × 76 and a unit size of 0.13 mm is introduced to segment the input image from the foreoptics. The CIFS system consists of the foreoptics, lenslet array and imaging spectrometer, as shown in Fig. 14. The foreoptics images the object at infinity onto the lenslet array, which lies at the focal plane of the foreoptics. The lenslet array segments the image of the object to create a pupil image array at the focal plane array of the lenslet array, which is very close to the rear of the lenslet array and is difficult to see clearly in Fig. 14.
The spectral resolution of the IFS system is evaluated by pupil image point array, and the spot diagram of pupil images on the image surface is shown in Fig. 15. Four sampled fields of view are utilized to characterize the spectral resolution, including three on-axis points and one off-axis outermost point. The spectral resolution is within 4.0 nm at 760 nm, within 1.2 nm at 580 nm, and within 0.2 nm at 400 nm.
Fig. 15. Spot diagram of pupil image: (a) image space field (0 mm, 0 mm); (b) image space field (12.4 mm, 0 mm); (c) image space field (17.5 mm, 0 mm); (d) image space field (17.5 mm, 5 mm).
3.3 Simulation experiment of spectral imaging and performance evaluation
The simulation experiment of spectral imaging of the CIFS is performed based on the optical design software Zemax. And the waveband is from 465 nm to 606 nm, which is available with Zemax. Figure 16 shows the principle of spectral imaging of the CIFS. An object with a length-width ratio of approximately 2:1 and a pixel number of 784 × 396 is shown as Fig. 16(a). The rotation angle of the lenslet array is set to 3.7°. The object is firstly imaged onto the lenslet array by the foreoptics as Fig. 16(b) shows. Then, the lenslet array subdivides the image and achieves a pupil image array on its focal plane, as shown in Fig. 16(c). The pupil image array subsequently passes through the imaging spectrometer, and the raw data shown as spectral bar array is captured on the sensor, as Fig. 16(d) shows.
Fig. 16. Schematic diagram of principle of the simulation spectral imaging experiment of the CIFS. (a) Object (b) Image after the foreoptics (c) Intermediate pupil image after the lenslet array (d) Raw data captured on the sensor.
After wavelength calibration, the relationship between the wavelength and the pixel position is obtained. The centroid coordinates of pupil images of any wavelengths are extracted. Further, all the information contained in the pupil image with specific wavelength is extracted from the raw data captured on the sensor.
To evaluate the spectral resolution of the CIFS, Fig. 17 shows the pupil images with three different wavelengths and the partially magnified pupil images are accordingly presented. As shown in the partially magnified images, the pupil image between 465 nm and 464.2 nm, 535 nm and 533.8 nm, and 606 nm and 603.6 nm are distinguished clearly. The spectral resolution is within 4 nm as evaluated theoretically above.
Fig. 17. Pupil images with different wavelengths. (a) pupil image with 465 nm and 464.2 nm. (b) pupil image with 535 nm and 533.8 nm. (c) pupil image with 606 nm and 603.6 nm.
Processed with MATLAB, the two-dimensional images at dedicated wavelengths are obtained by combining the pixels in every sub-image. Figure 18 shows 36 reconstructive spectral images of different wavelengths, ranging from 465 nm to 606 nm and with a spectral interval of about 4 nm.
Fig. 18. Reconstructive images at different wavelengths. The wavelength is from 465 nm to 606 nm, and the spectral interval is about 4 nm.
4. Conclusion
In this paper, a cemented-curved-prism based integral field spectrometer (CIFS) is proposed, which utilizes a hemispherical lens to achieve imaging with high numerical aperture and high spatial resolution, and a cemented-curved-prism to improve the spectral resolution without introducing complex aberrations. In addition, the design idea of aplanatic imaging and sharing-optical-path lays the foundation for CIFS to exhibit high resolution imaging in compact structure. We analyze the relationships between parameters of optical elements and system resolution, obtaining refractive index requirements of the hemispherical lens and the cemented-curved-prism for high spatial and high spectral resolution imaging of the system, which provides guidance for the construction of CIFS. Simulation results show that the imaging spectrometer achieves pupil matching with a 1.8 f-number lenslet array, and the spectral resolution of the IFS system is better than 4 nm, providing 268 × 76 spatial sampled points with 403 spectral channels in the wavelength band of 400 to 760 nm. The effectiveness of CIFS is fully verified through the simulation experiment of spectral imaging based on Zemax. The pupil images show that the spectral resolution is as evaluated theoretically. Reconstructed spectral images of different wavelengths further demonstrate the excellent performance of the proposed CIFS. The CIFS paves the way for the development of integral field spectrometers exhibiting both high spatial and high spectral resolution imaging with high numerical aperture.
Funding
National Natural Science Foundation of China (61340007, 61405134); National Defense Basic Scientific Research (JCKY2018414C013); Natural Science Foundation of Jiangsu Province (BK20161512); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare no conflicts of interest.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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.
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