Analytical design of athermal ultra-compact concentric catadioptric imaging spectrometer

Jiacheng Zhu; Jiacheng Zhu; Weimin Shen; Weimin Shen

doi:10.1364/OE.27.031094

1. Introduction

Imaging spectrometer has been used in remote sensing to acquire both spatial and spectral information of ground scene, providing vital information about homeland security, disaster prevention, environment monitoring, etc. In order to acquire information more efficiently and accurately, long slit and large relative aperture are usually needed to increase its swath and signal-to-noise ratio (SNR). However, these two features will lead to the rapid increase of its volume and then its cost of development and launch. Therefore, it is very expected to solve the contradiction between long slit, large relative aperture and its compactness.

The concentric spectrometers including Offner and Dyson forms are commonly adopted in hyperspectral remote sensing missions. They have high imaging quality without keystone and smile distortions and their volume and weight are smaller than those of previous generation imaging spectrometers such as AIS [1] and AVIRIS [2].

In 1973, Abe Offner [3] proposed a scanner for unit magnification lithography composed of two concentric spherical mirrors, which provides high quality images in an annular field [3–5]. Mertz [6] replaced the convex mirror of the Offner configuration with a convex diffraction grating to form the Offner spectrometer. It was then detailedly analyzed and improved by Kwo [7] and Lobb [8,9]. It inherits the performance of the Offner configuration, and its primary aberrations are eliminated in an annular field. The form was fully developed by Chrisp [5] in 1999. He separated the concave mirror into primary and tertiary mirrors to increase the degrees of freedom for aberration control. In 2006, Prieto-Blanco et al. [10] performed a thorough analytical design of Offner-Chrisp spectrometer, demonstrating that astigmatism is its main residual aberration. To avoid obscuration, slit and image of the Offner-type spectrometer have to be off-axis, and its F-number is usually greater than 2.5. An example of long-slit Offner-Chrisp spectrometer is the JPL Hyperspectral and Infrared Imager (HyspIRI) [11] design for 0.38–2.5µm wavelength range with dispersive width of 6.36mm. Its slit is 48mm long and its F-number is 2.8. Size of its optical system is 200×100×320mm. Another example of Offner-type spectrometer is the ESA Fluorescence Imaging Spectrometer (FLORIS) [12] for FLEX mission. Its high-resolution imaging spectrometer works under the Littrow condition with a corrected meniscus lens lengthening its slit and broadening its dispersive width. It has a slit of 44mm long and a F-number of 3.1. Its package size is about Φ220×400mm. It can be seen from the above two examples that even if the Offner-type spectrometers can meet the requirements of long slit and large relative aperture, their volumes are still large.

Dyson spectrometer [13] is developed from the Dyson relay [14] by replacing the concave mirror with the concave diffractive grating. Because of no central obscuration, it can have larger relative aperture and more compact structure than the Offner-type. In 2014, JPL reported the design of Compact Wide Swath Imaging Spectrometer (CWIS) [15] for the solar reflected spectrum (0.38–2.50µm) with wide swath (1600 pixels). It has fast optical speed (F/1.8) and long slit (48mm), and its size is Φ125×325mm. CWIS has the same slit length as HyspIRI, and its relative aperture is much larger than the latter, but its volume is only 60% of the latter. The Dyson-type spectrometer is much better than the Offner-type in improving SNR and reducing volume. However, in the VNIR band, the application of Dyson spectrometer is not as popular as the Offner-type. Three reasons result in the situation. First, the slit and the imaging plane are too adjacent to the hemisphere lens in the traditional Dyson configuration, resulting in inconvenience to install the detector or fold the optical path. Second, the inherent stray light caused by the hemisphere lens will influence the imaging quality heavily. Third, it is susceptible to the temperature and hard to be applied in complex working environments. These problems increase the difficulty and cost of developing a Dyson spectrometer.

An increase in requirement for solving the contradiction between long slit, large relative aperture and small size has motivated the improvement of existing spectrometer forms and the emergence of new technologies. The spectrometer form, a double-pass reflective triplet (RT) with a flat grating [16,17] can have quite small volume and large relative aperture. Yuan et al. [17] detailedly analyzed the RT spectrometer and then designed, aligned and tested the prototype. Their instrument has fast optical speed (F\2), perfect image quality and low smile and keystone distortions. However, they used field combination of three sub-modules to enlarge the coverage area, rather than an individual imaging spectrometer with large FOV and long slit.

Most recently, Reimers et al. [18] introduced freeform surfaces both at and away from the aperture stop to correct aberrations and designed freeform spectrometers based on Offner-Chrisp configuration. Their designs obtain increased slit length and compactness, showing a 2× increase in slit length or a 5× reduction in volume than a non-freeform counterpart. They also showed significant benefit of having freeform gratings, but the extremely high surface precision and finish are needed. The added complexity of manufacturing a grating on a freeform substrate may limit its widespread use at present.

In 2018, Chrisp provided a novel imaging spectrometer form for size, weight and power limited applications [19]. The form utilizes a catadioptric lens and a flat immersion grating, controlling primarily astigmatism and eliminating systematic errors. His design has small volume with F number of 3.3 and slit length of 28mm. However, as its relative aperture and slit length further increase, its aberrations especially the astigmatism will increase rapidly.

As a precursor to the work presented in this paper, Warren [20] proposed a monolithic spectrometer consisting of two catadioptric body portions joined together. His design can obtain increased relative aperture and decreased size due to the compression of the aperture angle of incident beam. However, anastigmatism for long slit was not taken into consideration in his concept. In addition, performance of such catadioptric system is susceptible to the operating temperature, thus, extra temperature control measures are needed for his designs to avoid thermal defocus, which will also lead to the increase of its volume and cost. Therefore, athermalization of spectrometer are needed to stabilize its performance for complex operating environment and size limited applications. Generally, matching the thermal expansion coefficients of mechanical and optical materials of reflective spectrometers can realize athermalization [21]. However, catadioptric spectrometers with several dielectric materials may complicate the method for matching mechanical materials, and athermalization will be incomplete for large temperature range.

In this paper, we proposed a novel athermal Ultra-Compact concentric catadioptric Imaging Spectrometer (UCIS) possessing large relative aperture and long slit. It is a monolithic system made of optical components, and it does not require mechanical components for connecting and fixing. It has the advantages of large relative aperture, long slit, high imaging quality, compact structure and excellent thermal adaptability. In Section 2, we expound the principle of the UCIS, and discuss the method to lengthen its slit through analyzing its residual astigmatism by tracing the chief ray. In Section 3, we present its athermalization design through preferring lens materials with different optical properties to compensate thermal defocus. In Section 4, design procedure and a design example of the UCIS is given, and it is compared with Offner-Chrisp and Dyson spectrometers with the same specifications. The conclusions are given in Section 5.

2. Theory of the UCIS

Layout of the UCIS is shown in Fig. 1(a). It comprises three optical components, i.e. a hemisphere lens L1, a meniscus lens L2, and a thick lens L3. Diffraction grating G is etched into the center of the convex surface of L2, and the concave surface M of L3 is a reflecting surface. All of three components are glued together in a concentric layout with the common center C. The entrance slit and the array detector are arranged onto the plane surface of L1.

Fig. 1. (a) Sketch of the UCIS. (b), (c) Ray tracing of chief ray from an arbitrary point on the slit showing location of the meridional and sagittal images.

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The UCIS inherits from the Offner configuration and keeps concentric, astigmatism is still its main residual aberration [10]. We mainly discuss its astigmatism characteristics and anastigmatism condition in this paper. To analyze the astigmatism of this system, locations of its meridional and sagittal images were solved by tracing the chief ray, and expression of the astigmatism was derived. Then influences of the refractive indexes and radii of lenses on the astigmatism will be analyzed, which will guide the selection of lens materials and the determination of lens radii for achieving the best anastigmatism.

Figures 1(b) and 1(c) show the trace of chief ray from an arbitrary point O on the slit. Because of the diffraction on G, the incident plane y_OCz and the emergent plane y_ICz are non-coplanar unless O is on the center of the slit. In the incident plane y_OCz, I_M1 and I_S1 are meridional and sagittal images of M, respectively. We define φ as object polar angle, φ_S1 as sagittal image polar angle, respectively. The incident and refraction angles at each surface are shown in the figure. Simple geometrical considerations lead to the relations:

(1)$$\varphi = 2{\theta _3} + {\theta _2} + {\theta _1} - 2{\theta _2}^{\prime} - {\theta _1}^{\prime},$$

(2)$${\theta _2}^{\prime} = {\theta _4}.$$

Refractive indexes of L1, L2, and L3 are n₁, n₂, and n₃, respectively. Radii of their spherical surfaces are R₁, R₂, and R₃, respectively. According to the Rowland’s condition [10] and the Snell’s law,

(3)$$\textrm{OC} = {R_1}\sin {\theta _1} = \frac{{{n_2}}}{{{n_1}}}{R_1}\sin {\theta _1}^{\prime} = \frac{{{n_2}}}{{{n_1}}}{R_2}\sin {\theta _2} = \frac{{{n_3}}}{{{n_1}}}{R_2}\sin {\theta _2}^{\prime} = \frac{{{n_3}}}{{{n_1}}}{R_3}\sin {\theta _3}.$$

In the emergent plane y_ICz, I_M2 and I_S2 are meridional and sagittal objects of M, respectively. I_M and I_S are meridional and sagittal images of the UCIS, respectively. φ_S2 is the sagittal object polar angle. φ_M and φ_S are meridional and sagittal image polar angles, respectively. When the meridional image meets Rowland's condition, the emergent chief ray is perpendicular to CI_M. According to geometrical relationships and the Snell’s law,

(4)$${\varphi _\textrm{M}} = 2{\theta _4}^{\prime} + {\theta _7}^{\prime} - 2{\theta _5} - {\theta _6} - {\theta _7},$$

(5)$${\theta _4}^{\prime} = {\theta _6}^{\prime},$$

(6)$$\textrm{C}{\textrm{I}_\textrm{M}} = {R_1}\sin {\theta _7} = \frac{{{n_2}}}{{{n_1}}}{R_1}\sin {\theta _7}^{\prime} = \frac{{{n_2}}}{{{n_1}}}{R_2}\sin {\theta _6} = \frac{{{n_3}}}{{{n_1}}}{R_2}\sin {\theta _6}^{\prime} = \frac{{{n_3}}}{{{n_1}}}{R_3}\sin {\theta _5}.$$

Astigmatism of the UCIS Astig can be expressed as:

(7)$$Astig = \textrm{C}{\textrm{I}_\textrm{M}}\tan ({{\varphi_\textrm{S}} - {\varphi_\textrm{M}}} ).$$

Relationships between parameters in plane y_ICz and y_OCz can be built through analyzing the diffraction of grating. Figure 2(a) shows the off-plane diffraction of the grating. The convex grating images I_M1 and I_S1 from y_OCz plane to I_M2 and I_S2 in y_ICz plane, respectively. Figure 2(b) is the view along z-axis. γ is the incident azimuth of chief ray on the grating, and γ’ is the diffractive azimuth. The grating equations are

(8)$$\begin{array}{l} \sin {\theta _4}^{\prime}\cos \gamma ^{\prime} - \sin {\theta _4}\cos \gamma = mg\lambda ,\\ \sin {\theta _4}^{\prime}\sin \gamma ^{\prime} - \sin {\theta _4}\sin \gamma = 0, \end{array}$$

where m is the diffraction order, g is the groove density of the grating, and λ is the wavelength. The following equation can be obtained by combining Eqs. (3), (6) and (8),

(9)$$\textrm{C}{\textrm{I}_\textrm{M}}\textrm{ = }\frac{{\sin \gamma }}{{\sin \gamma ^{\prime}}}\textrm{OC}\textrm{.}$$

Light path of the diffraction can be projected as the coplanar diffraction in the principal section of the grating, as shown in Fig. 2(c), and the specular reflection in the plane perpendicular to its principal section, as shown in Fig. 2(d). In Fig. 2(d), the meridional and sagittal image polar angles φ_S1x and φ_S2x satisfy:

(10)$$\begin{array}{l} \textrm{tan}{\varphi _{\textrm{S1x}}} = \textrm{tan}{\varphi _{\textrm{S1}}}\textrm{sin}\gamma ,\\ \textrm{tan}{\varphi _{\textrm{S2x}}} = \textrm{tan}{\varphi _{\textrm{S2}}}\textrm{sin}\gamma ^{\prime}. \end{array}$$

Fig. 2. Diffraction on the grating. (a) Path of off-plane diffraction on the grating; (b) View along the z-axis; (c) Projection in the principal section of grating; (d) Projection in the plane perpendicular to the principal section of grating.

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According to Young's formulas [10],

(11)$$\frac{1}{{\textrm{A}{\textrm{I}_{\textrm{S1x}}}}} + \frac{1}{{\textrm{A}{\textrm{I}_{\textrm{S2x}}}}} = \frac{{\cos \beta + \cos \beta ^{\prime}}}{{{R_2}}}.$$

In ΔACI_S1x and ΔACI_S2x, utilizing the law of sines and Eq. (11), we can deduce

(12)$$\sin \beta \tan {\varphi _{\textrm{S1x}}}\textrm{ = }\sin \beta ^{\prime}\tan {\varphi _{\textrm{S2x}}}.$$

Since the chief ray is specularly reflected in the xCz plane in Fig. 2(d), φ_S1x=φ_S2x, and then the following equation can be derived from Eqs. (10):

(13)$$\tan {\varphi _{\textrm{S2}}} = \frac{{\sin \gamma }}{{\sin \gamma ^{\prime}}}\tan {\varphi _{\textrm{S1}}}.$$

Similarly, in the incident plane and the emergent plane,

(14)$$\begin{array}{l} {\varphi _{\textrm{S1}}} = \varphi ,\\ {\varphi _\textrm{S}} = {\varphi _{\textrm{S2}}}. \end{array}$$

From Eqs. (13) and (14),

(15)$${\varphi _\textrm{S}} = \arctan \left( {\frac{{\sin \gamma }}{{\sin \gamma^{\prime}}}\tan \varphi } \right).$$

A specific solution of anastigmatism is φ_S=φ_M=φ=0, which means telecentric in both object and image space. Let the coordinate of an arbitrary object point be O(x, y). k =$\sqrt {{\textrm{x}^\textrm{2}}\textrm{ + }{\textrm{y}^\textrm{2}}} $/R₂ indicates the normalized off-axis value and k₁ =R₃/R₂, k₂ =R₁/R₂ are structural factors. According to Eqs. (1)–(9) and Eq. (15), with n₁, n₂, n₃, k, k₁, k₂ as variates, CI_M, φ'_S and φ'_M in Eq. (7) can be concretely expressed as:

(16)$$\textrm{C}{\textrm{I}_\textrm{M}} = k{R_2}\frac{{\sin \gamma }}{{\sin \gamma ^{\prime}}},$$

(17)$$\begin{array}{l} {\varphi _\textrm{S}} = \arctan \left\{ {\frac{{\sin \gamma }}{{\sin \gamma^{\prime}}}\tan \left[ {2\arcsin \left( {\frac{{{n_1}}}{{{n_3}}}\frac{k}{{{k_1}}}} \right) + \arcsin \left( {\frac{{{n_1}}}{{{n_2}}}k} \right)} \right.} \right.\\ \textrm{ }\left. {\left. { + \arcsin \left( {\frac{k}{{{k_2}}}} \right) - 2\arcsin \left( {\frac{{{n_1}}}{{{n_3}}}k} \right) - \arcsin \left( {\frac{{{n_1}}}{{{n_2}}}\frac{k}{{{k_2}}}} \right)} \right]} \right\}, \end{array}$$

(18)$$\begin{aligned} {\varphi _\textrm{M}} &= 2\arcsin \left( {\frac{{{n_1}k\sin \gamma }}{{{n_3}\sin \gamma^{\prime}}}} \right) + \arcsin \left( {\frac{{{n_1}k\sin \gamma }}{{{n_2}{k_2}\sin \gamma^{\prime}}}} \right) - 2\arcsin \left( {\frac{{{n_1}k\sin \gamma }}{{{n_3}{k_1}\sin \gamma^{\prime}}}} \right)\\ & \textrm{ } - \arcsin \left( {\frac{{{n_1}k\sin \gamma }}{{{n_2}\sin \gamma^{\prime}}}} \right) - \arcsin \left( {\frac{{k\sin \gamma }}{{{k_2}\sin \gamma^{\prime}}}} \right), \end{aligned}$$

where

(19)$$\gamma = \arctan \left( {\frac{x}{y}} \right),$$

(20)$$\gamma ^{\prime} = \arctan \left( {\frac{{{n_1}k\sin \gamma }}{{{n_1}k\cos \gamma + {n_3}mg\lambda }}} \right).$$

With Eq. (7) and Eqs. (16)–(20), we can analyze how refractive indexes and structural factors will influence the performance of the UCIS. We assume that m=−1, g = 200 Lp/mm, λ=0.6µm, and R₂=1. When L1, L2 and L3 use the same material (n₁=n₂=n₃=1.46), the system’s astigmatism distribution in a quarter of the object plane calculated by Eq. (7) is shown in Fig. 3. Let the normalized anastigmatism threshold be 0.0001, and assuming that the astigmatism is low when Astig is less than 0.001. It can be seen in Fig. 3 that there is an anastigmatic narrow annular field in the object plane, and the system is anastigmatic only if the slit is near this annular field, which limits the length of slit with low astigmatism.

Fig. 3. Anastigmatic region in a quarter of the object plane of UCIS. The normalized anastigmatism threshold is 0.0001, and we assume that the astigmatism is low when Astig is less than 0.001. The slit with low astigmatism is short. 4. Units of x and y axis are a.u. (n₁=n₂=n₃=1.46, k₁=1.92)

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When L1, L2 and L3 use different materials, their refractive indexes have a great influence on the distribution of astigmatism in the object plane. We use (n₁, n₂, n₃) to indicate the refractive indexes of L1, L2 and L3 with a minimum index of l = 1.46, a medium index of m = 1.60, and a highest refractive index of h = 1.74. Figure 4 shows all possible combinations of L1, L2 and L3 with two different refractive indexes: l and h. The structural factors are preferred to make the anastigmatic region as large as possible. There are two anastigmatic annular fields in Fig. 4(b), while there is only one or none in other figures. As shown in Fig. 2(b), if a₁ and a₂ are two anastigmatic annular fields, the entire slit can have low astigmatism when the center of slit is at a₁, and the edge of slit is at a₂. This is verified in Fig. 4(b). The UCIS allows a longer slit with low astigmatism when it has refractive indexes (l, h, l) and structural factors k₁=2.00, k₂=0.85.

Fig. 4. Anastigmatic regions in a quarter of the object plane for different combinations of two different materials of the UCIS. (n₁, n₂, n₃) respectively refers to the refractive index of L1, L2 and L3. The lower index is l = 1.46, and the higher refractive index is h = 1.74.

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Figures 5 shows all possible combinations of L1, L2 and L3 with three different refractive indexes: l, m, and h. There are two anastigmatic annular field in Fig. 5(b) and 5(d), respectively. Especially in Fig. 5(b), the astigmatism is low in a quite large circular region, where refractive indexes of L1, L2 and L3 are (l, h, m), and the structural factors are k₁=2.25, k₂=0.89. Its astigmatism along the slit is shown in Fig. 6, compared with a classic Offner spectrometer with n₁=n₂=n₃=1 and k₁=1.92. From this figure, astigmatism of the classic Offner spectrometer increases rapidly as the slit length increases, while the astigmatism is quite insensitive to slit length in the UCIS. Thus, the UCIS can reach ultra-long slit or ultra-compact structure if the slit is not quite long.

Fig. 5. Anastigmatic regions in a quarter of the object plane for different combinations of three different materials of the UCIS. (n₁, n₂, n₃) respectively refers to the refractive index of L1, L2 and L3. The minimum index is l = 1.46, the medium index is m = 1.60, and the highest refractive index is h = 1.74.

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Fig. 6. Astigmatism along the slits of two spectrometer types. Normalized off-axis value is 0.6 for Offner type and 0.38 for the UCIS. The abscissa is the normalized half-length of slit, and the ordinate is the normalized astigmatism. Astigmatism of the UCIS is quite insensitive to its slit length. Units of the abscissa and the ordinate are a.u.

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Combined with the above analyses, there will be two anastigmatic annular field in the object plane if n₂ is greater than n₁ and n₃, and the UCIS can have long slit and small volume by optimizing the structural factors.

3. Athermalization of the UCIS

Thermal adaptability is one of the most important performance of such catadioptric system. When the operating temperature changes, radii, thicknesses, and refractive indexes of lenses will change accordingly, resulting in thermal defocus and decline in image quality. We use the passive athermalization method, matching lens materials with different thermal expansion coefficients and thermo-optic coefficients to compensate the thermal defocus.

The UCIS can be divided into two parts: a refraction group and a reflection group, respectively shown in Fig. 7(a) and 7(b). Chief ray from the object point O to the image point I is shown in the figures. The center thickness of L1, L2, and L3 are d₁, d₂, and d₃, respectively. In the refraction group, chief ray from point O is sequentially refracted through the concave and convex surfaces of L2, respectively forming virtual images V₁ and V₂. Their image distances are l₁ and l₂, respectively. According to the Gauss formula, the image distances can be derived as:

(21)$${l_1} = \frac{{{n_2}{d_1}{R_1}}}{{{d_1}({{n_2} - {n_1}} )+ {n_1}{R_1}}},$$

(22)$${l_2} = \frac{{{n_3}({{l_1} + {d_2}} ){R_2}}}{{({{l_1} + {d_2}} )({{n_3} - {n_2}} )+ {n_2}{R_2}}}.$$

Fig. 7. Decomposition map of the UCIS. (a) The refraction group, (b) The reflection group. Red solid line is the chief ray from the object point O to the image point I. The chief ray passes through each surface successively to form virtual images V₁-V₆.

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In the reflection group, the chief ray is reflected by M with a virtual image V₃, then diffracted by G with a virtual image V₄, and then reflected by M again with a virtual image V₅. Image distances of V₃, V₄ and V₅ are l₃, l₄ and l₅, respectively.

(23)$${l_3} = \frac{{({{l_2} + {d_3}} ){R_3}}}{{2({{l_2} + {d_3}} )- {R_3}}},$$

(24)$${l_4} = \frac{{({{l_3} - {d_3}} ){R_2}}}{{2({{l_3} - {d_3}} )- {R_2}}},$$

(25)$${l_5} = \frac{{({{l_4} + {d_3}} ){R_3}}}{{2({{l_4} + {d_3}} )- {R_3}}}.$$

After the second reflection of M, the chief ray enters the refraction group again. V₅ is imaged to V₆ by the convex surface of L2, and then imaged to I by its concave surface. l₆ and l₇ are their image distances.

(26)$${l_\textrm{6}} = \frac{{{n_2}({{l_5} - {d_3}} ){R_2}}}{{{n_3}{R_2} + ({{n_2} - {n_3}} )({{l_5} - {d_3}} )}},$$

(27)$${l_\textrm{7}} = \frac{{{n_1}({{l_6} - {d_2}} ){R_1}}}{{{n_2}{R_1} + ({{n_1} - {n_2}} )({{l_6} - {d_2}} )}}.$$

The image I is focused on the array detector if l₇=d₁, and the system can obtain the optimum imaging quality. Thermal defocus appears if the change of l₇ is different from the change of d₁ as the operating temperature changes. We use thermal defocus coefficient (TDC) to show the temperature adaptability of the system:

(28)$$TDC = \left|{\frac{{\textrm{d}{l_7}}}{{\textrm{d}T}} - \frac{{\textrm{d}{d_1}}}{{\textrm{d}T}}} \right|,$$

where T is the temperature. TDC indicates the thermal defocus caused by per Kelvin change in operating temperature. It is related to the materials’ thermal expansion coefficients α₁, α₂, α₃ and thermo-optic coefficients dn₁/dT, dn₂/dT, dn₃/dT. TDC can be expressed in detail as:

(29)$$TDC = \left|\begin{array}{l} \frac{{\left[ {\frac{{\textrm{d}{n_1}}}{{\textrm{d}T}}{R_1}({{l_6} - {d_2}} )+ {n_1}{\alpha_1}{R_1}({{l_6} - {d_2}} )+ {n_1}{R_1}\left( {\frac{{\textrm{d}{l_6}}}{{\textrm{d}T}} - {d_2}{\alpha_2}} \right)} \right][{{n_2}{R_1} + ({{n_1} - {n_2}} )({{l_6} - {d_2}} )} ]}}{{{{[{{n_2}{R_1} + ({{n_1} - {n_2}} )({{l_6} - {d_2}} )} ]}^2}}}\\ - \frac{{{n_1}{R_1}({{l_6} - {d_2}} )\left[ {\left( {\frac{{\textrm{d}{l_6}}}{{\textrm{d}T}} - {d_2}{\alpha_2}} \right)({{n_1} - {n_2}} )+ ({{l_6} - {d_2}} )\left( {\frac{{\textrm{d}{n_1}}}{{\textrm{d}T}} - \frac{{\textrm{d}{n_2}}}{{\textrm{d}T}}} \right) + \frac{{\textrm{d}{n_2}}}{{\textrm{d}T}}{R_1} + {n_2}{\alpha_1}{R_1}} \right]}}{{{{[{{n_2}{R_1} + ({{n_1} - {n_2}} )({{l_6} - {d_2}} )} ]}^2}}} - {\alpha_1}{d_1} \end{array} \right|.$$

The UCIS is athermal when TDC = 0. To realize an athermalization design, the dependence of TDC with materials’ thermal expansion coefficients and thermo-optic coefficients were analyzed. Results are shown in Fig. 8. Variation of TDC with thermal expansion coefficients and thermo-optic coefficients are respectively shown in Fig. 8(a) and 8(b), where we assume R₁=64 mm, R₂=75 mm, R₃=150 mm, n₁=n₃=1.46, n₂=1.74. As show in Fig. 8(a), TDC varies significantly with α₂ and α₃, but there is a region where TDC remains zero. This means no thermal defocus regardless of the operating temperature only if α₁, α₂ and α₃ satisfy this specific condition. In Fig. 8(b), TDC is hardly influenced by thermo-optic coefficients and approximate zero when α₁, α₂ and α₃ are kept the same. Thus, only thermal expansion coefficients need to be considered when we select materials for an athermalization design.

Fig. 8. Maps of the TDC versus the (a) thermal expansion coefficients and (b) thermo-optic coefficients. α₁=10×10⁻⁶/ dT ℃, dn₁/dT = dn₂/dT = dn₃/dT = 5×10⁻⁶ /℃ in (a). dn₁/dT = 6×10⁻⁶/℃, α₁ =α₂ =α₃= 5×10⁻⁶ /℃ in (b).

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4. Design procedure and example

In a wavelength range of (λ_a-λ_b), according to Eq. (8), diffractive angles of λ_a and λ_b in the principal section of grating satisfy:

(30)$$\begin{array}{l} \sin {\theta _4}{^{\prime}_a} - \sin{\theta _4} = mg{\lambda _a},\\ \sin {\theta _4}{^{\prime}_b} - \sin{\theta _4} = mg{\lambda _b}. \end{array}$$

Combining Eqs. (30) and Eqs. (5)–(6), dispersive width h_spec of the spectral image can be expressed as:

(31)$${h_{spec}} = \textrm{C}{\textrm{I}_{\textrm{M }a}} - \textrm{C}{\textrm{I}_{\textrm{M }b}} = \frac{{{n_3}}}{{{n_1}}}{R_2}({\sin {\theta_4}{^{\prime}_a} - \sin {\theta_4}{^{\prime}_b}} ).$$

Then R₂ can be expressed as:

(32)$${R_2} = \frac{{{n_1}{h_{spec}}}}{{{n_3}mg({{\lambda_a} - {\lambda_b}} )}}.$$

Under anastigmatism and athermalization conditions, steps for designing an UCIS are shown in Fig. 9. Initial structure of the system can be obtained for the input specifications, λ_a, λ_b, h_spec, m, g and slit length. Lens materials are selected with n₂ the highest, and then R₂ can be calculated by Eq. (32). We can then get the astigmatism distribution in the object plane with Eq. (7). k₁ and k₂ are preferred for the largest anastigmatic region, thus R₁ and R₃ can be calculated. Substituting these structural parameters into Eq. (28), together with lenses’ refraction indexes and thermo-optic coefficients, map of the TDC versus the thermal expansion coefficients can be obtain. If α₁, α₂ and α₃ satisfy the athermalization condition (TDC≈0), radii and materials of lenses are then determined. If not, lens materials need to be reselected to satisfy the athermalization condition. Lateral displacement of slit can also be initially determined for achieving anastigmatism spanning the slit, and it may be slightly adjusted to avoid obscuration.

Fig. 9. Steps for designing an UCIS.

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Based on the above discussions and analyses, we have designed a high-quality UCIS system. Its initial structure was obtained with steps in Fig. 9, and then optimized by using ZEMAX optical design software. Table 1 shows its specifications and performance. Its spectral range is 0.4–1.0µm (VNIR band), F number is 2.25, slit length is 48 mm, and its spectral image spreads over 6 mm. Lens material is H-BaK3 (n = 1.547, α=6.2×10⁻⁶/℃) for L1 and L3, H-ZK7 (n = 1.613, α=6.1×10⁻⁶/℃) for L2. After optimization, the maximum distance between centers of three spherical surfaces is 0.2 mm. To obtain detector clearance, we shortened L1 by 3 mm in the forward direction. Layout of the designed UCIS is shown in Fig. 10(a). The system has excellent imaging quality with smile and keystone distortions less than 0.13µm. Figure 11(a) shows geometrical spot diagrams for images at five wavelengths and five object heights. The maximum RMS spot diameter is 7.59µm. All the spots are enclosed in a square pixel of side 15µm. This gives a spectral resolution of 1.5 nm.

Table 1. Characteristics of the UCIS Designed in the Text

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Fig. 10. Designs of different spectrometers with the same specifications shown in Table 1. (a) The UCIS. (b) The Offner-Chrisp spectrometer. (c) The Dyson spectrometer.

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Fig. 11. Geometrical spot diagrams of (a) the UCIS, (b) the Offner-Chrisp spectrometer and (c) the Dyson spectrometer.

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Tolerances of the system decide the difficulty and cost of the fabrication and alignment for the system. We set up the tolerances of the designed UCIS by using software CODE V’s tolerance analysis ability, and we utilize the diffractive MTF as the nominal criterion. Surface’s radius, irregularity, decenter, tilt, lens’s thickness, decenter, tilt, and material’s index, abbe for each element are concerned. The tightest five tolerance items and their effects on the MTF are given in Table 2, and other tolerance items not listed are routine (surface radius tolerance ± 1 fringe, surface irregularity ± 0.3 fringe, surface tilt ± 0.03deg, surface decenter ± 0.03 mm, lens tilt ± 0.03deg, lens decenter ± 0.03 mm, lens thickness ± 0.05 mm). It is noted that the worst offender is the irregularity of the grating substrate, i.e. the convex surface of L2. And other four offenders are all about L3. Therefore, the surface precision of L2, the fabrication and alignment precision of L3 should be guaranteed.

Table 2. The tightest five tolerance items of the designed UCIS

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By the final analysis, spanning full field and spectrum, the MTF has a maximum decrease of 0.075 for all tolerances concerned. These tolerances also result in a smile and keystone error below 8% of a pixel for the 97th percentile. The reduction of performance is acceptable and the designed UCIS is not difficult for fabrication and alignment.

In order to verify that the UCIS enables increased compactness, an Offner-Chrisp spectrometer and a Dyson spectrometer with the specifications in Table 1 are also designed. Designs for comparison are respectively shown in Figs. 10(b) and 10(c) with the same plotting scale. The designed Offner-Chrisp spectrometer has a package size of 250×160×340 mm, an increase of 10× in volume than the UCIS and doesn’t reach an equal imaging quality. Its geometrical spot diagrams are shown in Fig. 11(b), and the maximum RMS spot diameter is 13.2µm. The Dyson design includes a plano-convex lens made from CaF2 and a meniscus lens that is fused silica. Such design results in a package size of Φ82×255 mm, much smaller than the Offner-Chrisp type. Volume of the Dyson design is similar to the UCIS, but its length is still 1.9 times longer than the UCIS. Its geometrical spot diagrams are shown in Fig. 11(c), and the maximum RMS spot diameter is 11.8µm. Its slit is quite close to the image plane, with a clearance of only 13 mm, while the UCIS has a clearance of 50 mm between the slit and image. In conclusion, the designed UCIS can greatly reduce the volume without degrading the imaging performance, which is really important in remote sensing applications.

Furthermore, to verify the athermalization of the designed UCIS, imaging quality in a large temperature range from −30 ℃ to 70 ℃ was analyzed. Results show that the maximum RMS spot diameter is 7.71µm at −30 ℃ and 7.45µm at 70 ℃, which indicates excellent thermal adaptability. Thermal performance of Offner-Chrisp type and Dyson type are also analyzed for comparison. Their mechanical and optical materials in the model are given in Table 3. Typical thermal expansion coefficients between −30 ℃ and 70 ℃ are also given. MTF of three spectrometer types at −30 ℃, 20 ℃, and 70 ℃ are shown in Fig. 12. It is noted that MTF curves of the UCIS almost invariable at different temperature, and the lowest MTF at 33.3 Lp/mm is 0.82 at −30 ℃, 0.83 at 20 ℃ and 70 ℃. MTF of the Offner-Chrisp spectrometer drops from 0.75 at 20 ℃ to 0.66 at −30 ℃ and 70 ℃, although it utilizes the athermal match of fused silica and invar. Thermal performance of Dyson spectrometer is also not as good as the UCIS. Its MTF drops from 0.83 at 20 ℃ to 0.63 at −30 ℃ and 0.68 at 70 ℃. Therefore, thermal adaptability of the designed UCIS is far superior to other two types.

Fig. 12. MTF of three spectrometer types at −30 ℃, 20 ℃, and 70 ℃. The evaluated wavelength is 0.7µm.

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Table 3. . Mechanical and optical materials of two types

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5. Conclusions

We presented a novel pushbroom imaging spectrometer, the UCIS. Its anastigmatism condition was discussed by tracing its chief ray and mapping the astigmatism distribution. Results show that the UCIS enables a wide anastigmatic range for long slit when the refraction index of L2 is the highest. We also pointed that athermalization can be realized by matching lens materials with different thermal expansion coefficients. Thermo-optic coefficients of materials do not have significant effect on the thermal defocus. Under anastigmatism and athermalization conditions, a specific example for the VNIR band was designed. It confirms that the UCIS gives an excellent imaging quality and a negligible decline in imaging quality from −30 ℃ to 70 ℃. Results also show at least a 10× reduction in volume than classic Offner-Chrisp spectrometers when considering equal spectral bandwidth, F number, slit length and spectral resolution. Compared with Dyson spectrometer, the UCIS still has the advantage of almost half the length and larger clearance between slit and image. It is very suitable for hyperspectral remote sensing applications requiring wide swath, high SNR and small size.

Funding

National Key Research and Development Program of China (2016YFB0500501-02); Priority Academic Program Development of Jiangsu Higher Education Institutions.

References

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Specifications and Performance	Value
Spectral range /µm	0.4–1.0
F number	2.25
Slit length /mm	48
Dispersion width /mm	6
Groove density / lines·mm⁻¹	226.4
Diffraction order	−1
Lateral displacement of slit /mm	29.8
R1 /mm	45.84
R2 /mm	67.09
R3 /mm	135.03
Center thickness of L1 /mm	45.79
Center thickness of L2 /mm	21.45
Center thickness of L3 /mm	67.80
Concentricity /mm	0.2
Package size /mm	90×100×135
Smile /µm	0.13
Keystone /µm	0.13
RMS spot diameter @ 20 ℃/µm	7.59
RMS spot diameter @ −30 ℃/µm	7.71
RMS spot diameter @ 70 ℃/µm	7.45

Parameter	Tolerance	Decrease of MTF
G irregularity	±0.2 fringe(P-V)	0.043
L3 surface tilt about X	±0.02 deg	0.022
L3 element tilt about X	±0.02 deg	0.018
L1 and L3 index	±0.0006	0.010
L3 element Y-decenter	±0.02 mm	0.008

Elements	Offner-Chrisp type	Dyson type
Grating	F_Silica α=0.51×10⁻⁶/℃	F_Silica
Spherical mirrors	F_Silica	——
Lenses	——	F_silica & CaF2 α=18.9×10⁻⁶/℃
Mechanical component	Invar α=0.71×10⁻⁶/℃	Invar

Specifications and Performance	Value
Spectral range /µm	0.4–1.0
F number	2.25
Slit length /mm	48
Dispersion width /mm	6
Groove density / lines·mm⁻¹	226.4
Diffraction order	−1
Lateral displacement of slit /mm	29.8
R1 /mm	45.84
R2 /mm	67.09
R3 /mm	135.03
Center thickness of L1 /mm	45.79
Center thickness of L2 /mm	21.45
Center thickness of L3 /mm	67.80
Concentricity /mm	0.2
Package size /mm	90×100×135
Smile /µm	0.13
Keystone /µm	0.13
RMS spot diameter @ 20 ℃/µm	7.59
RMS spot diameter @ −30 ℃/µm	7.71
RMS spot diameter @ 70 ℃/µm	7.45

Parameter	Tolerance	Decrease of MTF
G irregularity	±0.2 fringe(P-V)	0.043
L3 surface tilt about X	±0.02 deg	0.022
L3 element tilt about X	±0.02 deg	0.018
L1 and L3 index	±0.0006	0.010
L3 element Y-decenter	±0.02 mm	0.008

Analytical design of athermal ultra-compact concentric catadioptric imaging spectrometer

Abstract

1. Introduction

2. Theory of the UCIS

3. Athermalization of the UCIS

4. Design procedure and example

5. Conclusions

Funding

References

Cited By

Figures (12)

Tables (3)

Equations (32)

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