## Abstract

Increasing the size of low-orbiting space telescopes is necessary to attain high-resolution imaging for Earth or planetary science, which implies bigger and more complex imaging systems in the focal plane. The use of homothetic imaging systems such as the Spot and Pleiades push-broom satellites would lead to prohibitive linear focal plane dimensions, especially for IR missions requiring large-volume cryostat. We present two optical TMA telescopes using an image-segmentation module based on astronomical image slicer technology developed for integral field spectroscopy, made of a set of freeform mirrors defined by Zernike polynomials. Each telescope has a linear 1.1° field of view; the first one considers a matrix detector and the second one considers several linear TDI detectors currently used in space missions. We demonstrate that such systems provide efficient optical quality over the full field and offer a substantial gain in terms of volume of the focal plane arrays.

© 2017 Optical Society of America

## 1. INTRODUCTION

A dimensioning parameter for Earth-observation low-orbiting imaging systems is the focal plane dimension. For the next generation of VIS/IR instruments using the TDI push-broom mode, the need for high angular resolution over a wide field of view leads to an excessively large linear focal plane. For instance, the homothetic Pleiades 1.1° FoV telescope would lead to 700 mm image size in the focal plane to get 25 cm spatial resolution on Earth (Fig. 1 shows Pleiades focal plane dimensions). Moreover, the IR channel would require the use of a heavy, large, and high-power-consumption cryostat, which may be prohibitive.

We propose an innovative approach to focal plane arrangement based on the integral field unit (IFU) technology developed for ground-based [1,2] and space spectrometers [3,4] for the James Webb Space Telescope (JWST). The idea is to consider the IFU principle in a reverse way, i.e., to subdivide the linear field of view with segmenting mirrors and re-image it onto a 2D array TDI detector or several linear TDI detectors currently used in space missions. We named this approach a segmentation module, and it allows for a small, low-weight, low-power-consumption focal plane ideally suited for VIS and IR imaging applications.

The main issue concerns the gap of FoV between an IFU device for astronomical observation and an Earth-observing telescope. As shown in Table 1, the telescope field of view is almost 1500 times higher than the IFU for the Super Nova Acceleration Probe (SNAP) or the Near Infrared Spectrograph (NIRSpec). The very wide FoV creates huge optical aberrations but, as we will demonstrate, freeform mirrors allow us to reduce it significantly. A growing work in freeform optics has been ongoing for a few years, especially on wide-field optical design [5–7], fabrication, metrology [8], and assembly [9]. Earth-observation instruments will take advantage of these developments.

Some developments have been carried out on field segmentation through the DARPA Aware program [10] to reach a wide field at high-resolution objective. A monocentric multiscale camera has been realized and would increase the target discrimination and search in daytime and nighttime conditions during military operations. In Section 5, we compare the characteristics of the telescopes we present hereafter with this camera.

## 2. OPTICAL PRINCIPLE AND OPTIMIZATION PROCESS

The optical principle is shown in Fig. 2 and is made of two successive elements:

- • An unoptimized classical TMA Korsch telescope where mirrors are simply defined by conic constants. An accessible pupil plane is defined to set a deformable mirror (DM) [11,12] whose main aim is to compensate for thermo-elastic drifts as well as zero-gravity bending effects. This gives a more complementary way of correction than M2 displacement capability for the correction of undesired misalignment.
- • A segmentation module of magnification $m$ to reach the required spatial resolution on Earth. It acts as a sub-field repositioning system, and slices the total field into several sub-fields that are then re-imaged on the detector. The slicing of the field is operated right after the Korsch focal plane in order to generate an overlap between sub-images, thus facilitating image post-processing reconstruction. It is made of a set of freeform slicing mirrors, hereinafter called ms1, and a set of freeform focusing mirrors, hereinafter called ms2. Zernike polynomials up to the fifth order have been considered for the definition of these mirrors.

This solution is interesting if the Korsch focal plane dimension is small, allowing us to reduce segmentation module mirror size and complexity. In this sense, the segmentation module magnification $m$ is higher than 1, but does not have to be too high to avoid a fast Korsch. To get an effective telescope, we have to find a tradeoff between the number of sets of mirrors in the segmentation module, the complexity of the shape of the optics, and the magnification of the module. The first step is to do a parametric study, in which every parameter of the module is varied like in Table 2.

The high number of sets of mirrors increases strongly the mass of the system and makes it very complex, so two sets of mirrors were chosen. As mentioned previously, the module magnification has to be higher than 1 but not too high to avoid a fast Korsch telescope. Values of 3 or 4 help to reduce incident angles on mirrors in the module, but imply dealing with a very fast Korsch, which may be too sensitive to misalignment and would inflate the cost of its integration. We decided to fix the magnification $m=2$. These two choices led to the use of more complex optics to correct field aberrations. We investigated the different surface shapes according to the list shown in Table 2 by increasing step by step the complexity. From spherical to $XY$ polynomials, optical performance is not good enough, so the use of freeform mirrors defined by Zernike polynomials is the ultimate solution. Thanks to the asymmetry of the polynomials and a smart selection of some of them, we have more degrees of freedom to correct field aberrations, enabling us to attain near-diffraction-limited telescopes.

The second step is to optimize efficiently the telescope by following this optimization process:

- 1. Optimization of the Korsch telescope; the segmentation module does not yet exist;
- 2. Design of the segmentation module and independent optimization from the Korsch;
- 3. Global optimization: Korsch + segmentation module.

We propose two optical telescopes with a segmentation module based on freeform optics to correct such aberrations. The two 1.3 m diameter telescopes cover a scan-field of 1.1°, which, for a low-orbiting imaging telescope at 700 km altitude, corresponds to a swath width of 13.4 km. The angular resolution is 73.7 mas to get a 25 cm spatial resolution on Earth. We consider the VIS panchromatic channel of spectral range 490–800 nm. The main telescope parameters are shown in Table 3.

## 3. TELESCOPE 1: OPTICAL DESIGN AND PERFORMANCE

#### A. Optical Layout and Focal Plane Arrangement

This solution uses only one matrix detector, a new generation of CMOS TDI portion-by-portion matrix detector ($5k\times 5k$, 8 μm pixel size) currently under development at the French Space Agency, CNES. The general view of the optical system is shown in Fig. 3.

The segmentation module is made of a total of 26 small mirrors, divided into two sets of mirrors:

- • The first set of mirrors is made of 13 mirrors. They are positive segmenting mirrors, named ms1. They slice the $x\text{\hspace{0.17em}}\text{field}=1.1\xb0$ linear FoV into 13 sub-fields of 0.1023°. The light is reflected through the second set of mirrors.
- • The second set of mirrors is made of 13 mirrors. They are positive focusing mirrors, named ms2. They reflect the sub-fields and form 13 sub-images on the sensor.

A couplet of one segmenting mirror plus one focusing mirror slice and re-image one sub-field on the matrix sensor. In the final system, there are 13 configurations. In a first step, we design and optimize only three configurations: ${C}_{0}$ (central-field), ${C}_{3}$ (mid-field), and ${C}_{6}$ (edge-field). The objective is to demonstrate the concept of field slicing for space telescopes, but not to design a complete system. Configurations are indexed from ${C}_{-6}$ to ${C}_{6}$, positive and negative indexed configurations being symmetrical. Each configuration considers a sub-field $(x\text{\hspace{0.17em}}\text{field},y\text{\hspace{0.17em}}\text{field})=(0.1023\xb0,0.00256\xb0)$ corresponding to an image size of $40.0\text{\hspace{0.17em}}\mathrm{mm}\times 1.0\text{\hspace{0.17em}}\mathrm{mm}$. By positioning the slicing mirrors 32.4 mm out of the focal plane, we assure an overlap of 7.5 mm between two adjacent sub-images. For the optimization of these three configurations, we consider 10 field points (see Fig. 4): one at the center of each edge, two at $\pm 0.7*0.05115\xb0$, and two at $\pm 0.05115\xb0$ around the central field.

Note that the $x$ axis is the across-track direction (perpendicular to the on-Earth projection of the satellite motion) and the $y$ axis is the along-track direction.

The main advantage of this solution is the focal plane compactness. All the 1.1° FoV is imaged onto a $40\times 40\text{\hspace{0.17em}}\mathrm{mm}$ size matrix detector, and the 13 sub-images are positioned on top of each other; it is 17 times smaller than the classical TMA push-broom telescope focal plane. The price to be paid is the addition of few mirrors along the optical train. However, this can replace the splitting flat mirrors used in the Pleiades telescope just before the focal plane, as described in Fig. 1.

#### B. Optical Performance

### 1. Wavefront Error

For estimation of the image quality, we use wavefront error (WFE) and the modulation transfer function (MTF). For the sub-fields corresponding to configurations C0, C3, and C6, the results of WFE are shown in Fig. 5 and the numerical values are summarized in Table 4.

These false-color representations of the WFE maps clearly show a good homogeneity of performance over the field and between each configuration. The mean WFE over the three configurations is 37.5 nm RMS, which is quite high. In this study, the mirror placed in the pupil relay is flat. In the real system, it will be a deformable mirror to first compensate for thermo-elastic drifts as well as zero-gravity bending effects, and then to improve global performance of each configuration by correcting aberrations that are constant across the field. The typical required WFE for VIS observations is about 20 nm RMS. The dynamic range of existing DMs developed by the French Space Agency in recent years is high enough to reach 20 nm RMS and at the same time to correct errors on the primary mirror. We consider the quality of the module to be high enough to be combined with an active system to get a high-resolution and low-distortion image.

### 2. Modulation Transfer Function

MTF is an important parameter for Earth imaging; it characterizes the capability of the telescope to resolve observed scene details. The MTF is the module of the optical transfer function defined as the Fourier transform of the point spread function (PSF). The telescope pupil is circular, so the PSF would be an Airy disk for a diffraction-limited instrument. The pixel size is 8 μm, corresponding to a cutoff spatial frequency of 62.5 cycles/mm; this is called the Nyquist frequency of the detector. In Fig. 6, we present normalized MTF values for the 10 field points (cf. Fig. 4) according to the image spatial frequency up to the Nyquist frequency of the detector.

As for WFE in the previous section, we have a good performance homogeneity close to the diffraction limit over the field by and between configurations. MTF values are up to 0.1 at Nyquist frequency, where the diffraction limit value is 0.22. In addition, it is necessary to mention that the design has a low distortion of less than 1.1% per image.

#### C. Mirror Shape

The segmentation module mirrors are defined using biconic Zernike polynomials and the following sag equation:

Zernike polynomials have more contribution in slicing mirrors than focusing mirrors. Field aberration correction increases with the telescope FoV so that the deviation at configuration C6 is around 6 times higher than on-axis configuration C0. The compensation is mainly done by the first set of mirrors, the slicing mirrors. We show a tendency as to the deviation patterns between mid-field C3 and off-axis C6 configurations; the deviation patterns are similar and deviation values increase through the field, making us confident about the optimization of the intermediate configurations.

#### D. MTF Tolerancing Study

A preliminary tolerancing study has been performed to check the sensitivity of the segmentation module for each configuration. We consider the MTF parameter in tangential and sagittal modes at the Nyquist frequency 62.5 cycles/mm and Nyquist/2 frequency 31 cycles/mm. We applied perturbation on slicing and focusing mirrors, corresponding to tip/tilt and shifts of respectively 100 μrad and 50 μm along the axis $\pm X$ and $\pm Y$; it is approximately 3 to 5 times the achievable precision during the integration process.

In Fig. 9, we clearly see that the segmentation module mirrors are very tolerant to tip/tilt and shifts, and the maximal MTF value deviation from the nominal does not exceed 0.025; it occurs for the off-axis configuration C6 when the slicing mirror is tilted. Globally, the position of slicing mirrors is slightly more sensitive. This is because the reflected beam is deflected so that the beam footprint on the focusing mirror is shifted, and also because freeform slicing mirrors have a higher deviation from the sphere than focusing mirrors. But keep in mind that the MTF value at the Nyquist frequency exceeds 0.1, with a diffraction-limited value of 0.22. We can consider the 0.025 maximum deviation to be negligible. In this case, freeform optics does not impact the tolerancing budget.

## 4. TELESCOPE 2: OPTICAL DESIGN AND PERFORMANCE

#### A. Optical Layout and Focal Plane Arrangement

This solution uses nine TDI linear detectors already qualified for space missions. The general view of the optical system is shown in Fig. 10.

The segmentation module is made of a total of 18 small mirrors, divided into two sets of mirrors:

- • The first set of mirrors is made of nine mirrors. They are positive segmenting mirrors, named ms1. They slice the linear $x\text{\hspace{0.17em}}\text{field}=1.1\xb0$ into nine sub-fields of 0.1432°. The light is reflected through the second set of mirrors.
- • The second set of mirrors is made of nine small mirrors. They are positive focusing mirrors, named ms2. They reflect the sub-fields and form nine sub-images on nine TDI linear sensors.

As for Telescope 1, a couplet of one segmenting mirror plus one focusing mirror slice and re-image one sub-field on one linear sensor. In the final system, there are nine configurations. In the first step, we design and optimize only three configurations: ${C}_{0}$ (central-field), ${C}_{2}$ (mid-field), and ${C}_{4}$ (edge-field). Configurations are indexed from ${C}_{-4}$ to ${C}_{4}$, positive and negative indexed configurations being symmetrical. Each configuration considers a sub-field of $(x\text{\hspace{0.17em}}\text{field},y\text{\hspace{0.17em}}\text{field})=(0.1432\xb0,0.0064\xb0)$ corresponding to an image size of $91.0\text{\hspace{0.17em}}\mathrm{mm}\times 1.0\text{\hspace{0.17em}}\mathrm{mm}$. By positioning the slicing mirrors 100 mm out of the focal plane, we assure an overlap of 14 mm between the two adjacent sub-images. For the optimization of these three configurations, we consider 10 field points (cf. Fig. 4): one at the center of each edge, two at $\pm 0.7*0.07162\xb0$, and two at $\pm 0.07162\xb0$ around the central field.

Note that the focal plane dimension is 150 by 250 mm in one plane, instead of 700 mm long in two planes for the homothetic focal plane of Pleiades shown in Fig. 1.

#### B. Optical Performance

### 1. Wavefront Error

WFE results in the focal plane are shown in Fig. 12 and numerical values are summarized in Table 6.

These false-color representations of the wavefront error clearly show a good homogeneity of performance over the field by configuration but a higher disparity between each configuration. The mean wavefront error is 36 nm RMS, which is of the same magnitude as for Telescope 1. Typically, 20 nm RMS is also targeted in that case. As for Telescope 1, the deformable mirror may be used to improve the performance and reach the 20 nm RMS goal. Furthermore, worse performance occurs for the configuration C2 than for C4; this reflects the sensitivity during the optimization process in the Zernike polynomial order selection. We increase mirror shape complexity step by step by adding polynomials, and the performance doesn’t converge through the optimum in the same way according to the polynomial selected.

### 2. Modulation Transfer Function

In this solution, the Nyquist frequency is 38.5 cycles/mm. In Fig. 13, we present normalized MTF values for the 10 field points (cf. Fig. 11) according to the image spatial frequency up to the Nyquist frequency of the detector.

We obtain good performance homogeneity over the field by and between configurations. A slight degradation occurs, nonetheless, off-axis. MTF values are up to 0.11 at the Nyquist frequency where the diffraction limit value is 0.22. Global performance is slightly better than for Telescope 1, angles of incidence on mirrors are reduced, and the incident aperture on the segmentation module is $F/\#=14$ instead of $F/\#=8.6$ for Telescope 1. In addition to the image quality just shown, it is necessary to mention that the design has a very low distortion: less than 0.4% by image.

#### C. Mirror Shape

The segmentation module mirrors are defined by Zernike polynomials and the following sag equation:

Field aberration correction is mainly done by the first set of mirrors. PV deviation of slicing mirrors is higher than for Telescope 1 by a factor 2 to 4, and lower by a factor 1 to 6 for focusing mirrors. Note that mirrors are twice the size of those in Telescope 1. We also show a good tendency on the deviation patterns between mid-field C2 and off-axis C4 configurations, with similar deviation patterns and increasing deviation values through the field, making us very confident about the optimization of the intermediate configurations.

## 5. COMPARISON TO MONOCENTRIC MULTISCALE CAMERA

In Section 1, we mentioned that some developments have been carried out on field slicing with the realization of a monocentric multiscale camera for terrestrial military operations [10]. The purpose of this DARPA Aware program is to realize a wide FoV, high resolution, compact, and lightweight gigapixel-class camera for day/night vision. The camera is made of a simple monocentric glass objective followed by microcameras using spherical or aspherical glass optics which have overlapping fields of view. Here, the complexity is transferred from the objective, which has been greatly simplified, to the focal plane with more than 1000 optics and 200 sensors. Contrarily, the purpose of our study is to simplify the telescope focal plane and reduce its size, which leads to an increase in the complexity of the optics. Table 8 compares the main characteristics of Telescopes 1 and 2 with the monocentric multiscale camera.

Telescope 1 and 2 see 1.1° FoV compared to the 120° FoV for the monocentric camera. In Telescope 1 and 2, each camera (named “configuration” in previous sections) considers a larger sub-field relative to the total FoV than that for the monocentric multiscale camera, implying an increase in the complexity of the optics. This is why freeform optics are required for the segmentation module mirrors we present in this paper. Moreover, our systems consider only 1 matrix sensor or 9 linear sensors, whereas the monocentric camera uses 226 matrix sensors. These microcameras are made of 7 lenses for a total of 1582 lenses, which is obviously impossible in space. These two systems differ on many aspects which arise from the objectives of the mission, and we clearly distinguish the two opposite trends of optical conception. However, it highlights the field slicing and overlapping concept as a solution to provide a wide-field, high-resolution imaging system.

## 6. DISCUSSION AND CONCLUSION

The emergence of innovative manufacturing technologies allows us to envision optical designs using freeform mirrors. This approach revolutionizes the way optical designs are optimized, and offers new parameters of optimization.

We propose two optical TMA telescope designs working on a drift-scan mode for high-resolution planetary imagery using an image-slicer-based technology to reduce the volume and mass of the focal plane. An innovative segmentation module slices the linear FoV and reimages it on the telescope focal plane in a very compact way. The first solution presents an imager using only one 40 mm side matrix detector, while the second solution uses several linear detectors. With the use of freeform mirrors, we can reach near-diffraction-limited performance. The first solution is more compact, uses only one detector, and has smaller mirror size. Concerns about tolerancing, mirror manufacturing, and alignment strategies should be considered carefully to reach a good optical quality.

Some improvement may be made to reach diffraction-limited performance. We could increase the complexity of the shape of the Korsch telescope mirrors, first by using aspherics and, if necessary, freeform surfaces. Second, Zernike polynomials are widely known as an orthonormal function basis connected with wavefront aberrations. The classical Zernike polynomials are defined over a circular aperture, while slicing mirrors are rectangular in both designs. 2D Legendre polynomials or other polynomials as described in [13] might be used. Another solution would be to use curved linear detectors to correct more efficiently field-dependent aberrations, especially the field-curvature aberration. The Laboratory of Astrophysics of Marseille (LAM, France) and the CEA Leti (France) are working together on variable curved detector technology and are able to curve continuously from convex to concave VIS and IR matrix detectors [14,15]. This technology will provide another degree of freedom for field aberration correction (especially astigmatism and field curvature), and help future designs to be more compact, with simpler mirror shapes.

## Funding

DGA/Aix-Marseille Université (AMU) (2014.60.00.40); French Space Agency, Centre National d’Etudes Spatiales (CNES) (R-S15/OT-001-117); Centre National de la Recherche Scientifique (CNRS).

## Acknowledgment

This study has been conducted thanks to a Ph.D. fellowship from DGA/AMU as part of the Research & Technology project led by the French Space Agency (CNES). Submission is covered by CNRS funding.

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