Innovative focal plane design for large space telescope using freeform mirrors

Wilfried Jahn; Marc Ferrari; Emmanuel Hugot

doi:10.1364/OPTICA.4.001188

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.

Table 1. Comparative Table of IFU Characteristics and the Proposed Segmentation Module

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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.

Fig. 1. Pleiades focal plane. Linear TDI detectors are positioned on each box face. Light split is done inside by folding and splitting mirrors. The inner volume is about $400 \times 100 \times 80 mm$ [credit CNES].

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Fig. 2. Optical principle of the reverse image-slicing telescope.

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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.

Table 2. Investigated Parameters for the Parametric Study

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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 $X Y$ 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.

Table 3. Main Characteristics of the Two Telescopes

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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 ( $5 k \times 5 k$ , 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.

Fig. 3. Top: layout of the unfolded TMA Telescope 1. Bottom: zoom on the unfolded segmentation module layout. Colors represents sub-fields, and each sub-field is a configuration as explained later. Green is configuration C0, red is C3, and orange is C6.

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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 $x field = 1.1 °$ 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 $(x field, y field) = (0.1023 °, 0.00256 °)$ corresponding to an image size of $40.0 mm \times 1.0 mm$ . 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 $\pm 0.7 * 0.05115 °$ , and two at $\pm 0.05115 °$ around the central field.

Fig. 4. Scheme of the optimized configurations we consider and respective image position on the matrix detector. Green is configuration C0, red is C3, and orange is C6.

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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 $40 \times 40 mm$ 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.

Table 4. Wavefront Error Indicator of the Three Configurations

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Fig. 5. Image quality: RMS wavefront error of configurations C0, C3, and C6 from top to bottom.

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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.

Fig. 6. Modulation transfer function for 10 points in the sub-fields corresponding to configurations C0, C3, and C6 (from top to bottom).

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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:

z = \frac{c_{x} x^{2} + c_{y} y^{2}}{1 + {[1 - (1 + k_{x}) c_{x}^{2} x^{2}]}^{1 / 2} - {[1 - (1 + k_{y}) c_{y}^{2} y^{2}]}^{1 / 2}} + \sum_{i = - 1}^{N = 19} A_{i} Z_{i} (ρ, ϕ),

where

c_{x}

and

c_{y}

are the curvature of the surface along the

x

and

y

axes, respectively;

k_{x}

and

k_{y}

are the conic constants (equal to 0 in our case);

N

is the number of considered Zernike coefficients;

A_{i}

is the coefficient of the

i

th Zernike Standard polynomial;

x

,

y

are the coordinate in lens units;

ρ

is the normalized radial coordinate; and

ϕ

is the angular coordinate. To assess complexity of the designed freeform surfaces, we use the difference between the calculated surface sag and the mean spherical surface with null Zernike values. Numerical values are given in Table 5. The mean radius of curvature is calculated as the mean of tangential and sagittal radii of curvature. Results are shown in Figs. 7 and 8.

Table 5. PV Deviation in μm from the Mean Spherical Surface of Mirrors

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Fig. 7. Residuals after subtraction of the mean spherical surface for slicing mirrors of configurations C0, C3, and C6.

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Fig. 8. Residuals after subtraction of definition sphere for focusing mirrors of configurations C0, C3, and C6.

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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.

Fig. 9. MTF value deviation at Nyquist frequency from nominal solution due to tip/tilt and shift (respectively filled and striped bar) on slicing and focusing mirrors (respectively upper and lower diagram).

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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.

Fig. 10. Top: layout of the unfolded TMA Telescope 2. Bottom: zoom on the unfolded segmentation module layout. Green is configuration C0, red is C2, and orange is C4.

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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 $x field = 1.1 °$ 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 $(x field, y field) = (0.1432 °, 0.0064 °)$ corresponding to an image size of $91.0 mm \times 1.0 mm$ . 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 $\pm 0.7 * 0.07162 °$ , and two at $\pm 0.07162 °$ 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.