II.A.

Non-freeform designs

In this sections designs that use off-axis sections of rotationally symmetric surfaces are presented. The parent surfaces are rotationally symmetric polynomial aspheric surfaces, which are described by a polynomial expansion of the deviation from a conic surface. They already provide very good performance and we use these designs as starting points for the freeform optimization in the next section.

The definition of a freeform surface varies in literature, we consider a surface freeform when it lacks any rotational symmetry. Other definitions arise due to the complexity of the manufacturing procedures. Some of the surfaces in the present section are off-axis aspheres with large deviation from the best fit sphere (BFS) and they could be considered freeform (although the surface still has symmetry) because the complexity of the surface shape requires freeform manufacturing techniques. The surface would not be manufactured off-axis, but instead it would be centred during the ultraprecise fabrication process, e.g. using servo assisted diamond turning.

As it was mentioned in the previous section we will focus on the designs that are based on rectangular apertures, specifically, the ones illustrated in Table 3. Using requirements and design constrains outlined in Table 1, two optical designs were created; both based on a Three Mirror Anastigmat (TMA) using off-axis aspherical components. First, the layouts of the design with 200mm x 100mm rectangular entrance pupil (also the primary mirror) are depicted in Figure 1. The size of the deployable primary mirror equals the larger side of a 2U rectangular volume and the rest of the optical components are located inside the 2U volume. From the 3D layout it becomes apparent that there is plenty of space for e.g. non-optical components inside the 2U volume.

Figure 1

The side view (left) and 3D view of the off-axis asphere based TMA with a 200mm x 100mm rectangular primary mirror. The 200mm x 100mm primary is deployable, the other mirrors are fixed.

The Strehl ratio and the MTF curves of the design are shown in Figure 2. The image quality is very good, the design provides diffraction limited performance over the complete 1.1 x 1.1 degree² FOV (Strehl ratio > 0.8). The MTF on the other hand reveals that the contrast (and resolution) is not sufficient in the direction, which is perpendicular to the shorter side of the 200mm x 100mm aperture. Comparing the tangential and sagittal MTF curves, in tangential direction the MTF stays above 0.2 for the full field, but in sagittal direction the cut-off frequency is around 35 cycles per mm, which is lower than the Nyquist frequency.

Figure 2

On the left the RMS Strehl Ratio map of the design presented above is shown. The Strehl Ratio is better than 0.85 everywhere in the FOV, the worst being in the centre. In the right the PSF cross section in the centre of the FOV (worst case) is shown, FWHM is ~20 μm.

In the following a design based on two rectangular entrance pupils (also acting as primary mirror segments) is presented. The larger rectangular mirror is deployable and has a 200mm x 90mm size, the smaller primary mirror segment is fixed with the rest of the optical system, and it has a 200mm x 40mm size. All optical components are fixed and located in the 2U volume, except the deployable primary mirror. Two layouts of the design are shown in Figure 3 and the red 100mm x 100mm square corresponds to the cross section of the 2U volume. In order to have a completely unobscured optical system and avoid any vignetting, a very minor field offset is applied and also the primary mirror segments are smaller than what was prescribed in section I. M1-a and M1-b are the off-axis sections of the same parent optical axis and the concept is the same for M2-a and M2-b.

Figure 3

The side view (left) and 3D view (right) of the off-axis asphere based TMA with two primary mirrors. One deployable with 200mm x 90mm size (M1-a) and one fixed with 200mm x 40mm size (M1-b). The red 100mm x 100mm box is the cross section of the 2U volume.

Although the entrance pupil area is increased with respect to the previous design, which ideally would improve the resolution and MTF performance in sagittal direction, due to large geometrical aberrations and reduced Strehl ratio, the resolution and MTF actually become worse. Some improvements can be achieved by tilting the focal plane surface and optimizing the surface shape of the fold mirror. Also, M1-a-and M2-a surfaces can be optimized independently from M1-b and M2-b to further improve image quality, but it still remains worse than the diffraction limit over a significantly large area of the field of view (the Strehl ratio map and the MTF curves for the design are shown in Figure 6).

Figure 4

The side view (left) and top view (right) is shown. By tilting M3, tilting/decentering M4 and at the same time optimizing with freeform surfaces, the image quality can be recovered.

Figure 5

3D layouts of the design are shown. The two largest mirrors are the primary segments, the rectangular is a deployable and the U-shaped is fixed. By increasing the area of the primary mirror, the MTF can be improved significantly.

Figure 6

The Strehl ratio (left) and the MTF curves (right) are shown for the non-freeform (top row) and freeform (bottom row) designs. The freeform optical design offers diffraction limited performance except the corners of the square FOV and the MTF performance is also very good, the average MTF is higher than 0.2 at the Nyquist frequency higher and higher than 0.4 at the half of the Nyquist frequency.

II.B.

Freeform designs

The design shown in Figure 3 is the starting point for our freeform optimization. We set the following targets for the freeform optical system. On one hand, we would like to reduce the geometrical aberrations in the system and recover diffraction limited performance over the full FOV. Additionally, by tilting and decentering various optical components, we aim at creating a spatial arrangement, which improves mounting and packaging of the instrument and also decrease scattered light. We would like to investigate the possibility of further improving the MTF performance by increasing the size of the fixed primary segment. Finally, we investigate optical design that make use of square and TDI detectors as well, so we optimize using different FOV and focal plane shapes.

In Figure 4 and Figure 5 the layout of the first freeform based optical design is shown. By tilting M3, both M4 and the detector can be placed outside of the symmetry plane of the optical system (Figure 4, right). Having achieved that, both the size of M1-a (from 200mm x 90mm to 200mm x 100mm) and M1-b can be increased. In the case of the latter, a U-shaped aperture is created. The out of plane arrangement offers the possibility of placing baffles at various locations and thus achieves a more efficient scattered light reduction than before. In the optimization process, M2 and M4 were transformed into freeform surfaces, M1-a and M1-b still have the same parent optical surface, which is a rotationally symmetric asphere, M3 is a (centred) rotationally symmetric asphere.

In the following the optical performance of the freeform design is compared to the rotationally symmetric design (Figure 3 in section II.A). The Strehl ratio and MTF curves are shown in Figure 6 for both designs and it becomes apparent that the freeform optical design improved significantly. It offers diffraction limited performance almost all over the square FOV and the MTF performance is also very good. The average MTF is higher than 0.2 at the Nyquist frequency higher and higher than 0.4 at the half of the Nyquist frequency. The design improved a lot with respect to the non-freeform optical design especially in contrast.

Another freeform based design is developed, which is based on a 2k x 128 stage TDI detector and thus have a very elongated focal plane area. In Figure 7 the 3D layout and the footprint on the freeform surface is shown. The M2 and M4 mirrors are merged into one freeform mirror, which is a monolithic mirror with multiple freeform surfaces. Similarly to the previous freeform based design, M1-a and M1-b have the same parent optical surface, which is a rotationally symmetric asphere and M3 is a (centred) rotationally symmetric asphere. The image quality is shown in Figure 8, it is diffraction limited over the full field of view and the MTF is also very promising.

Figure 7

A freeform based design with a 2k x 128 (stage) TDI detector is shown on the left. The freeform surface itself is the M2 and M4 in one surface. The footprint on the surface is shown on the right. M1-a is deployable, the other optical surfaces are fixed in the 2U volume.

Figure 8

The Strehl ratio and the MTF curves are shown for the freeform based optical design, which is based on the TDI detector.

II.C.

Freeform surface types and manufacturing

We used two surface types that are both readily available in Zemax OpticStudio™, the optical design software that we were using during the freeform optical design optimization. The Zernike Standard Sag is defined by the same polynomial as the one that we used in the non-freeform optimization, plus additional aspheric terms defined by the Zernike Standard coefficients. The second surface type is described by a base radius of curvature and a sequence of Chebyshev polynomials, which is orthogonal over a rectangular area and so fits our purposes due to the rectangular mirror apertures in our designs. Still, most of the time, the Zernike Standard Sag surface type is proved to be more useful, probably because the starting point design is a TMA type optical system. Even, when the Chebyshev polynomials outperform the Zernike polynomials, a very high number of polynomial orders and coefficients are necessary to describe the surface, which makes the optimization process less efficient and global minima finding methods are needed to be implemented to avoid local minima in the optimization process. As usually in freeform optical design optimization, the careful selection of the field configuration, which essentially defines the input ray bundle is very important due to the necessity to achieve proper sampling on the freeform surface and homogeneous optical performance over the field.

There are several well established methods for freeform manufacturing. Small freeform lenses make use of developments in plastic/glass moulding technologies, components with larger sizes are realized by ultra precision milling, grinding and polishing. Diamond turning, a very common technique, can be successfully applied in many cases, but it has its limitations due to various factors, like temperature drifts, residual vibrations or undesired machining forces. If low surface form error (<500nm PV) and/or roughness (<3-5nm) is needed local figure correction techniques have to be used, e.g. ion beam figuring (IBF) or magnetorheological finishing (MRF). Applications, which require broad wavelength range utilize freeform mirrors, usually additional layers (electroless Ni) are applied to the substrate (Al, SiC) to increase local figure correction quality [2]. For the optical systems described in the present paper, the latter manufacturing methods would provide the performance that is necessary to achieve specifications.