Melanie Roulet

Carolyn Atkins

Emmanuel Hugot

Sabri Lemared

Simona Lombardo

Marc Ferrari

3D printing for astronomical mirrors

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INTRODUCTION

3D printing could revolutionise the manufacture of optics and telescope structures. This new manufacturing process was initiated in the early 1980s, [START_REF] Bird | Exploring the 3d printing opportunity[END_REF] first used with plastics the technology offers now a wide range of materials from ceramics to metals as well as composite materials. Many studies started to investigate 3D printed optics both for space and ground telescopes and the results are very promising 2 . [START_REF] Atkins | Additive manufactured x-ray optics for astronomy[END_REF] In this paper we focus on space optics. The main requirements regarding the launch phase for space optics are their rigidity and their mass. 3D printing could provide lightweight structures which are not fabricable with traditional mechanical manufacturing; therefore the scope for mirror and lightweighting design has become substantially wider.Combined with topology optimisation, this technology is even more powerful to create the optimum shape for the given requirements. 3D printing could also be combined with polishing processes, like stress polishing for the creation of free-forms supersmooth optics. In Section 1 is described the design phase of a printed warping harness for the manufacturing of the WFIRST off axis parabolas. In this case 3D printing is used to create innovative design for the warping harness able to generate a force distribution corresponding to astigmatism and coma deformations on the substrate. In Section 2, direct 3D printed substrates were investigated. This study project aimed to explore the surface quality of the printed samples after printing and with simple post processing, compare the different materials and their behaviour after printing. melanie.roulet@lam.fr In the first part of this paper we focus on the Wide Field Infrared Telescope (WFIRST) [START_REF] Spergel | Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report[END_REF] working in the infrared, in Figure 1(left). We focus on the development of a stress polishing method for the mirrors of the coronagraphic instrument. These mirrors, off axis parabolas, must have an high surface quality in order to provide the minimal amount of errors and then not degrade the final image sharpness. More specifically, mid and high spatial frequency ripples must be avoided on the surfaces, at the scale of the frequency were the instrument is seeking planets. For that purpose, we use the stress polishing method, which principle consists in applying a spherical polishing on a warped substrate, in order to imprint the shape of the warping applied onto the reflective surface. The idea is to create printed harness to imprint astigmatism and coma deformation. The use of 3D printing allows using complex shape structure and also lightweight structure for the harness to generate non symmetrical force distribution.

In a second part of this study, we focus on X-ray field and the Lynx design concept, in Figure 1(right). 5 X-ray needs extreme precision optics and optimum mass. [START_REF] O'dell | Toward large-area sub-arcsecond x-ray telescopes ii[END_REF] We use 3D printing to investigate an alternative method to create high surface quality lightweight mirrors. At the same time using topology optimisation we can reduce the weight and the bulkiness. In this way we compared the 3D printing process and the material and we studied the feasibility of lightweight structures manufactured by 3D printing. We investigated surface quality and roughness after the printing and after applying post-processing.

OFF AXIS PARABOLA

WFIRST coronagraph is the case study to develop a new fabrication process for off axis parabola. This instrument uses eight off axis parabola as relay optics. Additive manufacturing allows to think about innovative designs for the warping harness, with hollow structure or innovative shape. [START_REF] Spergel | Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report[END_REF]

Zernike decomposition

The decomposition of the surface shape have been done with Zernike polynomials. [START_REF] Noll | Zernike polynomials and atmospheric turbulence *[END_REF] The aberrations are sorted with an increasing radial order n and a decreasing azimuthal order m, as shown in Table

Stress polishing

Stress polishing is a polishing technique developed by the German astronomer Bernhard Schmidt in the 1930's. [START_REF] Lemaître | New procedure for making schmidt corrector plates[END_REF] The process consists in applying forces through a warping harness on the substrate during the polishing (cf Figure 2 (left)). Then, after the polishing process, (cf Figure 2 (right)) the warping harness is removed and the substrate comes back to its initial position. The warping harness must generate a warping function equal to the inverse of the required final shape. With this technique we can manufacture low cost -high performance optics, as long as the mechanical deformation is handled and precisely controlled, and within the breakage limit of the mirrors substrates. Finite element optimisation is crucial to reach the required quality.

Astigmatism mirror

To manufacture astigmatism shape we use the combination of forces drawn in Figure 2 (middle). Two pairs of equal and opposite forces are applied on a ring underneath the substrate. This combination gives a variation of the radius of curvature in two orthogonal directions which is defined by the Equation 1. The first mode of this equation gives a terms in ρ 2 cos(2θ) which corresponds to astigmatism 3 in Table 1.

9 ω(ρ, θ) ∝ ∞ m=2,6,10,... 1 m(m -1) + 2(1 + v) (1 -v)(m -1)m 2 - ρ 2 m(m + 1) ρ m cos(mθ) (1)

Simulation and Result

The substrate was simulated with zero expansion glass ceramic material, commonly known as Zerodur . This material has great capacities of polishing so we can obtain extremely smooth surface with residual roughness lower than 1 nm RMS. Additionally Zerodur is often used in space application due to its very low CTE, 0 +/-0.02 10-6/K. [START_REF] Schott | Thermal expansion of zerodur[END_REF] For modelling the combination of forces in Figure 2 (middle) we applied four forces on the ring: Two opposite forces in the negative vertical direction and perpendicularly two forces on the positive vertical direction in Figure 3 (left). From Patran/Nastran analysis, the node displacements in z-axis direction are shown in Figure 3 (right). The node displacement are symmetrical both on x-axis and y-axis. Comparing this simulation to the one which has been done in the PhD thesis by E.Hugot [START_REF] Hugot | Astronomical optics and elasticity -Active Optics for future Extremely Large Telescopes and their instrumentation[END_REF] we obtained as expected displacement in microns. The aberrations decomposition displayed pure astigmatism 3, Zernike numbers 5 and 6 in Table 1 with no residuals aberrations.

Off axis parabola

An off axis parabola (OAP) is an optical surface which can be defined by a combination of two Zernikes aberrations added to a parent spherical surface: astigmatism and coma. As this is the combination of a even aberration (astig 3) and an odd aberration (coma 3), the idea to generate this surface shape is to break the symmetry of the warping harness. Nevertheless breaking symmetry is not enough to manufacture pure astigmatism 3 and coma 3, the design also generates unwanted aberrations, like trefoil 5. In this design, five optimisation parameters are considered. We carried out a set of simulations in order to delete the trefoil 5 aberrations.

For the simulation we use the same conditions as previously, which includes Zerodur material, forces underneath the mirror's edges and centre fixed for boundary condition. We do the simulation with approximately 100,000 nodes and the meshing uses hexahedral element. We perform the simulation using the dimension of the blank test NASA model, with a diameter of 70mm. We converged on functional point where the trefoil 5 is cancelled out and the required design parameters are respected. Now the challenge is to control the ratio between astigmatism and coma to cover all the optical requirement for the WFIRST off axis parabola.

Discussion

With parametric study we find a functional point for the blank test mirror, where the warping harness generates pure astigmatism and coma. The challenging part in this study was to respect the required magnitude of astigmatism and coma and remove the residuals and also to respect the dimensions (diameter, thickness). After testing several design we were able to converge towards the solution which respects all the previous requirements.

The next step will be to carry out the same simulation and parametric study for the eight off axis parabolas respecting their own magnitudes and dimensions.

In the same time, we should verify our simulation data by the creation of the first prototype. Then the measurement of the fringes pattern with the interferometer will confirm or not our simulations and so our warping harness design. Then we could go further to the design of the next off axis parabola models.

LIGHTWEIGHT STRUCTURE

In this section, we were interested in the weight and cost constraints which apply on mirrors for space telescope mission as Lynx design concept, keeping the high precision of the structure. We are interested into 3D printing and lightweight structures to create new lightweight design for future space optics.

Mirrors for X-ray telescopes were our study case, the goal was to investigate the ability to fabricate a high quality mirror surface like the precision Chandra X-ray Observatory mirrors, [START_REF] Weisskopf | Chandra x-ray optics[END_REF] but to use 3D printing to develop a lightweighted mirror to reduce weight towards the next generation of high spatial resolution X-ray telescope, Lynx.

Prototype simulation

In this first section is developed the comparison of two lightweight designs. One traditional lightweight structure, the honeycomb design and one new, the arch design. Honeycomb design is a common pattern for lightweight structures, it can be manufactured with mechanical tools contrary to the arch design, which can only be manufactured by 3D printing, in Figure 5. We will compare the properties of a mechanical manufacturing lightweight design (honeycomb) with a 3D printed lightweight design (arches). Both designs were tested with finite element analysis. The prototypes are circular with a diameter of 40mm and 6mm thickness. We chose this configuration, far from the shape of X-ray mirror because polishing of circular flat surfaces is simpler than curved and rectangular surface. Both design had 65% mass reduction compare to a full cylinder of 6mm thickness. The honeycomb model had no base while the arch design had a base, that doesn't impact the result on the nodes displacement. A pressure of 3500Pa, representing the polishing pressure, was applied on the reflecting surface, the base was fixed and the material is aluminium. Figure 6 shows the nodes displacement of the reflecting surface in the z-axis. The maximum displacements are found on the edge where there were no stiff borders and a lack of pattern. In honeycomb design this maximum displacement was -23.8nm while for the arches design it is -15.7nm. Moreover with arch design there were less displacements between the arches whereas in the honeycomb design we can see deeper node displacements.

The arch design showed better surface quality after a polishing phase than the honeycomb. This configuration is stiffer.

Manufacturing

3D printing method

All the samples were manufactured with 3D printing at CA Models. There are different manufacturing methods. Below is presented the two methods used for the prototyping. Stereo-lithography (SLA) process is an additive manufacturing process, ultraviolet laser is focus on to a vat of photo-polymer resin (liquid) which is then solidified and forms a single layer of the desired 3D shape, in Figure 7 (left). This process is repeated until the shape is completed. It can create 3D object in plastic and ceramic-like materials. [START_REF] Stampfl | Additive Manufacturing Technologies[END_REF] Selective laser Sintering (SLS) and Metal laser Sintering (MLS) are different methods to SLA. This technology uses a laser to fuse and bond small grains of material, in Figure 7 (right). The laser traces the patterns of each cross section of the design onto a bed of powder. After one layer is built, the bed lowers and another layer is built on the top of the existing layers. This process continues until the 3D design is completed. It can print object in metals, polymer and composite material. [START_REF] Stampfl | Additive Manufacturing Technologies[END_REF] Glass filled nylon is a composite material. It is made by mixing polyamide powder and glass beads. The surface of the material is white and slightly porous. Compared to native polymer, glass filled improves mechanical properties of rigidity and strength by reduced the non-lubricated wear properties. This material is largely used in 3D printing with Selective Laser Sintering. [START_REF] Chung | Processing and properties of glass bead particulate-filled functionally graded nylon-11 composites produced by selective laser sintering[END_REF] Bluestone is a trade name for a ceramic-like material manufactured with stereo-lithography process. This material is not used in optics. It is a new composite material resulting of the development of 3D printing. Its Young's modulus is 7.6 GPa and this material shows exceptional stiffness. [START_REF] Camodels | Data sheet -Bluestone[END_REF] AlSi10Mg is aluminium alloy. AlSi10Mg is printing with MLS which is similar process than SLS. It offers good strength, hardness and dynamic properties. Its Young's modulus is around 70 GPa. This material has excellent machinability so we could apply polishing on its surface to reduce the roughness. 17

Material

Post processing

The initial goal to evaluate the surface quality of the samples immediately post-print and after some basic smoothing processes in order to quantify the surface quality prior to grinding and polishing. Samples were processed at different levels, the sanded sample was sanded by hand, the skim sample was sanded with finer grit and the blasted samples was bead blasted. Glass filled nylon is not strong enough to be post polished so there is only the raw sample. Table 2 summarises the samples and post-processing treatments. Checked boxes mean that the sample has been done. From the raw samples of each material we added a step of coating. The coating is added to provide a mechanically harder surface to polish to ensure good reflective properties. Bluestone and glass filled nylon were coated with pure nickel and the AlSi10Mg with nickel phosphor (NiP), as shown in Table2. Nickel phosphor is commonly used in X-ray optics manufacture due to its very good polishing properties. The coating process for nickel phosphor is made by a catalytic electroless process at high temperature and for that reason it was not applied to the glass filled nylon and Bluestone samples. Indeed their mechanical properties change when material reach high temperature. Only Bluestone and glass filled nylon the samples are coating with pure nickel, at room temperature, which has also good properties in polishing. Pure nickel and nickel phosphor have roughly the same density so both coatings have similar weight.

Material/Post polishing Raw Sanded Skim Blasted Pure Ni coating NiP coating AlSi10Mg X X X Bluestone X X X X X Glass filled Nylon X X

Surface quality

Only non destructive measurements have been performed on the samples. [START_REF] Atkins | Additive manufactured x-ray optics for astronomy[END_REF] We measured the surface profile, the roughness and waviness. A Taylor Hobson Form Talysurf Intra measured the height of the surface at given locations along a line, in Figure 9 (right). To sample the whole surface measurements were taken in 3 different directions, in Figure 9 (left), 0 • , 45 • and 90 • directions. The measurement was performed three times for each direction in order to have more accuracy in the surface profile. Figure 10 shows the average surface profile for AlSi10Mg, Bluestone and glass filled nylon raw sample in the 0 • direction. The amplitude of the surface profile varies depending of the materials. Glass filled nylon surface is a lot rougher than AlSi10Mg and Bluestone. Furthermore on these last one we observed waviness pattern, which depicts approximately 6 peaks. This is a consequence of their orientation within the 3D printer and the arches lightweight structure.

Surface profilometry

Fourier analysis

In order to remove the waviness pattern from previous data, Fourier analysis was applied. The Fourier decomposition is a way to represent a function as the sum of simple sine waves. It is used in signal treatment to isolate narrow band component of a compound waveform. The top right picture in Figure 11 is a zoom of the AlSi10Mg raw sample for low frequencies. There is two obvious peaks which depicts the waviness pattern explained in section 3.3.1. The larger bump were in the diagonal (45 • ) direction and in the vertical direction (0 • ) the bumps had slightly lower magnitude. On Bluestone spatial frequencies these patterns can be seen in all directions. The bumps magnitude and period were similar, which obviously leads to a diagonal pattern with larger bumps. As we expected from the surface profile measurement there were no evident pattern for the Glass filled nylon.

In order to only investigate the roughness we used the ISO standard 4288-1996 definition. [START_REF] Hobson | Exploring surface texture: A fundamental guide to the measurement of surface finish[END_REF] This standard gives for an anticipated roughness the necessary cut-off to highlight the roughness, in our case the cut off is between 0.8mm and 2.5mm. Considering these filters, roughness is below 1mm-2mm, waviness between 2mm-10mm and form is above 10mm. These are approximate values and they vary between measurement length and expected roughness.

Discussion

For all the samples and directions we calculate the Root Mean Square (RMS) and Peak to Valley (PV) values [START_REF] Hobson | Exploring surface texture: A fundamental guide to the measurement of surface finish[END_REF] to compare the surface quality of the samples. In Figure 12 is presented the RMS in function of PV for each sample, without filter, with 2.5mm filter and 0.8mm filter. We take the mean of RMS and PV of the three orientation to obtain only one value per sample. In blue there are all the Bluestone samples, in purple all the AlSi10Mg samples and in yellow the glass filled nylon sample. Then in the graph the post polishing process are also highlighted, the raw sample are squared, the sanded is triangular, the skim is diamond and the blasted are circular. Finally to show the impact of the filter there is a colour gradient, the lighter is the smaller filter and the darker is without filter.

As expected the raw samples without filter had the highest roughness. Particularly for glass filled nylon and AlSi10Mg, besides for these samples the filtering cut-off shows a lot of surface shape and waviness. From Figure 12 the gap between the non filtered and filtered data was huge, even between 2.5mm cut off and 0.8mm which attests the existence of a deep waviness pattern. While in Bluestone the first filter deleted the majority of the waviness pattern. Bluestone showed the best surface quality after printing without any post-processing. The AlSi10Mg post-processed sample showed very good surface quality, RMS under 4µm, the filtering showed there is still some waviness pattern after the processing but not to much because 2,5mm and 0.8mm cut were close. Blasted sample is 5 times better than raw in terms of PV and RMS. Finally, in comparison to all the samples AlSi10Mg blasted had the best surface in terms of roughness for very little effort in terms of post-processing.

For the post-process Bluestone samples the surface quality is better than the raw samples as expected. After applying the filter the waviness pattern is completely remove and all the data are located in a square between 20µm and 40µm of PV and 3µm-5µm of RMS so Bluestone shows very good result in term of roughness.

Topology optimisation

Topology optimisation is a mathematical method which optimises the design for a given set of loads, boundary conditions and constraints with the goal of maximising the performance of the system on pre-defined parameter, in our case the volume and so the weight. [START_REF] Bendse | Topology Optimization Theory, Methods, and Applications[END_REF] Figure 13. Optimum design given by COMSOL software after topology optimisation of the volume on a cube for 30% keeping mass (left) and 50% keeping mass (right)

Figure 13 shows the result of topology optimisation for a cube sandwiched between two plates. Aluminium was used in the simulation, the base was fixed and a uniform force was applied on the top. In Figure 13 (left) there is the result for 30% mass kept, that involves to have a large hole in the centre of the column and opening on the edges. For the model with 50% mass kept we obtained a small hole in the centre of the column but this design could not be printed in 3D. Indeed 3D printing can't print internal holes, the process needs to add support and so the support needs to be removed afterwards. With the model 30% as there were opening on the edges of the column the interior support could be removed.

This innovative technology will be used to design the next set of samples. We should take in account that all the solution from topology optimisation cannot be printed, so we should adapt the design to the requirements of 3D printing.

CONCLUSION AND PERSPECTIVES

In the first section of this paper, we demonstrated that we can manufacture off axis parabolas using stress polishing and 3D printing. We developed a new fabrication process simpler and faster than the previous one for space mirrors keeping the quality on the surface.

In the second part of this paper, surface quality of lightweight optics manufacture by 3D printing was investigated. Bluestone is desirable due to its smooth surface post-print and low density; however the high CTE would make it undesirable for environments with large temperature gradients. AlSi10Mg showed very good result after post processing.

The surface quality of of the polished coated samples was presented by Atkins et al in SPIE San Diego 2017. [START_REF] Atkins | Additive manufactured x-ray optics for astronomy[END_REF] All the polished samples are good enough for visible/optical wavelengths which is much better than required for infrared. Aluminium NiP would be classified as near high surface quality. They are not good enough for X-ray optic, so we will continue to develop and investigate the polishing treatment and the design. In the future we use topology optimisation computation to improve the design.

Figure 1 .

 1 Figure 1. WFIRST telescope (left) Lynx telescope (right) (credit: NASA)

  89

Figure 2 .

 2 Figure 2. Picture of grinding during polishing (left), Schematic of the applied force in case of astigmatism deformation (middle), Mirror on its deformation system (right) (credit: Hugot 2008 10 )

Figure 3 .

 3 Figure 3. CAD view of the mirror with the force positions (left), Top view of the displacement after the simulation [mm] (right)

Figure 4 .

 4 Figure 4. Top view of the displacement after the simulation [mm] (left), Zernikes decomposition in Matlab (right)

Figure 5 .

 5 Figure 5. Arch design CAD view (left) Honeycomb design CAD view (right)

Figure 6 .

 6 Figure 6. Comparison of the result from the simulations of Arch model (left) and honeycomb 4 model (right) in [mm]

Figure 7 .

 7 Figure 7. 3D printing method, Stereo-lithography (left) and Selective laser Sintering (right) (credit: proto3000, RP Canada)

Figure 8 .

 8 Figure 8. Prototypes picture, form left to right Glass filled nylon, Bluestone and AlSi10Mg

Figure 8

 8 Figure 8 introduces the different samples. Three materials have been used, from left to right: glass filled nylon, Bluestone and aluminium (AlSi10Mg).

Figure 9 .

 9 Figure 9. CAD view of the prototype design with the three direction measurements (left), Picture of the profilometry machine working on the AlSi10Mg raw sample (right)

Figure 10 .

 10 Figure 10. Average surface profile for raw samples on 0 • direction for AlSi10Mg (top), Bluestone (middle), Glass filled Nylon (bottom)

Figure 11 .

 11 Figure 11. Fourier analysis from the profilometry data. Spatial frequency for raw samples (left) in AlSi10Mg (top), Bluestone (middle) and Glassfilled nylon (bottom), Zoom on low frequencies for AlSi10Mg raw (top right) Picture of AlSi10Mg raw sample surface (low right)

Figure 12 .

 12 Figure 12. Summary of PV and RMS in microns for all the samples without filter, with 2.5 filter and with 0.8 filter

Table 1 .

 1 . Zernike Aberrations

	Number	Aberration	Expression	n m
	1	Piston	1	0 0
	2	Tilt x	ρ cos θ	1 1
	3	Tilt x	ρ sin θ	1 -1
	4	Spherical	2ρ 2 -1	2 0
	5	Astigmatism 3x	ρ 2 cos(2θ)	2 2
	6	Astigmatism 3y	ρ 2 sin(2θ)	2 -2
	7	Coma 3x	(3ρ 2 -2)ρ cos θ	3 1
	8	Coma 3y	(3ρ 2 -2)ρ sin θ	3 -1
	9	Trefoil 5x	ρ 3 cos(3θ)	3 3
	10	Trefoil 5y	ρ 3 sin(3θ)	3 -3
	11	Spherical 3	6ρ 4 -6ρ 2 + 1	4 0
	12	Astigmatism 5 x	(4ρ 2 -3)ρ 2 cos(2θ)	4 2
	13	Astigmatism 5 y	(4ρ 2 -3)ρ 2 sin(2θ)	4 -2
	14	Squad 7 x	ρ 4 cos(4θ)	4 4
	15	Squad 7 y	ρ 4 sin(4θ)	4 -4
	16	Coma 5 x	(10ρ 4 -12ρ 2 + 3)ρ cos θ 5 1
	17	Coma 5 y	(10ρ 4 -12ρ 2 + 3)ρ sin θ 5 -1

Table 2 .

 2 Summary of the samples

ACKNOWLEDGEMENTS

The authors would like to acknowledge the European commission for funding this work through the Program H2020-ERC-STG-2015 678777 of the European Research Council. In addition, part of this project has been funded by the UK Space Agency's National Space Technology Programme and C. Atkins acknowledges the following with respect to her research fellowship: this project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreement (GA) No. 665593 awarded to the Science and Technology Facilities Council.


Eduard R Muslimov

Emmanuel Hugot

Simona Lombardo

Melanie Roulet

Thibault Behaghel

Marc Ferrari

Wilfried Jahn

Advanced optical designs of curved detectors-based two-mirrors unobscured telescopes

Keywords: curved detector, unobscured telescope, aspheric mirrors, ghost analysis, design algorithm

In the present paper we consider a family of unobscured telescope designs with curved detectors. They are based on classical two-mirror schemes -Ritchey-Chretien, Gregorian and Couder telescopes. It is shown that all the designs provide nearly diffraction limited image quality in the visible domain for 0.4º×0.4º field of view with the f-number of 7. We also provide a brief ghost analysis and point on special features of the systems with curved detectors. Finally, the detector surface shape obtained in each case is analyzed and its' technological feasibility is demonstrated.

INTRODUCTION

Optical systems of unobscured telescopes have a number of obvious advantages like increase of the primary collecting area or exclusion of the diffraction on spiders. The simplest way to generate an unobscured design consists of the pupil shift or the field of view bias in an ordinary coaxial telescope design. Such a transformation for classical two-mirror designs was considered many times [START_REF] Bazhanov | Design of off-axial Gregory telescope design with freeform mirror corrector[END_REF][START_REF] Horton | A family of 2 mirror unobscured wide field telescope and collimator designs[END_REF] . Since the scheme symmetry in such an optical design is broken, some authors [START_REF] Horton | A family of 2 mirror unobscured wide field telescope and collimator designs[END_REF] propose to add more degrees of freedom by tilting the components in order to provide a high optical quality at high fnumber values.

On the other hand, it is known that one of the main factors limiting an off-axis use of the classical telescopes is the field curvature [START_REF] Bass | Handbook of optics: 2nd edition[END_REF] . However, the recent progress in curved detectors technology [START_REF] Ko | Ahemispherical electronic eye camera based on compressible siliconoptoelectronic[END_REF][START_REF] Tekaya | Hemi-spherical curved monolithic cooled and uncooled infrared focal planearrays for compact cameras[END_REF][START_REF] Dinyari | Curving monolithic silicon for nonplanar focal plane array applications[END_REF][START_REF] Guenter | Highly curved image sensors: a practical approach for improved opticalperformance[END_REF][START_REF] Chambion | Tunable curvature of large visible CMOS image sensors : Towards new optical functions and system miniaturization[END_REF][START_REF] Rim | The optical advantages of curved focal plane arrays[END_REF][START_REF] Dumas | Curved focal plane detector array for wide field cameras[END_REF] allows to compensate the field curvature by the detector shape. It means that the condition of the field curvature correction can be neglected when compensating the system's aberrations. It was shown by a number of authors [START_REF] Rim | The optical advantages of curved focal plane arrays[END_REF][START_REF] Dumas | Curved focal plane detector array for wide field cameras[END_REF] that using of a curved detector changes the optical design approach and makes it possible to reach a better image quality along with simplification of the optical design and increasing of the image illumination uniformity.

Thus the main goal of the present study is to demonstrate the prospective advantages of use of curved detectors in off-axis unobscured two-mirror telescopes. We consider three optical designs based on such well-known schemes as Ritchey-Chretien, Gregorian and Couder telescopes. In Section 2 we present the design algorithm applied for each of the schemes to produce an advances unobscured version. Section 3 presents analysis of the unobscured designs geometry and optical quality. In Section 4 the detector surface shape obtained in each case is analyzed. Section 5 contains the general conclusion on the study.

DESIGN PROCEDURE

For each of the considered two-mirror telescope schemes the following re-design procedure is used:

1. The initial optical scheme is scaled for the necessary focal length.

2. For the initial design the theoretical value of field curvature radius is calculated [START_REF] Wetherell | General Analysis of Aplanatic Cassegrain, Gregorian, and Schwarzschild Telescopes[END_REF] . In this particular case we assume that the radius corresponds to image surface of minimum wavefront error:

* eduard.muslimov@lam.fr; phone +33 4 91 05 69 18; lam.fr

𝑅 𝐼 ≈ -𝐷 𝑝 𝐹 𝑝 𝐹 𝑝 +𝜂 𝐹 𝑝 +𝐹-𝜂 (1)
Where Dp is the primary's diameter, Fp and F are the primary's and overall f-ratio values, respectively, and η is ratio of the back focal length to the primary's diameter. The value found according to (1) is substituted as the image surface curvature radius.

3. The entrance pupil is shifted by a distance allowing to exclude any central obscuration.

4. The entire system is numerically optimized with the standard tools implemented in Zemax software. The curvature radii (including the one of the detector surface), the conic constants and the distances are used as variables. The boundary conditions require maintenance of the focal distance and the principal geometry.

5. Tilt angle of each of the components is introduced and used as a variable parameter. The aperture off-axis shift is also set as a variable. The optimization is repeated. The additional boundary conditions on the marginal rays coordinates differences are used to avoid obscuration.

Below we present the designs derived from classical two-mirror systems by using of these procedure.

OPTICAL DESIGNS OVERVIEW

The design procedure described above was applied to three classical two-mirror telescope designs, namely Ritchey-Chretien, Gregorian and Couder schemes. For simplicity of comparison the principal optical parameters were the same in all the cases. They are presented in Table 1.

Table 1. Principal parameters of the telescopes optical designs.

Focal length, mm 1500 F-number 7 Field of view, º 0.4x0.4

Optical scheme type

Two-mirror unobscured off-axis telescope Detector surface Spherical

Optical schemes

All the optical schemes at different design steps are shown on Figure 1. Line a corresponds to the initial scheme (see p.1), line b corresponds to the off-axis design with curved detectors (see p.2-4), and line c represents the finalized design with tilted components and the real apertures. Note, that all the images have different scales. The actual dimensions for the finalized designs are: 216×330×438 mm 3 (X×Y×Z) for Ritchey-Chretien, 216×293×515 mm 3 for Gregorian,216×429×1715 mm 3 for Couder. Though in the Couder scheme we have to use an additional folding flat mirror to limit the telescope length. One can note that the components tilt angles in the Ritchey-Chretien type and Couder type designs are significant, while in the Gregorian telescope the components remain on the same axis even after the reoptimization.

The designed schemes, especially the Ritchey-Chretien based one, are interested in terms of ghost analysis. The result of non-sequential raytracing for this system is shown on Figure2. On one hand, the Ritchey-Chretien type design keeps such an advantage as possibility of simple internal baffling. If two simple screens are set next to the primary and secondary mirrors, the minimum angle of a ray passing through them is 13.8º and it falls outside of the detector sensitive area. The other beams from side sources are blocked. On the other hand, if we account for the detector surface reflectivity, a part light will be reflected back to the system and will reach the secondary (see the rays shown in yellow and violet on Figure 2). One of the reason why this undesired effect occurs is the influence of the detector shape on the reflected beam geometry. This parameter wasn't directly controlled during the optimization and it is still possible to read of this back reflection by means of the tilt angles adjustment. However, this simple analysis indicates an additional feature of the curved detectors. So an additional attention should be paid to the ghost analysis in a system with curved detectors.

A similar analysis was performed for the two other designs. In the Couder type system the light reflected back from the detector doesn't fall onto the secondary mirror due to the sign of the tilt angle. Also, in this case the folding mirror excludes possibility of a direct illumination of the detector. In addition, all the mirrors operate with relatively slow f-numbers, so there is enough space for internal baffles. In the Gregorian system the beam reflected back from the detector is partially blocked by the rear side of primary mirror, but a portion of it also can be re-reflected from the secondary. An internal baffling in the Gregorian type design is impossible as one can see from Figure 1.

Image quality

For estimation of the image quality standard spot diagrams are used. Since the optical schemes have no axial symmetry we consider a rectangular grid covering the square field of view. The reference value is the Airy disk radius, which is the same for all the presented designs and equal to 4.7 µm at wavelength of 550 nm.

The spot diagrams for Ritchey-Chretien type design are shown on Figure 3. The root-mean square (RMS) radii vary between 0.9 and 3.5 µm. The image quality is close to the diffraction limit for the entire field, though the image quality degrades rapidly towards the field corners. One also can note absence of a symmetry with respect to the XZ plane. In a similar way the spot diagrams for Gregorian type design are shown on Figure 4. The RMS radii are 0.1-3.4 µm. The spots burring at the field of view corners is even more notable, though the system remains diffraction limited. Since the aperture offset is smaller and the components tilt angles are negligible, the spot size distribution is more symmetrical. The spots radii distribution is more uniform, but the spots sizes exceed the Airy disk diameter because of uncompensated aberrations. So the telescope cannot be literally defined as a diffraction limited one, though it approaches the limit. The spot diagrams data is summarized in table 2. For comparison we provide also the values of spot sizes for a case of a flat detector. In this case the detector surface curvature was fixed at zero and the entire system was re-optimized. One can see that for the Ritchey-Chretien and Gregorian types the difference in the spot radii can be as big as an order of magnitude.

It emphasizes the importance of use of the curved detector in such a scheme. For the Couder type design the image surface steepness is much less, so the difference equals only to 13% in the worst case.

CURVED DETECTOR DESIGN

Since the focal length and the angular field of view is the same for all the considered designs, the detector dimensions are the same in all the cases and equal to 10.5×10.5 mm 2 . The sag map for each of the curved detectors is shown on Figure 6.

The key values defining the detector surface shape are also given in Table 3 below. Note that the radii values found after optimization are close to those computed analytically for the initial configurations. Such a spherical shape can be generated by mechanical bending of a back-thinned detector, attached on top of a thin metallic plate [START_REF] Hugot | Flexible focal plane arrays for VIS/NIR wide field instrumentation[END_REF][START_REF] Jahn | Innovative focal plane design for high resolution imaging and Earth observation: Freeform optics and curved sensors[END_REF] . This technology represents an extension of the active deformable mirrors technology, described in a number of sources [START_REF] Lemaître | Astronomical Optics and Elasticity Theory -ActiveOptics Methods[END_REF][START_REF] Laslandes | Mirror Actively Deformed and Regulated for Applications in Space: design and performance[END_REF] . This technology allows to bend a thin polished plate to the desired shape using a non-uniform load or a special distribution of the plate thickness. It was proven that with the same approach a commercial CMOS or CCD detector can be bent to the required radius of curvature and thus a fully functional curved detector can be manufactured.

Here we do not provide a detailed analysis of the calculated detectors shapes feasibility. Instead we make an approximate theoretical estimation of the maximum stress generated in the bending setup.

We assume that a thin silicon chip (a typical thickness is 100µm) is glued firmly on top of a circular metallic plate, so the chip almost doesn't affect the rigidity of the entire assembly and the stress and deformation distributions don't have discontinuities at the chip-to-plate interface. The plate diameter is equal to the detector's diagonal. Then we consider the simplest case of bending of the plate with supported edge by a uniform load. It may produce nearly spherical shape [START_REF] Timoshenko | Theory of plates and shells[END_REF] .

The maximum sag generated in such a bending setup is defined as

𝑤 𝑚𝑎𝑥 = (5+𝜈)𝑞𝑎 4 64(1+𝜈)𝐷 , ( 2
)
where ν is the Poisson ratio, q is the uniform load, a is the plate radius and D is the flexural rigidity defined as

𝐷 = 𝐸ℎ 3 12(1-𝜈 2 ) . ( 3
)
Here h is the plate thickness and E is the Young modulus.

The maximum stress is

𝜎 𝑚𝑎𝑥 = 3(3+𝜈)𝑞𝑎 2 8ℎ 2 . ( 4
)
Simply substituting the eq. ( 2) and ( 3) into (4) we can define the maximum stress for a given plate thickness and the required sag, i.e. to the required curvature. Let us assume that the bending facility is made of an Al alloy having ν=0.33 and E=69GPa. The results of computation with this simple analytical model for the detector curvature and linear field values taken from the optical designs are shown on Figure 7. The maximum stress values for a 1mm-plate are also given in Table 3. The maximum generated stress is much less than the tensile strength of silicon, which equals to 350 MPa. Obviously, this is an approximate estimation demonstrating that there is no risk of breaking the detector during its bending. A more precise modelling requires use of finite element analysis (FEA). Such a detailed study was carried out before for a small curved detector [START_REF] Jahn | Innovative focal plane design for high resolution imaging and Earth observation: Freeform optics and curved sensors[END_REF] .

CONCLUSIONS

In the present paper we demonstrated use of curved detectors in optical schemes of two-mirror unobscured telescopes. Each of the schemes is derived from a classical design (Ritchey-Chretien, Gregorian and Couder) by re-optimization with off-axis aperture, curved image plane and tilted components. It was shown that all the schemes provide nearly diffractive image quality over 0.4º×0.4º field with the f-number of 7. They also keep some advantages inherent for a specific geometry, like a possibility of internal baffling for the Ritchey-Chretien off-axis geometry. Meanwhile, it was found that a special attention should be paid to the ghosts' analysis if a curved detector is used.

We also provided a brief analysis of the detectors shapes and their feasibility. It is shown that the required curvature can be produced without generating any dangerous mechanical stress. We assume that such a simplified analytical estimation can be extremely useful in design of new optical schemes with curved detectors, because it can be used to define a boundary condition directly in the optical system optimization procedure. In some cases, it may be more convenient than use of precise calibration curves confirmed by FEA and experiments.

In general, the curved detector technology and the new optical designs, which becomes possible due to its use, are of interest for a number of application fields. However, the primary application of the curved detector should be optical devices for scientific research, especially the future astronomical instruments [START_REF] Thatte | The E-ELT first light spectrograph HARMONI: capabilities and modes[END_REF][START_REF] Muslimov | Fast, wide-field and distortion-free telescope with curved detectors for surveys at ultralow surface brightness[END_REF][START_REF] Szeto | Maunakea spectroscopic explorer design development from feasibility concept to baseline design[END_REF] , which require a perfect performance and operate in unusual modes.

Figure 1 .

 1 Figure 1. Generation of the unobscured two-mirror telescope designs with curved detectors: ainitial designs, boff-axis configurations with curved detectors, cfinalized designs.

Figure 2 .

 2 Figure 2. Non-sequential raytracing for the ghost analysis in the Ritchey-Chretien type scheme.

Figure 3 .

 3 Figure 3. Spot diagrams of the Ritchey-Chretien type telescope design.

Figure 4 .

 4 Figure 4. Spot diagrams of the Gregorian type telescope design.

Figure 5 .

 5 Figure 5. Spot diagrams of the Couder type telescope design.

Figure 6 .

 6 Figure 6. Sag maps of the curved detector surfaces: a -Ritchey-Chretien type design, b -Gregorian type design, c -Couder type design.

Figure 7 .

 7 Figure 7. Estimation of the maximum stress generated during the detector bending.

Table 2 .

 2 Spot diagrams data.

		Ritchey-Chretien	Gregorian	Couder
		On a curved surface
	Spot RMS radius in	0.9/3.5	0.1/3.4	1.7/2.9
	center/corner, µm
	Spot MAX radius in	1.5/5.4	0.2/4.8	5.3/7.7
	center/corner, µm
		On a plane
	Spot RMS radius in	10.1/7.6	17.1/6.6	1.9/3.0
	center/corner, µm
	Spot MAX radius in	14.5/14.3	22.7/12.3	6.0/7.0
	center/corner, µm

Table 3 .

 3 Curved detectors data

		Ritchey-Chretien	Gregorian	Couder
	Initial radius of curvature, mm	-125.6	59.5	-1358.9
	Optimized radius of curvature, mm	-95.8	65.6	-1314.3
	Sag, mm	0.29	0.43	0.02
	Maximum stress, MPa	1.77	2.62	0.13

ACKNOWLEDGEMENTS

The authors acknowledge the support from the European Research council through the H2020 -ERCSTG-2015 -678777 ICARUS program. This research was partially supported by the HARMONI instrument consortium. We also thank Christophe Gaschet, Bertrand Chambion and David Henry from Univ. Grenoble Alpes, CEA-LETI for their contribution to the curved detectors technology development.


Eduard Muslimov
email: eduard.muslimov@lam.fr

Emmanuel Hugot

Simona Lombardo

Melanie Roulet

Thibault Behaghel

Marc Ferrari

Wilfried Jahn

Christophe Gaschet

Bertrand Chambion

David Henry

Curved detectors for wide field imaging systems: impact on tolerance analysis

Keywords: curved detectors, image quality, tolerance analysis, wide field lens, off-axis telescope

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The curved detectors technology is a rapidly developing area, which in the nearest future can make a huge impact on performance of imaging optical systems and approaches to designs them. A number of functional devices, produced by different techniques were demonstrated by a number of research groups over the world [START_REF] Ko | Ahemispherical electronic eye camera based on compressible siliconoptoelectronic[END_REF][START_REF] Tekaya | Hemi-spherical curved monolithic cooled and uncooled infrared focal planearrays for compact cameras[END_REF][START_REF] Dinyari | Curving monolithic silicon for nonplanar focal plane array applications[END_REF][START_REF] Guenter | Highly curved image sensors: a practical approach for improved opticalperformance[END_REF][START_REF] Chambion | Tunable curvature of large visible CMOS image sensors : Towards new optical functions and system miniaturization[END_REF] .

The key advantages of use of the curved detectors are were demonstrated in a number of researches [START_REF] Rim | The optical advantages of curved focal plane arrays[END_REF][START_REF] Dumas | Curved focal plane detector array for wide field cameras[END_REF] . Firstly, when there is no need to correct the field curvature, a better image quality can be achieved. Secondly, the required quality can be reached in a simpler optical system. Thirdly, due to change of the angle of incidence to the focal surface, the image illumination can be substantially increased.

Obviously, the use of a curved detector should also affect the tolerances analysis for an optical system. Better image quality and/or simplification of the optical system design in the nominal state implies wider margins for the manufacturing and assembling errors. On the other hand, the optical system should become more sensitive to the detector parameters like the radius of curvature and its' linear and angular position. Thus, the main goal of this study is to quantify the effect, which introduction of a curved detector makes on tolerance analysis, and exemplify it by certain optical designs.

To provide a visual demonstration of the changes in performance and tolerances we consider two completely different optical systemsan axisymmetric lens and an off-axis telescope. For each of the systems we provide the optical quality and tolerances estimations. Then we perform a re-optimization with a curved detector and repeat the analyses to show the obtained gain.

The paper is organized as follows: in Section 2 we describe the demonstrative optical designs, Section 3 provides the image quality estimations for each case, in Section 4the tolerance analysis results are described and the conclusions on the study are given in section 5.

OPTICAL DESIGNS DESCRIPTIONS

We consider two completely different optical systems in order to make our study more general. However, we make an accent on fast optics with relatively wide field of view, which definitely benefits more from use of a curved detector.

The first demonstrative example is a wide-field axisymmetric lens system; the second one is a fast off-axis unobscured telescope. The main optical parameters of the systems are given in Table 1 and a more detailed description is provided below.

Table 1. The main parameters of the demonstrative optical designs.

Axisymmetric lens

Off-axis telescope Focal length, mm 25 250 F-number 3.5 2

Field of view, º 72 4x4

Axisymmetric lens system

The lens under consideration is based on the design presented in apatent [START_REF] Ikari | Photographic lens system[END_REF] . It is an all-spherical design consisting of 7 lens with 1 cemented pair and an internal aperture stop. As one can see, the lens operates with a wide field of view (FoV) and a moderate aperture. We consider 3 configurations of this design:

-Configuration A: the initial design with 7 lenses and a flat detector, -Configuration B: the same design re-optimized with a curved detector. During the re-optimization all the radii and thicknesses were set variable, the focal length was limited, the total length and the illumination on the FoV edge were limited. Also necessary boundaries on the center and edge thicknesses were applied.

-Configuration C: a new design obtained from the Configuration C after removing of 2 lenses and iterative optimization. The optimization conditions were the same as in the case B, but the target illumination was increased.

General views of the resultant optical designs are shown on Fig. 1.

Figure 1. General view of the axisymmetric lens systems (f'=25 mm, F/3.5, 72º): Ainitial configuration with 7 lenses and a flat detector; Bre-optimized configuration with 7 lenses and a curved detector; Cre-designed configuration with 5 lenses and a curved detector.

Off-axis telescope system

The off-axis telescope design represents a three-mirror anastigmate (TMA) [START_REF] Bass | Handbook of optics: 2nd edition[END_REF] . It is an all-reflective system consisting of three aspherical mirrors with common axis and decentered apertures. The primary and secondary mirrors are conicoids, while the tertiary is described by the even asphere-type equation with 3 additional coefficients. Since there is no possibility exclude any optical surfaces or elements, we consider only 2 configuration, which definition is analogous to that given in p.2.1:

-Configuration A: the initial design with a flat detector, -Configuration B: the same design re-optimized with a curved detector. In contrast with the previous example it is necessary to set limits on the marginal rays height in order to avoid obscuration or elements overlapping.

The general view of the optical system is shown on Fig. 2.

IMAGE QUALITY ANALYSIS

The image quality is assessed in a similar way for all the configurations of the both systems. We use spot diagrams and module transfer functions (MTF) to show the obtained in image quality.

Axisymmetric lens system

The spot diagrams for all the configurations of the demonstrative wide-field lens design are shown on Fig. On top of the visual advantage in terms of the optical quality, the curved detector-based designs provide higher image illumination (see Fig. 5). Configuration B has a higher illumination value than the A configuration for the entire FoV except of its edge, while configuration C provides gain of a factor up to 1.7.

Off-axis mirror system

Similarly, the telescopes spot diagrams are given on Fig. 6. The RMS radii are 3.4-10.1 and 2.6-6.2 microns for the configurations A and B, respectively. Figure 6. Spot diagrams: Ainitial configuration with a flat detector; Bre-optimized configuration with a curved detector.

The MTF plots shows even larger difference in the image quality between the two designs (see Fig. 7). Finally, we should mention that the difference in image illumination is negligible because the angular FoV is relatively small.

TOLERANCE ANALYSIS

The tolerance analysis was performed in a similar way for all the designs. We make an assumption about the detector pixel size and require the spot RMS radii to be less than this value. We used only the standard analysis tools implemented in Zemax software. In order to account for simultaneous change of the design parameters we use the probability to achieve the target spot size found with the Monte-Carlo analysis as the main criterion. The aim is to guarantee the probability of 80% or more; if appears to be impossible with reasonable tolerances values, we tried to maximize the probability. For each of the systems the analysis was performed starting from the configuration providing the best quality. Then the tolerances were picked up to the next configuration and tightened or widened in an iterative process. In all the cases just a simple focus compensator is used and the reference wavelength is 532 nm. We must also note that truncation was applied when defining some of the tolerances.

Axisymmetric lens system

For the lens design we assume that the detector 10 has 24.6x24.6 mm 2 sensitive area with the pixel size of 24x24 µm 2 . The demonstrative lens system contains 13 refractive surfaces, a physical diaphragm the detector, so it is inconvenient to present the full tolerance results here. Instead we provide some analytics visualizing the obtained gain in the tolerances. Table 1 shows the maximum and minimum values of the tolerances describing the shape, refractive properties and positioning of all the optical surfaces. It's clear that the minimum values increase in the configurations with curved detectors. The gain can be as much as factor of 4 for the 7-lens design and factor of 40 for the 5-lens design. The drawback of these change in tolerances is an expectable increase of requirements for the detectors shape and position. However, if we consider the quantitative change of these values (see Table 2) we find that: the tolerance on the curvature radius is relatively wide; the linear position should be more precise but the accuracy is achievable; the tolerances on tip and tilt becomes even wider because of the higher image quality. The only parameter which can become challenging is the local irregularity of the detector surface. The obtained values are definitely measurable, but their manufacturability strongly depends on the technology used. It should be emphasized that the obtained change in the tolerances is related not only to the smallest values, but for distribution of all the tolerances on design parameters. To visualize this change we use stacked histograms (see Fig. 8- 10). Note that the values are arranged into three groups, which can be referred to as "high precision", "moderate precision" and "standard precision". Also, some of the parameters are scaled to fit the same range. In general, one can note that with the increase of the image quality obtained due to introduction of the curved detectors all the tolerances gradually move from the "high/moderate precision" zone to the "standard precision" zone.

Off-axis mirror system

For the off-axis telescope we assume that the detector [START_REF]Prime 95B 25mm Scientific CMOS Camera Datasheet[END_REF] has size of 17.7x17.7 mm 2 with a 11x11 µm 2 pixel. In this case we can provide the full description of the tolerancing results, since there are only three optical surfaces. They are presented in Table 4. As one can note, a trade-off of the tolerances takes place here. The largest values corresponding to the aspheres vertex radii becomes smaller, while the smallest one like the surface irregularity and angular misplacements become larger. The main point is that the change of the radii tolerance doesn't affect the manufacturing and testing complexity, while the change of the tilt and surface error margins make it much simpler. In a similar way we provide the tolerances on the detector parameters (see Table 5). One can see that the values become slightly smaller, but nor of them approach the limits of manufacturability or testability. Finally, the Monte-Carlo results are given on Fig. 12. The conventions are the same as those used on Fig. 11. Both of the designs meet our requirements for the probability and both of the curves almost converge around the pixel size mark. However, the configuration B in general provides a higher probability to obtain the desired optical quality.

Figure 12. Results of the tolerances Monte-Carlo analysis for the off-axis telescope (the dotted lines mark the corresponding limits).

CONCLUSIONS

In the present paper we explored the influence of introduction of curved detectors into wide-field imaging optical systems. The expected effects are substantial increase of the image quality which allows to widen the margins of the manufacturing precision requirements; and tightening of the tolerances on the shape and position of the detector itself. However, we clearly shown on examples of two different optical designs that the first effect is superior over the second one. While the requirements related to the detector remain feasible, the required level of precision for the optical elements can be decreased by an order of magnitude. Therefore, on top of the gain in the optical quality and image illumination the designs with curved focal planes have such an advantage as better manufacturability and simplicity of assembling. In our opinion, this last advantage is often missed in discussions of the curved detectors prospects. Particularly, this feature can change the estimative cost of building of a new optical system with a curved detector in comparison with a one using a standard flat detector. In general, we believe that development of the curved detectors technology and the corresponding optical design techniques can be useful for the future astronomical instruments like E-ELT HARMONI [START_REF] Thatte | The E-ELT first light spectrograph HARMONI: capabilities and modes[END_REF] , MSE [START_REF] Szeto | Maunakea spectroscopic explorer design development from feasibility concept to baseline design[END_REF] or Hector [START_REF] Saunders | Very fast transmissive spectrograph designs for highly multiplexed fiber spectroscopy[END_REF] .

One of the factors of risk, which was demonstrated in this study is the relatively high requirement to the detector's shape local errors occurring in some cases. The possibility to measure and control this value should be a subject of a separate research.

Figure 2 .

 2 Figure 2. General view of the off-axis telescope (f'=250 mm, F/2, 4ºx4º).

3 .

 3 The rootmean square (RMS) radii of the spots are 9.6-33.6, 10.4-18.7 and 9.1-18.6 microns for the configurations A, B and C, respectively. One can see that use of the curved detector allows to obtain a notable increase of image quality at the FoV edge. Moreover, decrease of the optical components number almost doesn't change the geometrical aberrations.

Figure 3 .

 3 Figure 3. Spot diagrams (the colors correspond to wavelengths of 486, 587 and 656nm): Ainitial configuration with 7 lenses and a flat detector; Bre-optimized configuration with 7 lenses and a curved detector; Cre-designed configuration with 5 lenses and a curved detector.

Figure 4 .

 4 Figure 4. Module transfer functions (the colors correspond to field positions): Ainitial configuration with 7 lenses and a flat detector; Bre-optimized configuration with 7 lenses and a curved detector; Cre-designed configuration with 5 lenses and a curved detector.

Figure 5 .

 5 Figure 5. Relative illumination on the image surface: Ainitial configuration with 7 lenses and a flat detector; Breoptimized configuration with 7 lenses and a curved detector; Cre-designed configuration with 5 lenses and a curved detector.

Figure 7 .

 7 Figure 7. Module transfer functions (the colors correspond to field positions): Ainitial configuration with a flat detector; Bre-optimized configuration with a curved detector.

Figure 8 .

 8 Figure 8. Tolerances overview for configuration A (initial configuration with 7 lenses and a flat detector).

Figure 9 .

 9 Figure 9. Tolerances overview for configuration B (re-optimized configuration with 7 lenses and a curved detector).

Figure 10 .

 10 Figure 10. Tolerances overview for configuration C (re-designed configuration with 5 lenses and a curved detector).Finally, Fig.11shows the Monte-Carlo analysis results. The grey line corresponds to the pixel size, the dotted vertical lines correspond to the nominal spot size increased by 5%. As one can see, the configuration A provides only 20% probability even with the strict manufacturing requirements. The configurations B and C are close to each other and completely fulfill our requirements to the probability.

Figure 11 .

 11 Figure 11. Results of the tolerances Monte-Carlo analysis for the axisymmetric lens (the dotted lines mark the corresponding limits).

Table 2 .

 2 Extremal values of the tolerances for the axisymmetric lens.

		Configuration A	Configuration B	Configuration C
		Min	Max	Min	Max	Min	Max
	Radius, mm	0.005	0.1	0.02	0.5	0.05	0.5
	Thickness, mm	0.005	0.1	0.015	0.1	0.02	0.2
	Element decenter, mm	0.02	0.1	0.02	0.1	0.05	0.1
	Element tilt, º	0.02	0.1	0.02	0.1	0.05	0.1
	Surface decenter, mm	0.02	0.1	0.02	0.1	0.05	0.1
	Surface tilt, º	0.02	0.1	0.05	0.1	0.05	0.1
	Surface irredularity, µm	0.03	0.25	0.06	0.25	0.02	0.22
	Refraction index	0.0001	0.0001	0.0001	0.0001	0.004	0.004
	Abbe number	0.05	1	0.1	2	1	2

Table 3 .

 3 Detector parameters tolerances for the axisymmetric lens.

		Configuration A Configuration B Configuration C
	Radius of curvature, mm	∞	61.5	62.0
	Radius tol., mm	-	1.0	1.0
	Irregulerity tol., mm	0.01	0.001	0.004
	Angular position tol., º	0.05	0.1	0.2
	Linear position tol., mm	1	0.1	0.2

Table 4 .

 4 Values of the tolerances for the off-axis telescope.

		Radius, mm	Surface irreg., µm Distance, mm	Decenter, mm	Tip/Tilt, º
			Configuration A
	Primary	1.0	0.1	0.2	0.2	0.02
	Secondary	1.0	0.007	0.2	0.2	0.05
	Tertiary	0.6	0.1	0.2	0.1	0.01
			Configuration B
	Primary	1.0	0.1	0.2	0.2	0.04
	Secondary	0.5	0.05	0.2	0.2	0.05
	Tertiary	0.2	0.1	0.2	0.1	0.03

Table 5 .

 5 Detector parameters tolerances for the off-axis telescope.

		Configuration A Configuration B
	Radius of curvature, mm	∞	709.4
	Radius tol., mm	-	1.0
	Irregulerity tol., mm	0.1	0.1
	Angular position tol., º	0.2	0.1
	Linear position tol., mm	2	0.5

ACKNOWLEDGEMENTS

The authors acknowledge the support of the European Research council through the H2020 -ERC-STG-2015 -678777 ICARUS program. This research was partially supported by the HARMONI instrument consortium. We also would like to thank Pierre Baron from Altechna Company for fruitful discussions, which helped us to state the aim of this study.


Eduard Muslimov
email: eduard.muslimov@lam.fr

Emmanuel Hugot

Simona Lombardo

Melanie Roulet

Marc Ferrari

Freeform optics complexity estimation: comparison of methods

Keywords: freeform optics, optical design, manufacturability, interferometric testing, asphericity gradient

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Imaging freeform optics is a rapidly emerging field, which allows to build optical systems with previously unachievable performance. Use of the freeform surfaces allows to achieve a better aberrations correction, and therefore to increase the resolution and/or create optical systems with a smaller f-number, wider field of view and smaller dimensions. However, use of the freeform optics requires a certain revision of the optical design process. Particularly, it is necessary to estimate and control the freeform surface complexity during the design and optimization in order to obtain a manufacturable and cost-effective solution. The complexity criteria, which were previously introduced for rotationally-symmetric aspheres based on the conic sections cannot be directly applied to freeforms [START_REF] Jeu | Criterion to appreciate difficulties of aspherical polishing[END_REF][START_REF] Foreman | Simple numerical measure of the manufacturability of aspheric optical surfaces[END_REF] .

The common way to demonstrate the freeform complexity is deviation from the best-fit sphere. This method is simple, but it can lead to misevaluation of manufacturability, because many of the freeforms manufacturers do not use spherical blanks, but grind directly to the freeform shape [START_REF] Van Ligten | Diamond Grinding of Aspheric Surfaces on a CNC 4-Axis Machining Centre[END_REF] . In addition, there are different ways to quantify this deviation from the basic geometry. The results may vary depending on the assumptions about the clear aperture shape and size, as well as on usage of the root-mean square (RMS), peak-to-valley (PTV) or integral values.

Another approach to estimation of the surface complexity is derived from the capability to measure the exact shape of the surface. The most of the manufacturers emphasize that measurement is the key point in the aspheres and freeforms fabrication chain. In this case the complexity is assessed by the surface slope steepness [START_REF] Kumler | Designing and specifying aspheres for manufacturability[END_REF][START_REF] Ma | Applying slope-constrained Q-type aspheres to develop higher performance lenses[END_REF][START_REF] Yuan | Applying slope constrained Qbfs aspheres for asphericity redistribution in the design of infrared transmission spheres[END_REF] . This approach is related to the fact that the majority of optical testing techniques rely on interferometry, and therefore the slope is often represented as the fringes frequency. In contrast with the first approach this is directly related to the testing process but requires additional computations.

Thus the main goal of the present study is to compare different methods of the surface complexity estimation in the application to freeform surfaces and demonstrate a correlation between them, the advantages and shortcomings for each of them.

In order to cover different cases, we consider two different optical designs. The first one is a Korsch-type unobscured telescope scheme with three mirrors [START_REF] Bass | Handbook of optics: 2nd edition[END_REF] . As we show below, the design re-optimization with a freeform secondary leads to a solution with a slow freeform mirror, which has a rectangular aperture. The second example design is a two-mirror unobscured telescope based on Schwarzschild geometry [START_REF] Gross | Handbook of Optical Systems[END_REF] . In turn, this design uses a freeform with large deviations and a nearly circular aperture. A more detailed description of the designs is given in Section 2; Section 3 demonstrates the gain in the image quality achieved with the freeforms and the Section 4 provides the surface shape analysis. The general conclusions on the study are given in Section 5.

SAMPLE OPTICAL DESIGNS

For the demonstrative purposes we consider the two unobscured reflective telescope designs. In both of the cases we use a known design based on off-axis aspheres as the starting point. The main optical parameters for the initial and resultant designs are summarized in Table 1.

Korsch telescope

The Korsch-type design has a moderate f-number and a small field of view (FoV). The field is rectangular and has the aspect ratio of 8. Such a telescope could be used to feed a spectrograph. The general view of the optical design is presented on Fig. 1. The freeform surface is introduced on the second mirror, while the aperture stop is on the primary mirror. The freeforms described by the standard Zernike sag polynomials 9 . 7 modes, namely the defocus, the 3 rd order astigmatism coma and trefoil, the primary spherical and the 5 th order astigmatism are used. The entire system is reoptimized, but its principal geometry is maintained.

Schwarzschild telescope

The second sample design is also an unobscured telescope. It has a wide field of view and intended for imaging. The initial design is an off-axis part of the Schwarzschild scheme with two aspheric mirrors having approximately equal vertex radii. The first mirror is a conicoid and the second one is an even asphere with 3 additional coefficients. The aperture stop is on the secondary mirror. In contrast with the previous case the design was subject to a considerable revision. Firstly, each of the off-axis aspheres were turned into a surface, which has the vertex coinciding with the aperture center. Secondly, the a freeform was introduced on the secondary mirror. The shape is described by the standard Zernike sag polynomials as well. 11 modes, including all the YZ-symmetric terms from Z3 to Z22 are used. Finally, since the application of freeform allows to increase the image quality dramatically it was decided to change the f-number from 6.7 to 3. The entire optical system was also re-optimized with these new conditions and maintenance of the principal geometry. The general view of obtained design is shown on Fig. 2.

Figure 2. General view of the Schwarzschild-type telescope design (f'=500 mm, F/3, 20ºx8º).

IMAGE QUALITY ANALYSIS

Below we investigate the image quality parameters of all the optical design configurations mentioned before. It is evident that use of the freeform surfaces allows to achieve a substantial gain in the image quality.

Korsch telescope

The spot diagrams for the Korsch-type telescope designs are given on Fig. 3. For the initial configuration the spot radius RMS at 550 nm varies between 2.7 and 3.7 µm. For the optical system with a freeform the corresponding values are 0.85-0.92µm. One also can clearly see that in the second case the spot size is smaller than the Airy disk diameter, which is equal to 5.2 µm, i.e. the telescope becomes a diffraction-limited system. For these reason we do not provide other estimations of the image quality for this telescope.

Schwarzschild telescope

In a similar way we provide the spot diagrams for the second, Schwarzschild-type, telescope (see Fig. 4). The spot RMS radii for the initial configuration are 13.0-35.2 µm, while for the design with a freeform they are 10.3-13.6µm. So the gain in the spot size for the FoV corner is as big as factor of 2.6. One also should keep in mind the decrease of the Fnumber from 6.7 to 3. This telescope design provides relatively high image quality over the extended FoV. In order to visualize this quality and to facilitate comparison with other similar designs [START_REF] Agócs | Freeform mirror based optical systems for FAME[END_REF] we provide the field map (see Fig. 5). One can note that the wavefront aberrations remain relatively small and uniform for the entire field except of its corners.

Figure 5. RMS wavefront field map for the Schwarzschild-type design (the units on colorbar are waves).

FREEFORM SURFACES ANALYSIS

The freeform complexity analysis was performed in a similar way for all the designs and configurations. The actual surface was fitted with a spherical or toroidal surface. The latter shape can be useful if the freeform shape is dominated by the astigmatic term. It was supposed that the radii of curvature and all the coordinates of the center of fitting surface were free variables. The RMS deviation between the actual surface and the fitting one was minimized using a standard simplex-method [START_REF] Lagarias | Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions[END_REF] over the actual clear aperture. Then the residual deviation is estimated in different ways. The RMS and PTV sag deviations are calculated. Also the integral of the deviation, i.e. the volume of the deviation is computed. Finally, the slope of the deviation is calculated and converted into frequency of interference fringes. We assume that the reference wavefront matches the best fit surface so the fringes frequency is related to the residuals instead of the entire sag. The reference wavelength for the fringes frequency computation is 532 nm in all the cases.

Slow freeform in the Korsch design

Since the freeform mirror is out of the aperture stop position and the FoV is rectangular, the actual clear aperture of the surface is rectangular as well. The best fit sphere (BFS) has the radius of -383.063 mm and decentered by (0, 0.161, 0.004) in mm. The residual sag deviation is shown on Fig. 6A. One can note that the deviation is relatively small with the PTV value of only 11.6 µm. Fig. 6B demonstrates the vectorial field of the deviation gradient. The same computations were made for the case of fitting with a toroid.

For comparison we provide the results of analysis for the initial off-axis asphere, which was performed in the same way. All the data are summarized in table 2. Considering these data one can note a few points. Firstly, all the estimates show that the mirrors are relatively easy in fabrication. Secondly, comparing with a best fit tor is inapplicable for an ordinary asphere, but it can be useful for some aspheres. The third point is related to the relative change of the estimate. For instance, the RMS sag deviation from the BFS is the same for the ordinary asphere and the freeform. At the same time, the change of the integral deviation is 26% and that of the fringes frequency is 11%. In addition, the integral deviation is proportional to the amount of removed material and the fringes frequency is the parameter defining the interferometric testing possibility, while the RMS deviation does not directly correspond to any technological limitation.

So, for slow aspheres and freeforms computation of the RMS sag deviation is not a preferable way to estimate the surface complexity, because of its low sensitivity to the shape changes and absence of a direct connection with the fabrication or testing limitations.

Steep freeform in the Schwarzschild design

In contrast with the previous case, the freeform surface position coincides with the pupil, so the clear aperture is circular. So the computation was performed over this circular shape and with a radial grid. The rest of the procedure was similar to the one used for the previous design.

The BFS has radius of -1410.1 mm and its center coordinates are (0, 0.0004, 0) mm. The residual sag deviation is shown on Fig. 7 A. The PTV value reaches 185.8 µm. Typically, such a freeform is considered as a complex one. The deviation gradient map is shown on Fig. 7B. The rest of the complexity estimates are given in Table 3. It should be noted that for the BFS computations with the decentered ordinary asphere an additional tilt term was introduced.

One can see from the table that due to the decenter, which is large big as 634.5mm, the ordinary asphere has a huge asphericity. So regardless of the complexity estimation approach the aspherical secondary mirror in the starting point design appears to be extremely complex for fabrication. However, the results strongly depend on the reference surface. In contrast, the freeform can be considered as manufacturable surface, though the final decision depends on the material properties and the field of view of the interferometer. However, we should emphasize the sensitivity of the integral deviation value to the reference shape change.

CONCLUSIONS

In the present paper we provided two demonstrative optical design of unobscured telescopes with freeform mirrors. On the examples of these designs the possible gain in the imaging performance achievable with use of freeform surfaces is clearly shown.

Analyzing the shapes of freeforms and initial aspheres we compared different approaches for the surface complexity estimation. In general, we can conclude that all the presented freeforms are manufacurable with the currently available technologies.

We can note the following points regarding the complexity estimation, which can be useful for freeforms design practice:

1. Computation of the RMS sag deviation is not a preferable way to estimate the surface complexity.

2. It was shown before [START_REF] Ma | Applying slope-constrained Q-type aspheres to develop higher performance lenses[END_REF][START_REF] Yuan | Applying slope constrained Qbfs aspheres for asphericity redistribution in the design of infrared transmission spheres[END_REF] that the asphericity slope is a representative way to estimate the surface manufacturability, which is directly connected with the optical testing procedure. The presented analysis can confirm this point. However, the slope estimation requires either computation of the residual sag gradient, or using of specific surface equation 3. The sag deviation integral value is very sensitive to any changes of the actual and reference surfaces shapes and can be easily calculated.

4. For the cases when the design is driven by the astigmatism correction condition it can useful to use a toroid as a reference surface.

To summarize, one recommended way to estimate the surface complexity when comparing different designs or setting boundary conditions is using of the surface slope measured in the fringes frequency. But control over the surface slope may require new software tools or even new polynomial sets for freeforms design.Another alternative is to use then integral of sag deviation. It is notable for its sensitivity to the shape changes. Also computation of the value is easy even with the standard surface types and software tools. Finally, the value is related to the amount of removed material.

These recommendations can be of a special interest for the future high-performance instruments used in the fields of scientific research like astrophysics. For example, it can be used for the E-ELT HARMONI [START_REF] Thatte | The E-ELT first light spectrograph HARMONI: capabilities and modes[END_REF][START_REF] Muslimov | Design of pre-optics for laser guide star wavefront sensor for the ELT[END_REF] instrument and its adaptive optics system.

Figure 1 .

 1 Figure 1. General view of the Korsch-type telescope design (f'=500 mm, F/4, 0.8º x0.1º).

Figure 3 .

 3 Figure 3. Spot diagrams of the Korsch-type telescopes (the black circles show the Airy disk diameter; the reference wavelength is 550 nm): Ainitial configuration with off-axis aspheres; B -the resultant freeform-based design-optimized.

Figure 4 .

 4 Figure 4. Spot diagrams of the Schwarzschild-type telescopes (the reference wavelength is 550 nm): Athe initial configuration with off-axis aspheres at F/6.7; Bthe resultant freeform-based design at F/3.

Figure 6 .

 6 Figure 6. Residuals analysis for the freeform secondary in the Korsch-type design: Aresidual sag (the units are microns), Bslope of the residuals.

Figure 7 .

 7 Figure 7. Residuals analysis for the freeform secondary in the Schwarzschild-type design: Aresidual sag (the units are microns), Bslope of the residuals.

Table 1 .

 1 The main parameters of the demonstrative optical designs.

		Korsch telescope	Schwarzschild telescope
		Initial	Re-designed	Initial	Re-designed
	Surfaces	3 off-axis conic-based	2 off-axis conic-based	2 off-axis aspheres: 2 nd	Centered aspherical
	configuration	aspheres	aspheres and a freeform	order primary and 8 th	primary and centered
			secondary	order secondary	freeform secondary
	Focal length, mm	500	500	500	500
	F-number	4	4	6.7	3
	Field of view, º	0.8x0.1	0.8x0.1	20x8	20x8

Table 2 .

 2 Analysis of the Korsch telescope secondary mirror asphericity.

			Initial design,		Resultant design,
			off-axis asphere		freeform
	Fitting surface	Sphere	Toroid	Sphere	Toroid
	RMS deviation, µm	0.4	0.9	0.4	0.5
	PTV deviation, µm	10.1	21.2	11.6	11.1
	Integral	deviation,	1.04	2.28	1.31	0.94
	mm 3
	Slope		0.0010	0.0016	0.0012	0.0011
	Fringes frequency at	3.94	5.96	4.38	4.24
	532 nm, mm -1

Table 3 .

 3 Analysis of the Schwarzschild telescope secondary mirror asphericity.

			Initial design,		Resultant design,
			off-axis asphere	freeform
	Fitting surface	Sphere	Toroid	Sphere	Toroid
	RMS deviation, µm	141.3	108.2	15.2	23.6
	PTV deviation, µm	5.373•10 3	1.013•10 3	185.8	236.7
	Integral	deviation,	332•10 3	112•10 3	5.3•10 3	249•10 3
	mm 3
	Slope		0.5465	0.1030	0.0221	0.0249
	Fringes frequency at	2.05*10 3	387.2	82.9	93.5
	532 nm, mm -1

ACKNOWLEDGEMENTS

The authors acknowledge the support of the European Research council through the H2020 -ERC-STG-2015 -678777 ICARUS program. This research was partially supported by the HARMONI instrument consortium.