Active Optics in Astronomy: Freeform Mirror for the MESSIER Telescope Proposal

Mathematical and Computational Applications

Article Active Optics in Astronomy: Freeform Mirror for the MESSIER Telescope Proposal

Gerard Rene Lemaitre *, Pascal Vola * and Eduard Muslimov * Laboratoire d’Astrophysique Marseille (LAM), Aix Marseille Université (AMU) and CNRS, 38 rue Frédéric Joliot-Curie, 13388 Marseille CX 13, France * Correspondence: [email protected] (G.R.L.); [email protected] (P.V.); [email protected] (E.M.)

Received: 28 November 2018; Accepted: 20 December 2018; Published: 27 December 2018

Abstract: Active optics techniques in astronomy provide high imaging quality. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators. The aspheric mirror is obtained from a single uniform load that acts over the surface of a closed-form substrate whilst under axial reaction to its elliptical perimeter ring during spherical polishing. MESSIER space proposal is a wide-field low-central-obstruction folded-two-mirror-anastigmat or here called briefly three-mirror-anastigmat (TMA) telescope. The optical design is a folded reflective Schmidt. Basic telescope features are 36 cm aperture, f /2.5, with 1.6◦ × 2.6◦ field of view and a curved field detector allowing null distortion aberration for drift-scan observations. The freeform mirror is generated by spherical stress polishing that provides super-polished freeform surfaces after elastic relaxation. Preliminary analysis required use of the optics theory of 3rd-order aberrations and elasticity theory of thin elliptical plates. Final cross-optimizations were carried out with Zemax raytracing code and Nastran FEA elasticity code in order to determine the complete geometry of a glass ceramic Zerodur deformable substrate.

Keywords: telescope; active optics; aspherics; elasticity; optical design; deformable mirror

1. Introduction Technological advances in astronomical instrumentation during the second part of the twentieth century gave rise to 4 m, 8 m and 10 m class telescopes that were completed with close loop active optics control in monolithic mirrors [1], segmented primary mirrors [2] and segmented in-situ stressed active optics mirrors [3]. Further advances on ground-based telescopes allowed developments of adaptive optics for blurring the image degradation due to atmosphere. Both active and adaptive optics provided milestone progress in the detection of super-massive black holes, exoplanets and large-scale structure of galaxies. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators: a single uniform load acts over the monolithic surface of a closed-form substrate whilst under reaction to its elliptical perimeter ring. These freeform surfaces are the basic aspheric components of dispersive/reflective Schmidt systems. Both reflective diffraction freeform gratings and freeform mirrors have been investigated by Lemaitre [4]. MESSIER proposal is a wide-field low-central-obstruction folded-two-mirror-anastigmat or here called briefly three-mirror-anastigmat (TMA) telescope based on a folded reflective Schmidt optical design. It is dedicated to the survey of extended astronomical objects with extremely low surface brightness. The proposal name is in homage to French astronomer Charles Messier (1730–1817) (Figure1) who published the first astronomical catalogue of nebulae and star clusters. The catalogue was known as Messier objects—for instance, M1 for the Crab nebulae, M31 for the Andromeda galaxy

Math. Comput. Appl. 2019, 24, 2; doi:10.3390/mca24010002 www.mdpi.com/journal/mca Math. Comput. Appl. 2019, 24, 2 2 of 15 Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 2 of 15 andand so forth—and so forth—and distinguished distinguished between between permanent permanent and and transient transient diffuse diffuse objects, objects, then then helped helped in in sensingsensing or discovering or discovering comets. comets.

FigureFigure 1. Charles 1. Charles Messier Messier (1730–1817) (1730–1817) (Wikipedia). (Wikipedia).

MESSIERMESSIER optical optical design design would would provide provide a high a high imaging imaging quality quality without without any any diff diffractiveractive spider spider patternpattern in the in thebeams. beams. A preliminarily A preliminarily ground-based ground-based prototype prototype is planned is planned to serve to serve as a as fast-track a fast-track pathpathfinderfinder for fora future a future space-based space-based MESSIER MESSIER mission. mission. The The elliptical elliptical freeform freeform mirror mirror is generated is generated by by stressstress polishing polishing and and elastic elastic relaxation relaxation technique technique [4,5]. [4 ,5 ]. Super-polishingSuper-polishing of freeform of freeform surfaces surfaces is obtained is obtained from from spherical spherical polishing polishing after after elastic elastic relaxation. relaxation. PreliminaryPreliminary analysis analysis required required use use of ofthe the optics optics theo theoryry of of3rd-order 3rd-order aberrations aberrations and and elasticity elasticity theory theory of of thinthin elliptical elliptical plates. plates. The The fi finalnal cross-optimizations cross-optimizations were carriedcarried outout withwith ZemaxZemax ray-tracingray-tracing code code and andNastran Nastran FEA FEA elasticity elasticity code code that that provides provides accurate accurate determination determination of of the the substrate substrate geometry geometry of of a glassa glassceramic ceramic Zerodur Zerodur material. material.

2. Optical2. Optical Design Design with with a Re a Reflectiveflective Schmidt Schmidt Concept Concept A reflectiveA reflective Schmidt Schmidt design design is isa wide- a wide-fieldfield system system and and basically basically a a two-mirror two-mirror anastigmat. WithWith one onefreeform freeform surface surface correcting correcting all threeall three primary primary aberrations, aberrations, such asuch system a system has been has widely been investigated widely investigatedas well as as for well telescope as for or telescope spectrograph or sp designs.ectrograph The correctordesigns. mirrorThe corrector or reflective mirror diffraction or reflective grating is diffalwaysraction locatedgrating atis always the centre located of curvature at the centre of aspherical of curvature concave of a spherical mirror. concave mirror. ComparedCompared to toa aSchmidt Schmidt with with aa refractor-correctingrefractor-correcting element, element, which which is a is centred-system, a centred-system, a reflective a reflectiveSchmidt Schmidt necessarily necessarily requires requires a tilt a oftilt theof the optics. optics. TheThe inclination of of the the mirrors mirrors then then forms forms a a non-centred-systemnon-centred-system. For. Foran f an/2.5f /2.5focal-ratio, focal-ratio, the tilt the angle tilt angle of the of aspheric the aspheric primary primary mirror mirror is typically is typically of of ◦ aboutabout 10° 10andand somewhat somewhat depending depending on the on theFoV FoV size. size. For Fora re aflective reflective Schmidt Schmidt optical optical design, design, as MESSIER as MESSIER telescope telescope proposal proposal at f at/2.5,f /2.5, a supplementary a supplementary folding-folding-flatflat mirror mirror was was found found necessary. necessary. This This third third mirror mirror provides: provides: (i) an (i) easy an easy access access to the to thedetector detector for forminimal minimal central central obstruction; obstruction; and and (ii)(ii) avoidavoid any any spider spider inside inside the beamsthe beams that wouldthat would cause diffractivecause diffractiveartefact artefact cross-like cross-like images images at the focal at the image focal ofimage bright of stars.bright Thisstars. led This us led to a us three-mirror to a three-mirror design or designTMA or telescope.TMA telescope. InvestigationsInvestigations of ofthe the plane-aspheric plane-aspheric mirror mirror havehave beenbeen discussed discussed to defineto define the bestthe shapebest shape freeform freeformsurface surface of this of non-centred this non-centred system. system. The freeform The freeform surface surface is with is ellipticalwith elliptical symmetry symmetry and provides and providesbalance balance of the of quadratic the quadratic terms terms with respect with respect to the to bi-quadratic the bi-quadratic terms. terms. TemporarilyTemporarily avoiding avoiding the the folding-flat folding-flat mirror, mirror, let assumeassumea a two-mirror two-mirror non-centred non-centred system system where wherethe the primary primary mirror mirror M1 isM1 a freeformis a freeform and theand secondary the secondary mirror mirror M2 is aM2 concave is a concave spherical spherical surface of surfaceradius of ofradius curvature of curvatureR. The inputR. The beams input arebeams circular are circular collimated collimated beams merging beams merging at various at fieldvarious angles fieldof angles a telescope of a andtelescope define and along de thefine M1 along freeform the M1 mirror freeform an elliptical mirror pupil an elliptical due to inclination pupil due angle to i. inclinationA convenient angle valuei. A convenient of the inclination value angleof the allows inclination the M2 angle mirror allows to avoid the anyM2 obstructionmirror to avoid and provideany obstruction and provide focusing very closely to mid-distance between M1 and M2, then very near the distance R/2 from M1 (Figure 2).

Math. Comput. Appl. 2019, 24, 2 3 of 15 focusing very closely to mid-distance between M1 and M2, then very near the distance R/2 from M1 Math.(Figure Comput.2). Appl. 2018, 23, x FOR PEER REVIEW 3 of 15

Figure 2. Schematic of a reflective two-mirror Schmidt telescope. The centre of curvature of the Figure 2. Schematic of a reflective two-mirror Schmidt telescope. The centre of curvature of the spherical mirror M2 is located at the vertex of the plane-aspheric freeform mirror M1. The incident spherical mirror M2 is located at the vertex of the plane-aspheric freeform mirror M1. The incident beams have circular cross-sections with input pupil located at mirror M1. Deviation 2i occurs at the beams have circular cross-sections with input pupil located at mirror M1. Deviation 2i occurs at the principal rays. The semi-field maximum angle—here assumed circular—is denoted ϕm. principal rays. The semi-field maximum angle—here assumed circular—is denoted φm.

Let us denote xm and ym the semi-axes of the elliptic clear-aperture on M1 primary mirror. If the y-axisLet is us perpendicular denote xm and to y them the symmetry semi-axes plane of thex, ellipticz of the clear-aperture two-mirror telescope, on M1 primary one deﬁnes mirror. If the y-axis is perpendicular to the symmetry plane x, z of the two-mirror telescope, one deﬁnes

Ω = R/4ym (1) Ω = R/4ym (1) aa dimensionless dimensionless ratio ratio Ω,Ω where, where ym yism theis thesemi-clear-aperture semi-clear-aperture of the of beams the beams in y-direction, in y-direction, that is that half- is pupilhalf-pupil size in size y-direction. in y-direction. The focal-ratio The focal-ratio of theof two-mirror the two-mirror telescope telescope is denoted is denoted f/Ω.f /Ω. ItIt is is the the interesting interesting to toinvestigate investigate the the performance performance in the in field the field of view of viewof a 100% of a 100%obstructed obstructed two- mirrortwo-mirror telescope telescope which which then thenis a centred is a centred system. system. Comparing Comparing the blur the blurimages, images, the thefree free parameter parameter is theis theratio ratio that that characterize characterize the the meridian meridian profile profile section section of of the the primary primary mirror mirror M1. M1. This This ratio, ratio, or or aspect aspect 2 2 ratio,ratio, can can be be expressed expressed by by kk = rO2/r/mr2m wherewhere rOr isO isthe the radius radius of of null-power null-power zone zone and and rmrm thatthat of of clear clear aperture.aperture. The The blur blur residual residual sizes sizes as as a a function function of of parameter parameter kk areare displayed displayed by by Figure Figure 3 [[4,6].4,6]. In order to avoid 100% beam obstruction the primary mirror M1 must be tilted at a convenient inclination angle i. It can be shown that the shape ZOpt of the M1 freeform mirror is an anamorphic shape and expressed in first approximation by [4] s 3 2 2 2 1 2 2 2 2 2 2 ZOpt ' (h x + y ) − (h x + y ) , with h = cos i, (2) cos i 27Ω2R 8R3 where dimensionless coefficients could be considered as an under-correction parameter slightly smaller than unity (0.990 < s < 1). In fact, for a tilt angle of M1—that is a non-centred system—coefficients does not operates and must just be set to s = 1. A preliminary analysis of the two-mirror system shows that for four direction points of a circular field of view, the largest blur image occurs at the largest deviation angle of the field along the x-axis. In y-direction sideways blur images have an average size. The length from the vertex O of the M1 mirror to the focus F can be derived as [4]

1 3 OF = 1 + R (3) 2 26Ω2 showing that this distance is slightly larger than the Gaussian distance R/2. The optimal size of blur images for an f /4 two-mirror anastigmat telescope over the ﬁeld of view are displayed by Figure4.

Math. Comput. Appl. 2019, 24, 2 4 of 15 Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 4 of 15

FigureFigure 3. Off-axisOff-axis 5th-order 5th-order aberration aberration residuals of residuals a two-mirror of acentred-system two-mirror reflective centred-system Schmidt— reflective 2 2 Schmidt—100%100% obstruction—as obstruction—as a function of the a functionk-ratio, k = ofrO2 the/rm2, kthat-ratio, definesk = locationrO /rm of, the that null-power defines locationoptical of the null-powerzone. Plotted optical parameters zone. are: Plotted radius parameters of curvature are: R of radius M2 spherical of curvature concaveR mirror,of M2 Ω spherical of the aperture- concave mirror, Ωratioof the f/Ω aperture-ratio, angle of semi-fieldf /Ω of, angleview φ of, radial semi-field and tangential of view blurϕ, radial sizes ℓ andr, ℓt and tangential diameter blur of best sizes blur` r, `t and diametersize d. [1st of line best] Spot blur diagram size d.[ 1stof blur line] images Spot diagram on a sphere of blur centred images at the on centre a sphere of curvature centred atof theM1 centre of curvaturemirror that of provides M1 mirror stigmatism that provides on-axis. stigmatism [2nd line] Spot on-axis. diagram [2nd of lineblur] images Spot diagram on a sphere of blur centred images on a at the centre of curvature of M1 mirror that provides the best images. [Bottom line] Spots with k = 3/2 Math.sphere Comput. centred Appl. at2018 the, 23 centre, x FOR of PEER curvature REVIEW of M1 mirror that provides the best images. [Bottom line 5] of Spots 15 for field angles φ/φm varying from 0 to 1, obtained by using a slight under-correction factor of M1 with k = 3/2 for field angles ϕ/ϕm varying from 0 to 1, obtained by using a slight under-correction showingmirror s that= cos thisφm and distance defocus is 𝛥 fslightly. The three larger rows thanof spot the images Gaussian are at samedistance scale. R/ Lemaitre2. The optimal showed size of factor of M1 mirror s = cosϕm and defocus ∆f. The three rows of spot images are at same scale. Lemaitre that [6] the final diameterf/ of residual blurs—or angular resolution of reflective Schmidt—is then dResol = 3 blurshowed images that for [6 an] the 4 final two-mirror diameter anastigmat of residual telescope blurs—or overangular the fi resolutioneld of view of reflectiveare displayed Schmidt by—is Figure then φm2/256Ω3. 4. 2 3 dResol = 3 ϕm /256Ω . In order to avoid 100% beam obstruction the primary mirror M1 must be tilted at a convenient inclination angle i. It can be shown that the shape ZOpt of the M1 freeform mirror is an anamorphic shape and expressed in first approximation by [4] 𝑠 3 1 𝑍 − (ℎ𝑥 +𝑦) − (ℎ𝑥 +𝑦), 𝑤𝑖𝑡ℎ ℎ =cos 𝑖, (2) cos 𝑖 2Ω𝑅 8𝑅 where dimensionless coefficients could be considered as an under-correction parameter slightly smaller than unity (0.990 < s < 1). In fact, for a tilt angle of M1—that is a non-centred system— coefficients does not operates and must just be set to s = 1. A preliminary analysis of the two-mirror system shows that for four direction points of a circular field of view, the largest blur image occurs at the largest deviation angle of the field along the x-axis. In y-direction sideways blur images have an average size. The length from the vertex O of the M1 mirror to the focus F can be derived as [4] 1 3 𝑂𝐹 = 1 + 𝑅 (3) 2 2𝛺 Figure 4. Best residual aberrations of an f /4 two-mirror reflective Schmidt (Ω = 4) in the non-centred Figure 4. Best residual aberrations of an f/4 two-mirror reflective Schmidt (Ω = 4) in the non- system form. Inclination angle i = 5.7◦. Semi-field of view ϕ = 2.5◦. The under-correction parameter is centred system form. Inclination angle i = 5.7°. Semi-field ofm view φm = 2.5°. The under-correction just set to s = 1. The radius of curvature of the spherical focal surface is also given by R = OF in parameter is just set to s = 1. The radius of curvature of the spherical focal surface is also FoVgiven Equationby RFoV (3). = TheOF in largest Equation blur (3). image The largest corresponds blur image to that corresponds with highest to that deviation with highest of the deviation FoV. of the FoV.

Preliminarily analyses also show that the M1 mirror shape has opposite signs between quadratic and bi- quadratic terms. The shape is given by Equation (2) and represented by Figure 5 in either directions x or y in dimensionless coordinates of ρ.

Figure 5. The freeform primary mirror, of biaxial symmetry, is generated by homothetic ellipses having principal lengths in a cosi ratio. One shows that whatever x or y directions the null-power

zone is outside the M1 clear-aperture and in a geometrical ratio 𝜌 = 3/2 𝜌 − 1.224𝜌 since 𝑘 = 3/2.

Dimensionless coordinates z, ρ have been normalized for a clear semi-aperture ρ = 1, presently in the y-direction, with a maximum sag

z(1) = 3ρ2 – ρ4 = 2, (4)

which leads to algebraically opposite curvatures d2z/dρ2 for ρ = 0 and ρ = 1. The conclusions from the best optical design of an all-reﬂective two-mirror Schmidt telescope with optimal angular resolution are the followings [4,6]:

Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 5 of 15 showing that this distance is slightly larger than the Gaussian distance R/2. The optimal size of blur images for an f/4 two-mirror anastigmat telescope over the ﬁeld of view are displayed by Figure 4.

Figure 4. Best residual aberrations of an f/4 two-mirror reﬂective Schmidt (Ω = 4) in the non-

centred system form. Inclination angle i = 5.7°. Semi-ﬁeld of view φm = 2.5°. The under-correction parameter is just set to s = 1. The radius of curvature of the spherical focal surface is also given

by RFoV = OF in Equation (3). The largest blur image corresponds to that with highest deviation

of theMath. FoV. Comput. Appl. 2019, 24, 2 5 of 15

Preliminarily analyses also show that the M1 mirror shape has opposite signs between quadratic and bi- quadraticPreliminarily terms. analysesThe shape also is show given that by the Equa M1 mirrortion (2) shape and has represented opposite signs by betweenFigure 5 quadratic in either directionsand x bi-or y quadratic in dimensionless terms. The coordinates shape is given of ρ by. Equation (2) and represented by Figure5 in either directions x or y in dimensionless coordinates of ρ.

Figure 5. The freeform primary mirror, of biaxial symmetry, is generated by homothetic ellipses having Figure 5. The freeform primary mirror, of biaxial symmetry, is generated by homothetic ellipses principal lengths in a cosi ratio. One shows that whatever x√or y directions the null-power zone is having principaloutside the lengths M1 clear-aperture in a cosi ratio. and in One a geometrical shows that ratio whateverρ0 = 3/2 x ρormax y 'directions1.224ρmax thesince null-powerk = 3/2. zone is outside the M1 clear-aperture and in a geometrical ratio 𝜌 = Dimensionless coordinates z, ρ have been normalized for a clear semi-aperture ρ = 1, presently in 3/2 𝜌 − 1.224𝜌 since 𝑘 = 3/2. the y-direction, with a maximum sag Dimensionless coordinates z, ρ have been normalized for a clear semi-aperture ρ = 1, presently z(1) = 3ρ2 − ρ4 = 2, (4) in the y-direction, with a maximum sag 2 2 which leads to algebraically opposite curvaturesz(1) = 3ρ2d –z ρ/4d ρ= 2,for ρ = 0 and ρ = 1. (4) The conclusions from the best optical design of an all-reflective two-mirror Schmidt telescope which leadswith optimal to algebraically angular resolution oppositeare the curvatures followings d [24z/d,6]:ρ2 for ρ = 0 and ρ = 1. The conclusions from the best optical design of an all-reflective two-mirror√ Schmidt telescope 1. For a circular incident beam, the elliptical clear-aperture of the primary mirror is 3/2 times smaller to with optimal angular resolution are the followings [4,6]: that of the null-power zone ellipse. 2. The angular resolution dNC of a reflective non-centred two-mirror telescope is

3 3 dNC = i + ϕm ϕm (5) 256Ω3 2

3. The primary mirror of a non-centred two-mirror system provides the algebraic balance of second derivative extremals. Therefore, its central curvature is opposite to the local curvatures at edge.

3. Elasticity Design and Deformable Primary-Mirror Substrate Active optics preliminarily analysis of a Schmidt-type primary mirror M1 leads to investigate deformable substrates and use of stress mirror polishing (SMP). We consider hereafter the thin plate theory of plates with development to elliptical plates. Let consider the system coordinates of an elliptical plate and denote n the normal to the contour C of the surface. Equation of C is represented by (Figure6) Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 6 of 15

1. For a circular incident beam, the elliptical clear-aperture of the primary mirror is 3/2 times smaller to that of the null-power zone ellipse.

2. The angular resolution dNC of a reflective non-centred two-mirror telescope is 3 3 𝑑 = 𝑖+𝜑 𝜑 (5) 256Ω 2 3. The primary mirror of a non-centred two-mirror system provides the algebraic balance of second derivative extremals. Therefore, its central curvature is opposite to the local curvatures at edge.

C Math.of the Comput. surface. Appl. Equation2019, 24, 2 of C is represented by (Figure 6) 6 of 15

Figure 6. Top view of an elliptical vase form. The clear aperture of M1 mirror is included into a smaller Figuresurface 6. Top to that view of aof constant an elliptical thickness vase form plate. The delimited clear aperture by elliptical of M1 contour mirrorC is( dottedincluded line )into and a defined smaller by surfacesemi axeto that radii of ( ax constant, ay). An outerthickness ring plate is built-in delimited to the by plate elliptical at contour contourC where C (dottedn-directions line) and are defi normalned 0 byto semiC. The axe outer radii ring (ax, is ay delimited). An outer by ring a homothetic is built-in elliptic to the contour plate atC contourdefined C bywhere semi-axes n-directions radii (b xare, by ). normal to C. The outer ring is delimited by a homothetic elliptic contour C’ defined by semi-axes radii The bi-Laplacian equation of the flexure where a uniform load q is applied to the inner plate (bx, by). is [4,7] 4 4 4 4 2 2 4 4 The bi-Laplacian equation∇ z of≡ the∂ z /fl∂exurex + 2where∂ z/∂ xa uniform∂y + ∂ zload/∂y q= is qapplied/D, to the inner plate is (6) [4,7]where the rigidity D is [4,7] D = Et3/[12(1 − ν2)], (7) ∇𝑧≡𝜕𝑧/𝜕𝑥 +2𝜕𝑧/ 𝜕𝑥𝜕𝑦 +𝜕𝑧/𝜕𝑦 =𝑞/𝐷, (6) and E the Young’s modulus, ν the Poisson ratio and t the constant thickness of the plate over C. 0 where Thethe rigidity equation D ofis [4,7] homothetic ellipses C(ax, ay) or C (bx, by) can be expressed by quadratic forms. Ellipse curve C writes D = Et3/[12(1 – ν2)], (7) x2 y2 + − = 2 2 1 0. (8) and E the Young’s modulus, ν the Poissona ratiox a andy t the constant thickness of the plate over C. The equation of homothetic ellipses C(ax, ay) or C’(bx, by) can be expressed by quadratic forms. Remaining within the optics theory of third-order aberrations, such optical freeform surfaces can Ellipse curve C writes be obtained from elastic bending by mean of the following conditions: 𝑥 𝑦 • a flat and constant-thickness plate, t =constant+ −1=0., (8) 𝑎 𝑎 • a uniform load q applied to inner surface of substrate, • Remainingand a link within at the edgethe optics to an elliptictheory contourof third-orde expressedr aberrations, by a built-in such edge optical, or a freeformclamped edge surfaces, that is can be whereobtained the from slope elastic is null bending all along by the mean contour of theC followingof the plate. conditions: • a flat and constant-thickness plate, t = constant, Assuming that the three conditions below are satisfied, the analytic theory of thin plates allows • a uniform load q applied to inner surface of substrate, deriving a biquadratic flexure Z (x,y) in the form [4,7], • and a link at the edge to an ellipticElas contour expressed by a built-in edge, or a clamped edge, that is where the slope is null all along the contour C of the plate.! 2 x2 y2 Z = z 1 − − , (9) Elas 0 2 2 ax ay

where z0 is the sag at origin and where ax and ay are semi-axes corresponding to the elliptic null-power zone of the principal directions of contour. The ﬂexural sag z0 is obtained from substitution of Equation (9) into biharmonic Equation (6). This leads to

q a4 a4 = x y z0 4 2 2 4 . (10) 8D 3ax + 2axay + 3ay √ The null-power zone contour C is larger by a factor 3/2 to that of the clear-aperture (cf. Figure3). The dimensions of semi-axes ax and ay at C of the built-in ellipse – or null-power zone –relatively to that of clear-semi-apertures xm and ym are

2 2 2 2 2 2 ax = 3xm/2 = 3ym/2 cos i and ay = 3ym/2. (11) Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 7 of 15

Assuming that the three conditions below are satisﬁed, the analytic theory of thin plates allows deriving a biquadratic ﬂexure ZElas(x,y) in the form [4,7],

𝑥 𝑦 𝑍 =𝑧 1 − − , (9) 𝑎 𝑎 where z0 is the sag at origin and where ax and ay are semi-axes corresponding to the elliptic null- power zone of the principal directions of contour. The ﬂexural sag z0 is obtained from substitution of Equation (9) into biharmonic Equation (6). This leads to 𝑞 𝑎𝑎 𝑧 = . (10) 8𝐷 3𝑎 +2𝑎𝑎 +3𝑎

The null-power zone contour C is larger by a factor 3/2 to that of the clear-aperture (cf. Figure 3).

Math.The Comput.dimensions Appl. 2019 of semi-axes, 24, 2 ax and ay at C of the built-in ellipse – or null-power zone –relatively7 of 15 to that of clear-semi-apertures xm and ym are √ 𝑎 =3𝑥/2 = 3𝑦/2 cos 𝑖 𝑎𝑛𝑑 𝑎 =3𝑦/2. (11) From Equations (2) and (1), in setting s = 1 and x = 0 and for the built-in radius y = 3/2ym of the vase form, we obtain the amplitude of the flexure z0 in Equation (9), as From Equations (2) and (1), in setting s = 1 and x = 0 and for the built-in radius 𝑦=√3/2𝑦𝑚 of the vase form, we obtain the amplitude of the flexure z0 in Equation (9), as 9ym z0 = . (12) 119𝑦3 𝑧 = 2 Ω cos.i (12) 2Ω cos 𝑖 From Equations (10)–(12) we obtain, after simplification, the thickness t of the inner plate From Equations (10)–(12) we obtain, after simplification, the thickness t of the inner plate 2 /1/3 2(12 −(1 𝜐−υcoscos 𝑖)i) 𝑞q 𝑡=8Ωt = 8Ω y𝑦m.. (11) (13) 3(33 +(3 2+ cos2 cos 𝑖+3cos2 i + 3 cos4𝑖)i) 𝐸E

ThisThis defi definesnes the theexecution execution conditions conditions and and elasticity elasticity parameters parameters of an of elliptical an elliptical plate plate where where the the clearclear aperture aperture of primary of primary mirror mirror M1 M1uses uses a somewhat a somewhat smaller smaller area area than than the thetotal total built-in built-in surface surface as as delimiteddelimited by ellipse by ellipse C. C. TheThe conclusions conclusions from from a best a best optical optical design design of the of theprimary primary mirror mirror M1 M1of a of two-mirror a two-mirror Schmidt Schmidt telescope,telescope, as corresponding as corresponding to the to thepro profilefile displayed displayed by Figure by Figure 3, are3, are as asfollow follow [4]: [ 4]:

1. 1.A built-inA built-in elliptic elliptic vase-form vase-form is usef isul useful to obtain to obtain easily easilythe primary the primary mirror mirror of a two-mirror of a two-mirror anastigmat. anastigmat. √ 2. 2.The ellipticThe elliptic null-power null-power zone at zone the plate at the built-in plate built-in contour contour is 3/2 is times3/2 largertimes than larger that than of the that elliptic of the clear- elliptic apertureclear-aperture of the primary of the mirror. primary mirror. 3. 3.The Thetotal total sag of sag the of built-in the built-in contour contour is 9/8 is tim 9/8es times larger larger than than that thatof its of optical its optical clear-aperture. clear-aperture.

TheThe optimization optimization of ofa built-in a built-in substrate, substrate, of of cour course,se, requires losinglosing some some of of the the outer outer surface surface which whichthen then is not is usablenot usable by an by amount an amount of 33% of outside33% ou thetside clear-aperture the clear-aperture area. It area. is clear It is that clear a ﬂat that deformable a ﬂat deformableM1 substrate, M1 substrate, conjugated conjugated with an ellipticwith an inner elliptic contour, inner provides contour, most provides interesting most advantagesinteresting in advantagespractice within practice a built-in with condition a built-in(Figure condition7). (Figure 7).

Figure 7. Section of an elliptical closed vase form. This form is made two vase-forms oppositely jointed Figure 7. Section of an elliptical closed vase form. This form is made two vase-forms oppositely jointed together in a built-in link. The uniform load q is applied all inside the deformable substrate over the together in a built-in link. The uniform load q is applied all inside the0 deformable substrate over the elliptical contour C of semi-axe radii (ax, ay). Outer contour C can be made circular if (bx, by) are elliptical contour C of semi-axe radii (ax, ay). Outer contour C’ can be made circular if (bx, by) are sufficiently larger than semi-axe radii of C. sufficiently larger than semi-axe radii of C. An extremely rigid outer ring designed for a closed vase form, while polished flat at rest, allows the primary mirror to generate by uniform loading a bi-symmetric optical surface made of homothetic ellipses (Figure8). The case of a perfect built-in condition was applied to the design and construction of several aspherized grating spectrographs (4) such as UVPF of CFHT, MARLYs of OHP and PMO, CARELEC of OHP, OSIRIS of ODIN space mission. It was recently applied to the design and construction of aspherized gratings for the FIREBall balloon experiment [8] using a 1-meter telescope and multi-object spectrograph. The processing was as follows: (i) A plane reflective diffraction grating was first deposited on a deformable stainless steel matrix without stressing; (ii) the freeform bi-quadratic surface of this deformable matrix was bent by inner air pressure loading; (iii) the freeform gratings were obtained from resin replication during controlled stressing onto Zerodur vitro ceramic rigid substrate; and (iv) final aspheric shape of the gratings was obtained by elastic relaxation [4] (Figure9). Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 8 of 15

An extremely rigid outer ring designed for a closed vase form, while polished ﬂat at rest, allows the primary mirror to generate by uniform loading a bi-symmetric optical surface made of homothetic ellipses (Figure 8).

Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 8 of 15

An extremely rigid outer ring designed for a closed vase form, while polished ﬂ at at rest, allows Math.the Comput.primary Appl. mirror2019 to, 24 generate, 2 by uniform loading a bi-symmetric optical surface made of homothetic 8 of 15 Figure 8. Iso-level lines of a primary mirror M1 generating homothetical ellipses from perfect built- ellipses (Figure 8). in condition at contour C. Elliptic dimensions of the null-power zone over those of clear-aperture must be set in a ratio 3/2 . The algebraic balance of meridian curvatures is then achieved at centre and at clear-aperture.

The case of a perfect built-in condition was applied to the design and construction of several aspherized grating spectrographs (4) such as UVPF of CFHT, MARLYs of OHP and PMO, CARELEC of OHP, OSIRIS of ODIN space mission. It was recently applied to the design and construction of aspherized gratings for the FIREBall balloon experiment [8] using a 1-meter telescope and multi- object spectrograph. The processing was as follows: (i) A plane reﬂective diﬀraction grating was first deposited on a deformable stainless steel matrix without stressing; (ii) the freeform bi-quadratic surface of this deformable matrix was bent by inner air pressure loading; (iii) the freeform gratings FigureFigure 8. 8. Iso-levelIso-level lines lines of of a aprim primaryary mirror mirror M1 M1generating generating homothetical homothetical ellipses ellipses from perfect from perfect built- built-in were obtained from resin replication during controlled stressing onto Zerodur vitro ceramic rigid conditionin condition at contour at contour C. EllipticC. Elliptic dimensions dimensions of of the the null-power null-power zonezone over over those those of of clear-aperture clear-aperture must substrate; and (iv) final√ aspheric shape of the gratings was obtained by elastic relaxation [4] (Figure bemust set inbe aset ratio in a ratio3/2 .3/2 The . The algebraic algebraic balance balance of of meridian meridian curv curvaturesatures is isthen then achieved achieved at centre at centre and 9). atand clear-aperture. at clear-aperture.

The case of a perfect built-in condition was applied to the design and construction of several aspherized grating spectrographs (4) such as UVPF of CFHT, MARLYs of OHP and PMO, CARELEC of OHP, OSIRIS of ODIN space mission. It was recently applied to the design and construction of aspherized gratings for the FIREBall balloon experiment [8] using a 1-meter telescope and multi- object spectrograph. The processing was as follows: (i) A plane reﬂective diﬀraction grating was first deposited on a deformable stainless steel matrix without stressing; (ii) the freeform bi-quadratic surface of this deformable matrix was bent by inner air pressure loading; (iii) the freeform gratings were obtained from resin replication during controlled stressing onto Zerodur vitro ceramic rigid substrate; and (iv) final aspheric shape of the gratings was obtained by elastic relaxation [4] (Figure Figure 9. [Left] Active optics aspherization of gratings achieved by double replication technique of a 9). Figure 9. [Left] Active optics aspherization of gratings achieved by double replication technique of metala metal deformable deformable matrix. matrix. A quasi-constant A quasi-constant thickness thickness activeactive zone zoneis clamped is clamped – or built-in—or built-in – to a —torigid aouter rigid ringouter and ring is closed and is backside closed backside for air pressure for air pressureload control. load [Centre control.]. For [Centre FIREBall,]. For the FIREBall, whole deformable the whole matrixdeformable is an axisymmetric matrix is an axisymmetric piece expect piecefor its expect inner for built-in its inner contour built-in which contour is whichelliptical is. elliptical[Right] . Deformable[Right] Deformable matrix with matrix the with grating the gratingbefore beforereplication replication (LAM). (LAM).

ComparedCompared to to the the case case of of or or a a deformable deformable M1 M1 mirror—or mirror—or a a deformable deformable matrix—where matrix—where a a perfect perfect built-inbuilt-in conditioncondition leads leads to to some some outside outside loosen loosen optical optical ar areaea of of the the freeform freeform (a (a clear clear aperture aperture somewhat somewhat smallersmaller to to that that of of the the built-in built-in zero zero power power zone), zone), we we investigate investigate hereafter hereafter a a closed-formclosed-form substrate substrate withwith a a radially thinned outer elliptic cylinder. The optimized freeform surface for MESSIER proposal will then radiallyFigure thinned 9. [ outerLeft] Active elliptic optics cylinder aspherization. The optimized of gratings freeformachieved by surface double replication for MESSIER technique proposal of a will then provideprovidemetal almost almost deformable the telescope matrix. A theoreticalquasi-constanttheoretical angular thicknessangular active resolution resolution zone is byclamped by use use of– orof a built-in semi-built-ina semi-built-in – to a rigid edge outeredge condition conditionof theof themirror mirrorring substrate. andsubstrate. is closed backside for air pressure load control. [Centre]. For FIREBall, the whole deformable matrix is an axisymmetric piece expect for its inner built-in contour which is elliptical. [Right] 4.4. A A TMA TMADeformable TelescopeTelescope matrix Proposalwith Proposal the grating for for MESSIER before MESSIER replication (LAM). Our proposal of MESSIER experiment is named in honour to French astronomer Charles Compared to the case of or a deformable M1 mirror—or a deformable matrix—where a perfect Messier who started compiling in 1774 his famous Messier Catalogue of diffuse non-cometary objects. built-in condition leads to some outside loosen optical area of the freeform (a clear aperture somewhat Forsmaller instance to that the of Crab the built-in Nebulae, zero named power objectzone), ”M1,”we investigate is a bright hereafter supernova a closed-form remain substrate recorded with by a Chinese astronomerradially thinned in 1054.outer elliptic The present cylinder. MESSIERThe optimized proposal freeform is dedicatedsurface for MESSIER to the detection proposal of will extremely then low surfaceprovide brightness almost the objects. telescope A theoretical detailed descriptionangular resolution of the by science use of objectives a semi-built-in and edge instrument condition designof for thisthe space mirror mission substrate. can be found in [9]. A three mirror anastigmat (TMA) telescope proposal should provide detection of extremely low 4. A TMA Telescope Proposal for MESSIER surface brightness. MESSIER science goals require that any optical design should deliver a detection as a detection as low as 32 magnitude/arcsec2 in the optical range. A space-based telescope option should provide 37 magnitude/arcsec2 in UV at 200 nm. The preliminary ground-based telescope here investigated as a prototype should be useful for spectral ranges in blue, red and infrared regions. The main features require a particular optical design as follow:

1. a wide field three mirror anastigmat telescope with fast f-ratio, 2. a distortion-free field of view at least in one direction, 3. a curved-field detector for a natural curved reflective Schmidt, Math. Comput. Appl. 2019, 24, 2 9 of 15

4. a time delay integration by use of drift-scan techniques, 5. no spider in the beams can be placed in the optical train.

The ﬁnal space-based proposal, restrained to extremely low brightness objects in ultraviolet imaging, would be a 50-cm aperture telescope. For the optical range, our preliminary plan for MESSIER is to develop and build a 30-cm aperture ground-based telescope fulﬁlling all above features of the optical design. Preliminary proposed designs can be found in [5,10].

4.1. Optical Design and Ray Tracing Modelling Before studying a space model, we propose development of a ground-based prototype telescope is a three-mirror anastigmat (TMA) with an f-ratio at f /2.5. One may notice a classical design as a two-element anastigmat design where the first optical element is a refractive aspherical plate. Minimization of field aberrations is achieved by a balance of the slopes—that is balance of 1st-order derivative—where sphero-chromatism variations are dominating aberration residuals. Now using a mirror—instead of a refractive plate—as a first element M1 of a two-mirror anastigmat, the best angular resolution over the field of view is achieved by a balance of meridian curvatures—that is balance of the 2nd-order derivative—of the aspherical mirror or diffraction grating [4]. The proposed design is made of freeform primary mirror M1, followed by holed flat secondary mirror M2 and then a spherical concave tertiary mirror M3. The centre of curvature of M3 is located at the vertex of M1which is a basic configuration for a reflective Schmidt concept. The unfolded version, with only two mirrors, do not alter anastigmatic image quality (Figure 13). The optics parameters are optimized for a convex field of view (Table1). The three-mirror system gives unobstructed access to detector and avoid any spider in the field of view (Figures 10 and 11). Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 10 of 15

Table 1. MESSIER optical design parameters. Table 1. MESSIER optical design parameters.

TelescopeTelescope angularangular resolution resolution 2 arcsec2 arcsec On-axisOn-axis circular circular beambeam entrance entrance 256256 mm mm FocalFocal lengthlengthf 0 f’ 890890 mm mm Focal-ratio f/ f/ΩΩ f/2.5f/2.5 DeviationDeviation angle angle 2i 2i 22◦22° M1 Elliptic clear aperture 2xm × 2ym 356 × 362.7 M1 Elliptic clear aperture 2xm × 2ym 356 × 362.7 Angular FOV 2φ × 2φ 1.6° × 2.6° Angular FOV 2ϕmxmx× 2ϕmymy 1.6◦ × 2.6◦ Linear FOV 25 × 40 mm2 Linear FOV 25 × 40 mm2 M3 mirror curvature radius R3 −1769 mm M3 mirror curvature radius R3 −1769 mm UBK7 filters & SiO2 cryostat—Thicknesses 2 & 5 mm UBK7 filters & SiO2 cryostat—Thicknesses 2 & 5 mm Detector field curvature radius RFO −890 mm Detector field curvature radius RFO −890 mm

Figure 10. Schematic of Messier reflective Schmidt (M2 holed flat not shown). Figure 10. Schematic of Messier reflective Schmidt (M2 holed flat not shown).

Figure 11. [Left] MESSIER layout reﬂective TMA in the symmetry plane. [Right] 3-D view with detector (LAM).

The primary mirror surface M1 can be deﬁned by the aspheric anamorphic equation as derived from Zemax optics ray-tracing code—which is somewhat similar to Equaiton (2)—and denoted

𝑍 = + 𝐴𝑅(1−𝐴𝑃)𝑥 + (1+𝐴𝑃)𝑦 , (14) () where now in all this section with Zemax code (y, z) is the symmetry plane of the telescope. Restraining to the case of a simply bi-curvature surface for the ﬁrst quadratic term, then setting Kx = Ky = −1, denominator reduces to unity. The result from Zemax modelling −6 −1 −6 −1 optimization provides the four coeﬃcients, Cx = −3.598 × 10 mm , Cy = −3.467 × 10 mm , AR = 2.203 × 10−11 mm−3 and AP = −0.01854.

Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 10 of 15

Table 1. MESSIER optical design parameters.

Telescope angular resolution 2 arcsec On-axis circular beam entrance 256 mm Focal length f’ 890 mm Focal-ratio f/Ω f/2.5 Deviation angle 2i 22°

M1 Elliptic clear aperture 2xm × 2ym 356 × 362.7 Angular FOV 2φmx × 2φmy 1.6° × 2.6° Linear FOV 25 × 40 mm2 M3 mirror curvature radius R3 −1769 mm

UBK7 ﬁlters & SiO2 cryostat—Thicknesses 2 & 5 mm

Detector ﬁeld curvature radius RFO −890 mm

Math. Comput. Appl.Figure2019 10., 24 ,Schematic 2 of Messier reﬂective Schmidt (M2 holed ﬂat not shown). 10 of 15

FigureFigure 11. 11.[ Left[Left] MESSIER] MESSIER layout layout reﬂective reﬂective TMA TMA in in the the symmetry symmetry plane. plane. [Right [Right] 3-D] 3-D view view with detectorwith detector (LAM). (LAM).

TheThe primary primary mirror mirror surfacesurface M1M1 can be de deﬁnedﬁned by by the the aspheric aspheric anamorphic anamorphic equation equation as derived as derived fromfrom Zemax Zemax optics optics ray-tracing ray-tracing code—which code—which isis somewhatsomewhat similar similar to to Equaiton Equaiton (2)—and (2)—and denoted denoted

2 2 𝑍 = Cxx + Cyy + 𝐴𝑅(1−𝐴𝑃)𝑥 + (1+𝐴𝑃)𝑦 , 2 2 2 (14) ZOpt = q () + AR[(1 − AP)x + (1 + AP)y ] , (14) 2 2 2 2 1 + 1 − (1 + Kx)Cxx − (1 + Ky)Cyy where now in all this section with Zemax code (y, z) is the symmetry plane of the telescope. whereRestraining now in all this to sectionthe case with of a Zemaxsimply code bi-curvature (y, z) is the surface symmetry for the plane first of quadratic the telescope. term, then settingRestraining Kx = K toy the= − case1, denominator of a simply bi-curvaturereduces to surfaceunity. The for theresult first quadraticfrom Zemax term, modelling then setting −6 −1 −6 −1 optimization provides the four coefficients, Cx = −3.598 × 10 mm , Cy = −3.467 × 10 mm , Kx = Ky = −1, denominator reduces to unity. The result from Zemax modelling optimization provides AR = 2.203 × 10−11 mm−3 and AP = −0−.01854.6 − 1 −6 −1 −11 −3 theMath. four Comput. coefficients, Appl. 2018C, x23=, x− FOR3.598 PEER× 10REVIEWmm , Cy = −3.467 × 10 mm , AR = 2.203 × 10 mm 11 of 15 and AP = −0.01854. TheThetotal total sag sag of of clear clear aperture aperturein inx x-direction-direction (i.e., (i.e., off-symmetry off-symmetry plane) plane) for forxmax xmax= = 178 178 mm mm is is then then ZZOptOpt−−maxmax ==− 35.7035.70µ m.µm.Zemax Zemaxiterations iterations lead lead to RMS to residualRMS residual blur images blur inimages agreement in agreement with predicted with angularpredicted resolution angular (Figureresolution 12). (Figure 12).

Figure 12. Telescope image quality—curved FOV. (a) Spot diagram. (b) Diffraction encircled energy. Figure 12. Telescope image quality—curved FOV. (a) Spot diagram. (b) Diﬀraction encircled energy. Results from Zemax modelling and spot diagram one shows that the maximum RMS blur Results from Zemax modelling and spot diagram one shows that the maximum RMS blur residuals is φZemax = 5.5 µm or dNC = 1.27 arcsec. From Equation (5) and a same diagonal semi-FOV residuals is ϕZemax = 5.5 µm or dNC = 1.27 arcsec. From Equation (5) and a same diagonal semi-FOV (1.52◦) and same parameters in Table2, one also obtains for MESIER proposal the theoretical angular ◦ resolution(1.52 ) and [4 ,6same], parameters in Table 2, one also obtains for MESIER proposal the theoretical angular resolution [4,6], 3 3 = + = dNC 3 i ϕm ϕm 1.29 arcsec, (15) 256Ω 2 𝑑 = 𝑖+𝜑 𝜑 =1.29 arcsec, (15) showing that results from modelled and theoretical angular resolutions are similar. showing that results from modelled and theoretical angular resolutions are similar. One of the key requirements was to provide a free distortion design. Over a diagonal field

of view of maximum radius 𝜑 =1.5° it was shown that for a curved FoV distortion 𝛥𝜑/𝜑m remains smaller than 2 𝗑 10−4 and thus is negligible. The telescope image quality is estimated through its spot sizes in the diagram. Diagram indicates an image quality convenient for seeing limited conditions and close to the diffraction limit. Contribution of residual aberrations is 1.3 arcsec RMS over the field of view. Residual spot radius values take into account a UBK7 colour ﬁlter 2 mm thick where a spectral band can be selected in a filter wheel with 5 wavebands. An optional 5 mm thickness SiO2 cryostat window is planned for optical design of the ground-based prototype telescope (Table 2).

Table 2. Image quality of the telescope with use of filters.

Waveband [nm] Spot RMS radius in the FoV centre [µm] Spot RMS radius at the FoV edge [µm] 350–410 1.8 3.7 400–560 2.3 3.8 550–700 1.7 3.4 680–860 1.7 3.3 860–980 1.7 3.3

4.2. Elasticity Modelling of a Freeform Primary Mirror The requested aspheric surface with non-rotational symmetry of the primary mirror surface refers to an optical surface also called a freeform surface. The present freeform surface for our MESSIER primary mirror is made of homothetic-ellipse iso-level lines. This surface is to be designed through active optics methods where the deformable substrate is aspherized by a plane surfacing under stress – an active optics method also called stress mirror polishing (SFP). The principle uses a uniform load applied and controlled inside a closed vase form whilst a plane polishing is operated with

Math. Comput. Appl. 2019, 24, 2 11 of 15

One of the key requirements was to provide a free distortion design. Over a diagonal ﬁeld of view ◦ of maximum radius ϕm = 1.5 it was shown that for a curved FoV distortion ∆ϕ/ϕm remains smaller than 2 × 10−4 and thus is negligible. The telescope image quality is estimated through its spot sizes in the diagram. Diagram indicates an image quality convenient for seeing limited conditions and close to the diffraction limit. Contribution of residual aberrations is 1.3 arcsec RMS over the field of view.

Math. Comput.Residual Appl. 2018 spot, 23 radius, x FOR valuesPEER REVIEW take into account a UBK7 colour filter 2 mm thick wherea 12spectral of 15 band can be selected in a filter wheel with 5 wavebands. An optional 5 mm thickness SiO2 cryostat a full-sizewindow tool. is planned The fi fornal opticalprocess design delivers of the the ground-based required shape prototype after elastic telescope relaxation (Table2 ).of the load [5,6,10,11]. The closed vase form—orTable closed 2. Image biplate quality—is made of the telescopeof facing with twin use elliptical of filters. vase form blanks in ZerodurWaveband assembled [nm] together Spot RMS at the Radius end of in their the FoV outer Centre rings [µ throughm] Spot a RMSlayer Radiusof 100–150 at the µ FoVm thickness Edge [µm] 3M DP490 Epoxy, where elasticity constants are Poisson’s ratio ν = 0.38 and Young’s modulus E 350–410 1.8 3.7 = 659 MPa [12]. Elasticity constants of the blanks in Zerodur are Poisson’s ratio ν = 0.243 and Young’s modulus400–560 E = 90.2 GPa (Figure 13). 2.3 3.8 Modelling550–700 with Nastran code led us to 1.7 make cross optimizations with Zemax code. 3.4 The intensity of uniform680–860 loading q and final geometry of 1.7 the closed vase form mirror substrate are displayed 3.3 by Table 3. The final860–980 finite element modelling by Pascal 1.7 Vola display the displacements (Figure 3.3 14) and stresses (Figure 15).

4.2.Table Elasticity 3. Uniform Modelling load ofand a Freeformgeometry Primary of closed Mirror vase form deformable substrate of MESSIER primary mirror.The requestedSubstrate in aspheric Zerodur surface vitro-ceramic. with non-rotational Mirror aspheri symmetryzation is achieved of the primary after elastic mirror relaxation surface refers from SMP method and full-size tool plane polishing. to an optical surface also called a freeform surface. The present freeform surface for our MESSIER primary mirror is made of homothetic-ellipse iso-level lines. This surface is to be designed through active 5 optics methods whereUniform the load deformable of constant substrate pressure is aspherizedq by = 0.687 a plane × 10 surfacing Pa under stress—an active optics methodAxial alsothickness called of stress parallel mirror plates polishing (SFP).t = The 18 mm principle each uses a uniform load applied and controlledInner axial inside separation a closed vaseof closed form whilst plates a plane polishing20 mm is operated with a full-size tool. The ﬁnal processOuter delivers thickness the required of closed shape form after elastic relaxation56 of mm the load [5,6,10,11]. The closedInner vase form diameters—or closed of biplate elliptic—is cylinder made of facing2a twinx × 2a ellipticaly = 356 × vase 362.7 form mm blanks in Zerodur assembled togetherCylinder at the end radial of their thicknesses outer rings through atx layer, ty = of18 100–150and 18.33µ mmm thickness 3M DP490 Epoxy, whereDistance elasticity between constants middle are Poisson’s surface of ratio platesν = 0.38 and Young’sℓ = 38 mm modulus E = 659 MPa [12]. Elasticity constantsInner of and the outer blanks round-corner in Zerodur radii are Poisson’sRC ratio = 8 mmν = and 0.243 16 mm and Young’s modulus E = 90.2 GPa (Figure 13).

Figure 13. Elasticity design of primary mirror substrate as a closed vase form or closed biplate made of Figure 13. Elasticity design of primary mirror substrate as a closed vase form or closed biplate made of two identical vase vitro-ceramic material linked together with epoxy. The radial thicknesses (tx, ty) two identical vase vitro-ceramic material linked together with epoxy. The radial thicknesses (tx, ty ) and height ` of the outer cylinder provides a semi-built-in boundary which somewhat reduces the and height ℓ of the outer cylinder provides a semi-built-in boundary which somewhat reduces the size of inner ring radii (ax, ay) with respect to that of clear aperture (xm, ym). NB.: From anamorphose size of inner ring radii (ax, ay ) with respect to that of clear aperture (xm, ym). NB.: From xm/ym = ax/ay = tx/ty = cos i. anamorphose xm/ym = ax/ay = tx/ty = cos i. Modelling with Nastran code led us to make cross optimizations with Zemax code. The intensity of uniform loading q and ﬁnal geometry of the closed vase form mirror substrate are displayed by Table3.

Math. Comput. Appl. 2019, 24, 2 12 of 15

The ﬁnal ﬁnite element modelling by Pascal Vola display the displacements (Figure 14) and stresses (Figure 15).

Table 3. Uniform load and geometry of closed vase form deformable substrate of MESSIER primary mirror. Substrate in Zerodur vitro-ceramic. Mirror aspherization is achieved after elastic relaxation from SMP method and full-size tool plane polishing.

Uniform load of constant pressure q = 0.687 × 105 Pa Axial thickness of parallel plates t = 18 mm each Inner axial separation of closed plates 20 mm Outer thickness of closed form 56 mm

Inner diameters of elliptic cylinder 2ax × 2ay = 356 × 362.7 mm

Cylinder radial thicknesses tx, ty = 18 and 18.33 mm Distance between middle surface of plates ` = 38 mm Inner and outer round-corner radii R = 8 mm and 16 mm Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW C 13 of 15

FigureFigure 14. 14. DisplacementsDisplacements of of M1 M1 prim primaryary mirror mirror substrate substrate as as a a closedclosed vase vase form form withwith FEA FEA Nastran Nastran code. code.All elements All elements are hexahedra. are hexahedra. Boundaries Boundaries are expressedare expressed at the at the origin origin of displacementsof displacementsx =x y= =y z = 0 =and z = freedom0 and freedom along along three radialthree radial directions directions in plane in planex-y both x-y atboth back at surfaceback surface of the of ﬁgure. the ﬁgure. Total axial Totaldisplacement axial displacement 69.2 µm (LAM). 69.2 µm (LAM).

FigureFigure 15. 15. StressStress distribution distribution of of the the closedclosed vase vase form form duringduring stress polishing. The maximum tensile stress stressarises arises along along the internal the in roundternal cornersround corners of the elliptical of the elliptical rings with ringsσmax with= 6.92 σmax MPa. = 6 For.92 MPa. Schott-Zerodur For Schott-Zerodurthis value is much this smaller value thanis much the ultimatesmaller than strength the ultimateσ = 51 MPa strength for a 1-month σ = 51 MPa loading for timea 1-month duration [4]. loadingOne may time notice duration that at [4]. the One symmetry may notice plane that epoxy at the link symmetry reduces toplaneσ ' epoxy1 MPa link (LAM). reduces to σ ≃ 1 MPa (LAM).

ρm. One has shown [4] that with a perfect built-in condition – that is for a closed vase form with moderate ellipticity and a ring of large radial thickness, say, at least five times larger than the plate axial thickness—the flexure provides an elliptic null-power zone radius ρ0 where the size of the clear 3/2 ≃ aperture radius ρm is in the ratio ρ0/ρm =3/2 ≃ 1.224. This means that the useful optical area is convenient for ρ ∊ [ρm, ρ0] but unacceptable for ρ ∊ [ρm, ρ0]. From our cross optimizations with Zemax and Nastran codes the ring of the closed vase form provides a noticeable decrease in flexural rigidity, which is equivalent to a semi-built-in condition. Then, compared to a perfect built-in assembly and from our results described in latter subsection, the radii ratio between null-power zone ρ0 and clear aperture ρm has become after optimized modelling

Math. Comput. Appl. 2019, 24, 2 13 of 15

The general law of algebraically balancing the second-order derivatives applies on principal directions of a pupil mirror—as stated by Equation (4)—for a circular or elliptical clear aperture. In dimensionless coordinates and any x or y orthogonal directions, if the clear aperture radii is denoted ρm, then we must obtain algebraically opposite local curvature values for ρ = 0 and ρ = ρm. One has shown [4] that with a perfect built-in condition – that is for a closed vase form with moderate ellipticity and a ring of large radial thickness, say, at least five times larger than the plate axial thickness—the flexure provides an elliptic√ null-power zone radius ρ0 where the size of the clear aperture radius ρm is in the ratio ρ0/ρm = 3/2 ' 1.224. This means that the useful optical area is convenient for ρ ∈ [ρm, ρ0] but unacceptable for ρ ∈ [ρm, ρ0]. From our cross optimizations with Zemax and Nastran codes the ring of the closed vase form provides a noticeable decrease in flexural rigidity, which is equivalent to a semi-built-in condition. Then, compared to a perfect built-in assembly and from our results described in latter subsection, the radii ratio between null-power zone ρ0 and clear aperture ρm has become after optimized modelling

ρ0/ρm = 1.118 (16)

This ratio is significantly smaller than 1.224. Then from Zemax ray tracing optimizations, we now take benefit of an important result: the angular resolution is not substantially modified in using the optical surface also up to the inner zone of the ring. The flat deformable surface of the closed vase form requires use of stress polishing by plane super-polishing, which then avoids any ripple errors of the freeform surface. Closed vase form Messier primary mirror can be aspherized up to its inner part of the semi-built-in elliptic ring without any lost in optical area and angular resolution.

4.3. Optical Testing of MESSIER Freeform Primary Mirror Optical testing of the freeform primary mirror must be a precise measurement because of its anamorphic shape. From Equation (14) the clear aperture shape presents a total sag of ZOpt−max = −35.70 µm for xmax = 178 mm in x-direction, that is in the off-symmetry telescope plane. Several optical tests could be considered mainly based on a null-test system [6]. For instance this involves lens compensators [13] and computed-generated holograms [14] and their combinations. We adopted a singlet lens compensator as component already existing at the lab and providing exactly the correct compensation level of the rotational symmetry mode, that is, 3rd-order spherical aberration compensation of M1 primary mirror. This lens, also called Fizeau or Marioge lens, is plano-convex made in Zerodur-Schott with following parameters, axial thickness tL = 62 mm, R1L = ∞, R2L = 1180 mm, DL = 380 mm, used on elliptical clear aperture 356 × 362.7 mm. The axial separation to primary mirror is 10 mm. Remaining aberration is then a balanced anamorphose term to be accurately calibrated by He-Ne interferometry (Figure 16). Another important feature of MESSIER telescope proposal is the selection of a curved detector, RFOV ' −f = −R3/2 (cf. Table1), which allows a distortion free design over an active area of 40 × 25 mm2. This technology is presently under development by use of either variable curvature mirrors (VCMs) or toroid deformed mirrors [4,5]. References can be found on curved detectors in Muslimov paper [15]. Math. Comput. Appl. 2018, 23, x FOR PEER REVIEW 14 of 15

ρ0/ρm = 1.118 (16) This ratio is significantly smaller than 1.224. Then from Zemax ray tracing optimizations, we now take benefit of an important result: the angular resolution is not substantially modified in using the optical surface also up to the inner zone of the ring. The flat deformable surface of the closed vase form requires use of stress polishing by plane super-polishing, which then avoids any ripple errors of the freeform surface. Closed vase form Messier primary mirror can be aspherized up to its inner part of the semi-built- in elliptic ring without any lost in optical area and angular resolution.

4.3. Optical Testing of MESSIER Freeform Primary Mirror Optical testing of the freeform primary mirror must be a precise measurement because of its anamorphic shape. From Equation (14) the clear aperture shape presents a total sag of ZOpt−max = −35.70 µm for xmax = 178 mm in x-direction, that is in the oﬀ-symmetry telescope plane. Several optical tests could be considered mainly based on a null-test system [6]. For instance this involves lens compensators [13] and computed-generated holograms [14] and their combinations. We adopted a singlet lens compensator as component already existing at the lab and providing exactly the correct compensation level of the rotational symmetry mode, that is, 3rd-order spherical aberration compensation of M1 primary mirror. This lens, also called Fizeau or Marioge lens, is plano- convex made in Zerodur-Schott with following parameters, axial thickness tL = 62 mm, R1L = ∞, R2L = 1180 mm, DL = 380 mm, used on elliptical clear aperture 356 𝗑 362.7 mm. The axial separation to primaryMath. Comput. mirror Appl. is2019 10, 24mm., 2 Remaining aberration is then a balanced anamorphose term to14 ofbe 15 accurately calibrated by He-Ne interferometry (Figure 16).

FigureFigure 16. 16. Null-testNull-test singlet-lens singlet-lens aberration aberration compensa compensatortor for MESSIER for MESSIER freefo freeformrm primary primary mirror. mirror. (a) Scheme(a) Scheme mounting mounting of the of plano-convex the plano-convex lens lens that that compensate compensate for forspherical spherical aberration aberration of ofthe the primary primary mirrormirror surface. surface. (b ()b Simulated) Simulated He-Ne He-Ne interferogram interferogram of of rema remainingining non-axisymmetric non-axisymmetric fringes fringes to to be be calibrated.calibrated. (c ()c Interferences) Interferences can can be be obtained obtained from from a atwo-arm two-arm interferometer interferometer (LAM). (LAM). 5. Conclusions Another important feature of MESSIER telescope proposal is the selection of a curved detector,

RFOV ≃Modelling −f = −R3/ of2 freeform(cf. Table surfaces 1), which by allows development a distortion of active free opticsdesign techniques over an oractivestress area mirror of 40 polishing × 25 mmprovides2. This extremelytechnology smooth is presently surfaces. under Generated development from elastically by use of relaxed either plane variable or sphericalcurvature polishing mirrors (VCMs)with full-size or toroid aperture deformed tools mirrorsthese surfaces [4,5]. References are free from can ripplebe found errors. on curved detectors in Muslimov paper Applied[15]. to MESSIER TMA telescope proposal the best angular resolution of a non-centred optical designs corresponds to a primary mirror freeform optical surface made of homothetic elliptical isolevel 5.lines. Conclusions It has been shown that this leads to algebraic balance of local curvatures – that is balance of second derivatives of the optical surface Modelling of freeform surfaces by development of active optics techniques or stress mirror polishing closed vase form providesBeside extremely the facility smooth to obtain surfaces. easily Generated a freeform from aspheric elastically from relaxed elastic relaxationplane or spherical of a polishing naturally withprimary full-size mirror, aperture MESSIER tools these proposal surfaces would are free also from benefit ripple from errors. free distortion because of its curved detector curvedApplied FoV. to This MESSIER field curvature TMA telescope shall be proposal associated the to best a angular resolution. of a non-centred optical designsAuthor corresponds Contributions: toE.M. a primary investigated mirror the telescopefreeform optical optical design surface with Zemaxmade raytracingof homothetic optimization elliptical and developed a useful optical quasi-null-test optimization for the freeform mirror. P.V. developed an accurate finite element analysis of the closed vase form mirror with Nastran code for stress polishing and elastic relaxation technique. G.R.L. established the law that applies for the optimal shape to be given freeform mirrors and freeform gratings of reflective Schmidt systems, developed a raytracing code with Seidel modes freeform surfaces, investigated the telescope optical design with Zemax code anamorphic surface optimizations, and introduced the concept of closed vase form mirror. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Wilson, R.N. Reflecting Telescope Optics II; Astronomy & Astrophysics Library; Springer-Verlag: Heidelberg, Germany, 1996. 2. Nelson, G.E.; Gabor, G.; Hunt, L.K.; Lubliner, J.; Mast, T.S. Stressed mirror polishing—2: Fabrication of an off-axis section of a paraboloid. Appl. Opt. 1980, 19, 2341–2352. [CrossRef][PubMed] 3. Su, D.-Q.; Cui, X. Active optics in LAMOST. Chin. J. Astron. Astrophys. 2004, 4, 1–9. [CrossRef] 4. Lemaitre, G.R. Astronomical Optics and Elasticity Theory—Active Optics Methods; Astronomy & Astrophysics Library; Springer-Verlag: Heidelberg, Germany, 2009. 5. Muslimov, E.; Valls-Gabaud, D.; Lemaitre, G.R.; Hugot, E.; Jahn, W.; Lombardo, S.; Wang, X.; Vola, P.; Ferrari, M. Fast, wide-field and distortion-free telescope with curved detectors for surveys at ultra-low surface brightness. Appl. Opt. 2017, 56, 8639–8647. [CrossRef][PubMed] 6. Lemaitre, G.R. Sur la résolution des télescopes de Schmidt catoptriques. C. R. Acad. Sc. Paris 1979, 288, 297. 7. Timoshenko, S.P.; Woinovsky-Krieger, S. Theory of Plates and Shells; Engineering Mechanics Series; McGraw Hill: New York, NY, USA, 1959; p. 310. Math. Comput. Appl. 2019, 24, 2 15 of 15

8. Lemaitre, G.R.; Grange, R.; Quiret, S.; Milliard, B.; Pascal, S.; Lamandé, V. Multi Object Spectrograph of the FIREBall-II Balloon Experiment. In Proceedings of the OSA Conference on Optical Fabrication and Testing, Kona, HI, USA, 22–26 June 2014. 9. Valls-Gabaud, D.; behalf the MESSIER Consortium. The MESSIER surveyor: Unveiling the ultra-low surface brightness universe. Proc. Int. Astron. Union 2016, 11, 199–201. [CrossRef] 10. Lemaitre, G.R.; Wang, X.; Hugot, E. Reflective Schmidt Designs for Extended Object Detection in Space Astronomy—Active Optics Methods. In Proceedings of the 9th International Conference on Optical Design and Fabrication, Tokyo, Japan, 12–14 February 2014. 11. Lemaitre, G.R. Aphérisation par relaxation élastique de miroirs astronomiques dont le contour circulaire ou elliptique est encastré. C. R. Acad. Soc. Paris 1980, 290, 171–174. 12. Nhamoinesu, S.; Overend, M. The mechanical performance of adhesives for steel-glass composite façade system. In Challenging Glass 3—Conference on Architectural and Structural Applications of Glass; Louter, B., Veer, N., Eds.; TU Delft: Delft, The Netherlands, 2012. 13. Malacara, D. Optical Shop Testing, 3rd ed.; Wiley: New York, NY, USA, 2007. 14. Larionov, N.P.; Lukin, A.V. Holographic monitoring of off-axis aspheric mirrors. J. Opt. Technol. 2010, 77, 490–494. [CrossRef] 15. Muslimov, E.; Hugot, E.; Ferrari, M.; Behagnel, T.; Lemaitre, G.R.; Roulet, M.; Lombardo, S. Design of optical systems with toroidal curved detectors. Opt. Lett. OSA 2018, 43, 3092. [CrossRef][PubMed]

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