## Schmidt Correctors and Diffraction Gratings Aspherized by Active Optics

5.1 Various Types of Aspherical Schmidt Correctors

Active optics methods were originally applied to the figuring of refractive Schmidt correctors. Further developments followed with the development of reflective correctors and reflective-diffractive correctors. The various ways to obtain each of the three corrector types are different.

1. The methods developed for refractive plate correctors have been called elastic relaxation figuring or more commonly stress figuring.

2. Two different methods are practicable for obtaining reflective correctors, namely stress figuring and in-situ stressing.

3. Starting from a plane or a spherical grating, there are also two methods of obtaining reflective aspherized gratings: duplication on an active submaster and then stressing and second duplication (more commonly used), or simple duplication and stressing.

The main feature of these methods is the inherent smoothness of the aspherized surface that is obtained directly from spherical or plane surfaces. This avoids the high spatial frequency errors which are a signature of the classical zonal retouch process. Active optics methods allow obtaining telescopes and astronomical instruments of the highest intrinsic image quality.

5.2 Refractive Correctors

5.2.1 Third-Order Optical Profile of Refractive Correctors

The exact aspherical shape of the refractive corrector plate is represented by (4.11) and Table 4.1. The Kerber condition (4.14) of minimizing the spherochromatic variation, k = 3/4, gives the location of the null power zone at ro/rm = %/k = a/3/2. The chromatic dispersion is the highest at the radial zones rm/2 and rm, corresponding to maximal local slopes of opposite sign (Fig. 5.1).

G.R. Lemaitre, Astronomical Optics and Elasticity Theory, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-68905-8.5, © Springer-Verlag Berlin Heidelberg 2009

Fig. 5.1 Aspheric surface of a refractive Schmidt corrector plate. Kerber's condition k = 3/4 minimizes the axial spherochromatism. Optical shape Zopt 2kp2 — p4 = 3p2/2 — p4 where P = r/rm G [0,1 ]. The power of the plate is positive on-axis and null at p = yfk = 0.866. This condition provides the balance of the first derivative extremals in setting opposite maximum slopes at half-aperture p = 1/2 and clear aperture edge p = 1

Fig. 5.1 Aspheric surface of a refractive Schmidt corrector plate. Kerber's condition k = 3/4 minimizes the axial spherochromatism. Optical shape Zopt 2kp2 — p4 = 3p2/2 — p4 where P = r/rm G [0,1 ]. The power of the plate is positive on-axis and null at p = yfk = 0.866. This condition provides the balance of the first derivative extremals in setting opposite maximum slopes at half-aperture p = 1/2 and clear aperture edge p = 1

Let us represent the aspherical shape in third-order with A2, A4 coefficients in (4.11) reduced to their main part, namely A2 ~ M, A4 ~ — 1 /4 and under-correction parameter also reduced to s = 1. Denoting Q = f/d = R/4rm the telescope f/ratio and p = r/rm a dimensionless radius, theKerber condition k = 3/4 entails M = 3/27Q2 from (4.8). In this approximation, the refractive surface is

Zopt -- 28^—103 Q p2 - p4) rm with 0 < p < 1. (5.1)

5.2.2 Elasticity and Circular Constant Thickness Plates

With the case of refractive corrector plates, it is appropriate to consider the bending of constant thickness plates. The elasticity theory of small deformations of a circular plate when stressed by a uniform load q is due to Poisson [28]. The flexure ZElas is derived from Poisson's fourth-derivative equation v2V2ZEiaS - q/D = 0,

with the Laplacian and the flexural rigidity

where t, E, and v are the plate thickness, Young modulus and Poisson's ratio, respectively.

The general solution is represented by

Zßias = (q/64D) r4 + Ci r2 ln r +(C2 - Ci) r2 + C3 ln r + C4, (5.5)

where C1j2i3j4 are constants depending upon boundaries.

The radial stress or and tangential stress ot are derived from the corresponding bending moments Mr and Mt f d2Z v dZ\ ^ N

i dZ d2Z

The maximum values of stresses or and ot allow comparisons with the ultimate strength oult of the material for the validity in the application of active optics methods.

5.2.3 Refractive Correctors and the Spherical Figuring Method

B. Schmidt used the classical method by zonal retouch for making his corrector plate (in Chap. 4 [8]).1 The handwritten elasticity formulas found in Schmidt's personal papers all concern the flexure of beams.2 Though these formulas are not directly applicable to circular plates, he clearly emphasized that a much smoother profile could be obtained by surfacing a plate bent by partial vacuum while supported at its edge. Proposed by Schmidt as an aspherization concept for obtaining the best profile continuity, the elastic relaxation method - or stress figuring method - is now widely known as the basic method of active optics.

The active optics aspherization proposed by Schmidt only requires use of a full-size spherical tool and therefore must reject any local retouch. The theoretical problem was fully solved by Couder in 1940 [6], although he did not apply the method. The method was re-suggested for development by Chretien [3] who pointed out the advantage of accurate optical figures. Finally, Schmidt's spherical figuring method seems to have been first applied by Clark in 1964 [4] for a small plate and by Everhart in 1966 [8] in aspherizing a 29-cm clear aperture plate. In 1971, an

1 From A.A. Wachmann private communications to Erik Schmidt, it appears that B. Schmidt probably did not use the stress figuring technique for making his corrector plate although the plate thickness was thin enough for this to work.

2 B. Schmidt's personnal scientific papers were given to his nephew E. Schmidt in Palma de Mallorca by A.A. Wachmann, astronomer of Hamburg Observatory. By courtesy of E. Schmidt these papers were consulted by the author.

attempt was made for a 53-cm diameter plate [29]. However, this method presents some difficulties on realizing an accurate edge support and on controlling the tool curvature. Another active optics method which uses a plane figuring tool is now widely preferred (see next section).

The long delay between Schmidt's idea and its application is probably due to technical difficulties such as, for instance, the availability of finding an accurate enough pressure controller. The very simple process for aspherizing the corrector is to apply a partial vacuum under all the surface of the plate, which is in reaction at its edge, while figuring it with a tool of convenient curvature (Fig. 5.2).

The plate of thickness t is supported in reaction around the edge by an optically flat rim belonging to a bowl that can be rotated. The air is partially evacuated to a pressure p under the plate which is subjected to a uniform load q = po - p, where po is the atmospheric pressure. For a plate without hole on which a uniform loading is applied from center to edge, the constants C1 and C3 vanish in (5.5), as for C4 by choosing the coordinate origin at the plate vertex. Assuming that the plate edge rm is simply supported, and using the dimensionless radius p = r/rm, the elastic deformation is

ZElas = Et3{2l+V p2 - p ) rm ' 0 < p < 1 (5.8)

The accessible surface of the plate is ground and polished using a convex spherical tool.

Denoting m = rm/2RTool, where Rtoo1 is the radius of curvature of the tool, the equation of the just polished surface - still deformed by depression - is now that of the sphere

Zsphe = m(p2 + m2p4) rm with 0 < p < 1. (5.9)

When the plate is elastically relaxed by opening the bowl to the atmosphere, the rear surface will revert to a plane. The two surfaces of the plate have to be polished, so that one or both may be aspherized. Denoting p= 1 for one aspher-ized face and p = 2 for two surfaces having half asphericity on each, the front surface will become aspherical in a Kerber profile obtained from the active optics co-addition law

From identification of the coefficients in p2 and p4, the two parameters, RTool and t can be solved, leading for m, to the following third-degree equation

This equation always has a unique and positive real root. This root is much smaller than unity since Q2 > 2 and 0 < v < 1/2, thus, m3 is negligible. This is equivalent to saying that the coefficient of p4 in the expansion of the sphere is negligible relative to the coefficient of p4 of the elastic deformation. Since RTool/R = 1/8mQ, the radius of the spherical tool and the thickness of the plate can be expressed as functions of the mirror radius of curvature R

These relationships fully define the aspherization conditions. For example, with v = 1/5 and N = 3/2, the first one gives RtooI = 4.174pQ2R.

In order to avoid the breakage - implosion -, the radial and tangential stresses must be compared to the ultimate stress cult. These stresses are maximal and equal at the center of the plate,

96 Q2

2Ouit/3, showing that the highest asphericity is provided by the lowest possible loading. In a basic practical case of loading at q = 0.85 bar, i.e. allowing Sphe 3 corrections by under or over pressure during the end of the polishing, a conservative rule is to work at a stress value of half or two-thirds of cult. For instance, with fused silica (N = 1.48, v = 0.17, E = 77.5GPa, ffult ~ 76MPa), a ffult/2 working stress allows obtaining plates at an f-ratio Q ~ 1.78 if only one face is aspherized (p=1) or at f/1.77x2-1/3 = f/1.41 if both faces are figured with half asphericity on each (P= 2).

The imperative condition is that the supporting rim must define a plane to within a quarterwave geometry as well as for the edge of the plate which is the most difficult to achieve. In practice, the plate has to be somewhat larger than the rim size which would lead to a small correction on RTool in the above formula. The problem can be perfectly treated taking into account the small bending moment at rm due to the plate extra-radius exceeding the rim radius. This extra-radius of the plate introduces only a small and pure variation of curvature in the p2 term of ZElas i.e. has no effect on the p4 term. The previous theoretical curvature mode Cv 1 can be recovered by applying a small uniform load to the zone outside the rim.

5.2.4 Refractive Correctors and the Plane Figuring Method

In the method proposed by Schmidt, the main inconvenience is the control of the tool curvature in order to achieve a correct positioning of the null powered zone ro. Another difficulty is obtaining an axisymmetric geometry within a quarterwave criterion on the very edge of the plate. A final drawback comes from the figuring tool which has a different curvature each time one makes a corrector plate of different f-ratio. All these difficulties vanish with Lemaitre's two-zone stress figuring method [11-13,15] where flat tools of full aperture size are used and can be easily controlled by a pressing process onto a flat reference blank during the night. Compared to (5.10), the active optics co-addition law is now trivial,

RTool = <*>, Zsphe = 0 and Zopt + pZmaS = 0. (5.14)

The plate of outer radius r = b is supported on a metal ring at the clear aperture radius r = a = rm, which divides the surface into two concentric zones. A uniform load q1 is exerted on the inner zone, and a higher load q2 - close to the atmospheric load - is applied on the outer zone in the same direction as q1 (Fig. 5.3).

While the plate is under load, its outer surface is figured with a flat full-aperture tool. A peripheral ring connected to the support aids in centering the plate and also assures an air-tight seal by means of an o-ring or a modelling paste. This seal touches neither surface of the plate but only its edge, exerting a negligible radial force, so the edge of the plate is free to move axially.

Denoting a = rm the radius of clear aperture, p = r/a the dimensionless current radius, the ZElas solutions of inner zone zi, and outer zone z2, are derived from the general solution (5.5),

Fig. 5.3 Active optic aspherization of a refractive corrector plate by Lemaitre's two-zone method for plane figuring

Fig. 5.3 Active optic aspherization of a refractive corrector plate by Lemaitre's two-zone method for plane figuring

Z1 = (qxa4/64D)p4 + Xx p2 + X2, 0 < p < 1, (5.15a)

Z2 = (q2d4/64D)p4 + X3p2lnp + (X4-X3)p2 + X5 lnp + X6, 1 < p < b/a,

where the integration constants X1,...6 are determined from the continuity and boundary conditions.

Continuity conditions at the ring support:

{zi = 0 origin of deformation (5.16a)

z2 = 0 origin of deformation (5.16b)

d2z\/dp2 = d2z2/dp2 local curvature (5.16d)

Free edge boundary conditions:

dV2z2/dp = 0 shearing force (5.16e)

When solving the unknowns, the three parameters, Poisson's ratio v, radius ratio b/a and load pressure ratio n = q2/q1, define the profile geometry. For instance, with v = 1/5 and n = 1, a flexure array of ZElas(b/a) represents z1 and z2 for various b/a ratios (Fig. 5.4).

Since X1 is directly obtained from identification with (5.14), the plate thickness is not depending on q2 but only of q1, so that (5.12) becomes

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