Reflective Schmidt anastigmat telescopes and pseudo-flat made by elasticityGerard Lemaitre
Observatoire de MarseiFle, 2 Place Le Verrier, 13004-Marseille, Franceand Laboraroire dAstronomie Spatiale du Centre National de la Recherche Scienrfifque, AlIe& Peiresc, Les Trois Lucs,
13012-Marseille, France(Received 28 May 1976)
The recent need in space astronomy for wide-field telescopes having the best transmission and exempt fromthe chromatic aberration in uv, the severe conditions of the aspheric surface quality for the shortestwavelength, leads to new developments both in the theory and design of those surfaces. A two-surface all-reflective Schmidt telescope is analyzed (1) with Fermat's principle and (2) with Abbe sine condition. Adioptric elasticity method is presented to aspherize the primary mirror which is a nonaxisymmetric surface.Experimental results are given for an ff1.5, 180 mm focal length all-reflective Schmidt.
Considering an object at infinity, it is well known that,for any system of two separate coaxial mirrors, theSeidel spherical aberration and coma can be correctedby applying appropriate fourth-power asphericities. IfCl and C2 are the vertex curvatures of the mirrors ande the ratio of the axial separation to the focal length,one can define a class of aplanatic systems where theasphericity of each mirror depends on the parameter e.
In this class one can also define a subclass of two-mirror configurations for which Seidel astigmatism iscorrected along with coma and spherical aberration.Burch's' and Linfoot's2 investigations show that the mir-ror separation must be twice the focal length (e = 2) foran object at infinity. This range of two-mirror anastig-mats was described by Wynne 3 over a range of valuesof Petzval curvature p. If we define the Petzval curva-ture as positive when a surface is convex towards thedirection of propagation of incident light, the best knownand most often used configurations are l<p<m, curved-field Schwarzschild4; p= 1, Schmidt5 ; p = 0, flat-fieldSchwarzschild; p = - 1, monocentric Burch, 6
In the Schmidt system to be considered here, obtura-tion difficulties can be solved either (i) by folding thebeams with a tilted primary mirror, or (ii) by inter-changing the primary mirror with a refractive correc-tor. For these two possibilities, the vertex of the pri-mary component is at the center of curvature of thespherical secondary, according to the Seidel theorygiving the third-order anastigmatism.
In the Schmidt telescope configuration (ii), the re-fractive corrector will be achromatized giving a mini-mal sphero-chromatic aberration (Ross7 ). Ground-based astronomy widely uses this configuration forlarge and very large field surveys 8-10 and also for spec-trographic purposes for which variants such as solid,folded-solid, and semisolid cameras were developedP""2
In all cases the optical surfaces used belong to the so-called centered optical systems.
In configuration (i), the pseudo-flat primary becomesan off-axis mirror as previously proposed by Ep-stein.13 ' 5 This mirror must be a nonaxisymmetricalaspheric, and the optical surfaces used now belong tononcentered systems. But this configuration does notrequire any color correction, which conforms with thedesire for an exLension of the wavelength range into theuv in space astronomy (Courtes, Henize, Monnet 165 ' 8 ).Attainment of high resolution requires a more detailed
1334 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
investigation of the performance characteristics of theall-reflective Schmidt telescope' 9, 2-
In this paper we consider the fifth and the seventh-order aberrations of the all-reflective Schmidt tele-scope, known as an anastigmat in the third-order Seideltheory. The primary mirror is at the center of curva-ture of the secondary, and the resulting focus is situ-ated midway between the two mirrors.
The geometrical equations for the mirrors can be de-noted with conventional wvave-front aberration function2 2
as series
C = E Amnp" cosmt0, (1)
with mn and n integers, m _ n and m + n - I always odd,giving the order of aberration to be corrected,
Before folding the primary mirror we consider thecentered system (m =0) represented in Fig. 1, wherethe vertex of the primary is at the origin of the cylindri-cal coordinate system (D, p), the C axis being the axisof the primary. The resulting focal lengthf and theradius of curvature of the secondary are taken equal,respectively, to one and two unities.
I. HIGHER-ORDER STIGMATISM WITH SPHERICALSECONDARY
On axis, we derive the higher-power equation for theprimary taking the hypothesis of a perfectly sphericalsecondary. Let us call u the angle of the ray incidentat the focus and a the semideviation at the secondary(in the case of Fig. 1, u, a, u - a, and u - 2a arepositives), The condition for a sphere gives
- I 2sina = 2sinu (2)
If we call the distance FM'= l1 and M'M= 12, Fermat'sprinciple of constant light path requires that
11± 12+ C = constant = 3,
and by projection of Eq. (3) on the C and p axis,
C 12 COS (U -9a) -II cosu-1,
p = - 12 sin(u - 2ca) + 1 sinu .
(3)
(4a)
(4b)
The two lengths Z, and 12 can be given by the relations
Cop-right Q 1976 by the Optical Society of America 1334
p
-2 - 1 0
FIG. 1. Notation and basic geometry of an all-reflectingSchmidt system.
With polar coordinates (11, u) centered on the focus F,the equation for the secondary is
1/1l = ft+ (1 - t)2(1- t)I , (11)
so that
=-1 (1-2)(~ t+ 2(- t) )-l (12)
which leads to an aspherical mirror. The five firstA0 ,) coefficients giving meridional sections for the pri-mary and secondary, respectively, can be written fromEqs. (9), (10), and (12) as
g(0; 0; 1/26; 3/29; 3/210), (13a)
allowing one to write 11 and 12 as a function of the u pa-rameter
2 5 4 29 6 17.43 8 (6a+T. 6 - 5T 32. 2 9 u - 37 2 2 1 U, (a
19 2_ 1 4 6 7 6 44912=2 - 2 2 u - U U+ u +7 5 3 2 211 u * (6b,
The local equation for the spherical primary is, forp< 1,
-2 + 4 p E 4-i (7)
With the representation defined by Eq. (1), the twomirrors are represented by series whose Am, ,, coeffi-cients have m = 0 and n even. The five first AOon givingthe meridional sections for the primary and secondary,respectively, can be written from Eq. (6a) and (6b),and Eq. (7) as
t(0;0; 1/26;3/29; 5 x 32/214), (8a)
¢(- 2;1/22;1/26;1/29; 5/214) (8b)
With a 27r sr aperture angle at the focus (u= 7r/2), thecorresponding position and aperture radius for the pri-mary and secondary are, respectively,
g = 11i - 19,
p=7 VT- 11,and
C=- 1,
t(- 2; 1/22; 1/26; 1/29; 1/2"). (13b)
With a 27r sr aperture angle at the focus (u = 7r/2), thecorresponding position and aperture radius for the pri-mary and secondary are, respectively,
D = 1/32,
p=l,and
p = 12/7.
Comparing the two pairs of relations (8) and (13), onecan see that the stigmatic and aplanatic conditions leadto the same forms in the fifth-order theory. Similarly,for the secondary mirror, the departure from a sphereis given by the eight-power term AO8 , with AAO8 = 3/214.This difference is also the same for both primaries.Figure 2 gives an idea of the true forms of the geomet-rical ray traces and mirrors, first with the stigmaticcondition and second with the aplanatic condition (whichlimits the aperture angle to 271r sr).
In the fifth-order theory the only aberration is theelliptical astigmatism A24 p4 cos20; one can show thatA15=A 3 3 =0.
III. OFF-AXIS PRIMARY MIRROR
The primary mirror must be tilted so that the sec-ondary does not obstruct the light from the object field.
II. HIGHER-ORDER APLANATISM GIVINGNONSPHERICAL SECONDARY
For an object at infinity, the sine condition impliesthat each incident ray parallel to the axis intersects itsconjugate ray on a sphere of radius f = 1, which is cen-tered at the focus F. In addition to axial stigmatismwhich implies the absence of spherical aberration in allorders AO,,,, the circular coma must be corrected forall orders. This last condition determines all termsA,, 2,,1 in the wave-front function which depend on thefirst power of the off-axis distance. Hence, the Abbe sinecondition gives
p/sinu = constant = 1, (9)
which must be considered instead of Eq. (2). FollowingChretien2 3 and Baranne2 4 for a two-mirror aplanatictelescope, we denote t=sin2 (u/2) so that the exact sec-ond parametrical equation for the primary is
1 = 1J Opt(. c t) . V(1 . 66t)3(,N. 12, D m
1335 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
-2 -1 0
FIG. 2. Approximately true forms of the geometrical raytraces and mirrors, first (CW) with the stigmatic condition andsecond (C2) with the aplanatic condition. Abbe's sphere Z isrepresentated for an aperture angle of 27r sr.
Gerard Lemaltre 1335
I)
The minimum tilt angle (P is a function of the f ratioand of the nonvignetted field size. One must correct theoff-axis aberrations introduced by the rotation of theprimary, and therefore this primary must not be axial-ly symmetric.
Let T*(Ox*y*g*) be the trihedral for which the planeOx*y* is tangent to the primary mirror before the rota-tion (5 by the Ox* axis. The trihedral T(Ox y C) beingfixed to this mirror, the form 9M* of its surface beforethe rotation is given in the system of the trihedral T* by
FIG. 3. Notations used in the p rotation of the primary mir-ror.
For the odd compensations we can suppose, first ofall, that the primary mirror is a symmetrical aspheric,characterized by A0 4 = A&/cos p. Under these conditions,Eq. (15) shows that the first aberration is given by m 1and n= 7, which is a seventh-order circular coma. Todefine the coefficient A17 with the same degree of ap-proximation as previously, it is necessary to take into
6) account the inclination u - 2a of the marginal rays aswell as the resulting phase delay. In the plane of sym-metry Oy*g* this gives
y = ** - 4A& tan(Py ,,* .
The preceding equations give, for the circular comacoefficient,
A17 =A*216 tan(P.
IV. PSEUDO-PLANE PRIMARY MIRRORASPHERIZED BY ELASTICITY
In order to simplify the making of the primary mir-ror, which has a nonaxisymmetrical aspheric figure,we propose to use the dioptric elasticity method of as-pherization. Schmidt, who introduced the large-fieldreflecting telescope, was the first to have recourse toelasticity for the figuring of axisymmetrical aspher-ics,252 8 He developed his method entirely by experi-mentation, and in this manner produced a 30 cm diame-ter aspherized plate giving a correction for f/i, 75.Couder thoroughly analyzed the problem of elasticity30
applied to Schmidt's method. The dioptric elasticitymethod has since been used successfully by several op-ticians.3 1 3 3 We presently employ an elasticity methodwith two separate depressurized zones3 4'3 5 which hasrecently been applied to the construction of 50 and 62-cm-diam correcting plates for the Schmidt telescopesof the observatories of Lyon and Haute Provence.
It has been shown that the principle of elastic as-pherization of optics can be applied in a very generalmanner for the compensation of axial and off-axial aber-rations.3 6 Suppose that the primary mirror consists ofa circular plate having a thickness XC, this thicknessbeing axisymmetric but varying as a function of radiusp. Defining the flexural rigidity according to the con-vention of Timoshenko and Woinowsky-Krieger, 3 7
D = E3C 3/12(1 -v2),
In0
0COQ_
(14)
the value of the coefficient At,,2n behind given by the Eqs.(8a) or (13a).
In a first approximation, the equation of the inclinedprimary is, with respect to the initial trihedral T*,
9*= tan4pyM*+ +**/CoS 2(p. (15
The coordinate transformation equations being
x* =x,
y* = y cosp - C 8inqp, (16
* = y sino + D coso,
we obtain for the form of the mirror in the coordinatesystem of the trihedral T fixed to the mirror:
gM= 9**/Cosq p. (17)
For determination of the form of the folded primarymirror, we consider a circle of radius p in the planetangent to this mirror. A point on this circle is definedby the angle 0 with respect to the Oy axis (see Fig. 3).The coordinates p sinG and p cosO of this point projectinto the initial plane Ox*y* as p sinG and p cosG sine.Setting a = 2 sin2(, we can express the radius p* in thislast plane as
p*2
=p2
(1 - cos20 sin2
(p) = p2
(1 - a - acos20), (18a)
from which one derives, in a homogeneous approxima-tion,
Let us write A*in for the coefficients giving the stig-matism or the aplanatism of the centered system (cf.Secs. I and II). After the rotation by ( one obtains thenew coefficients from Eqs. (17) and (18),
One finds the coefficients for the even seventh-ordercompensations:
A04 = A(1 -sin3 ( + 3 sin4 )/cos (19a)
A2 4 = - A&4 sin2 2(1 2sin
2p)/coso, (19b)
A44 = A*04 8 sin4/cosp, (19c)
AO, = A&(1 - 3 sin 2 (p)/CoS p, (19d)
A, = - A&' sin2(p/cos 0, (19e)
(19f)
1336 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
M = Z Ao 2.p"*2nn=O, 4
(20)
(19g)
Ao8 =A*8/cos(p. (21)
G�rard Lemaltre 1336
where D is a function of radius in our case. E and vbeing the Young's modulus and the Poisson coefficient,respectively, we propose, for an elastic deformationt(p, 0) given according to Eq. (1), to search for thethickness profile SC(p) as well as the associated externalload configuration (functions of p and 0) that must beapplied to the plate to obtain this deformation.
In cylindrical coordinates, the radial and tangentialbending moments M1Zl and 91l, and the twisting moment9Re, are expressed by
Sp=-ZD[ap + V(l ap + pa l2)
1 a a2 1 a2
E= (I - Vpapa) D pair
(22a)
(22b)
(22c)
The radial and tangential shearing forces can be ex-pressed, respectively, as
20 X- a (vp
P6-sao0v2[Fp AF dmAdefdp
' I (pn,,)dp]do
[Jie + Ji edo]dp
pQde
p Jfllde
.pQp de
(23a)
(23b)
Let q be the exterior load applied by unit surface area,and p dO dp characterizing the elementary slice of thick-ness 3C. The equations expressing the equilibrium ofthe elementary slice are obtained by writing the equilib-rium equations of the torques with respect to the tan-gential axis (cot) and the radial axis (Uip) as well as theforces in the g direction (see Fig. 4), i. e.,
31,dp BP =P+ DP (R we-ap)
:::= - a + k"" I -2ap ?
q = - p [-p (pt ) , md v
(24a)
(24b)
(25)
I
-m,, dp
(M ', + PO do] dp
Each term of the aberration wave function being char-acterized by a pair (m, n), we attempt to define the con-ditions of a flexure C of the form pI cosmO. This hy-pothesis permits the calculation of the three moments(22a-c) as functions of v, m, n, and D. Next we con-sider a plate for which the flexural rigidity D has rota-tional symmetry and varies with radius according tothe expression[te + &emide]dp
J
D= E Ai p1i,J=1
(26)
where Ai and Cal are constant coefficients dependinguniquely on the external load q. Setting Am,,n= Al A,,m,,and introducing the rigidity defined by Eq. (26) in theEq. (22a-c) for the moments Mp, M,7, and Rp9, theshearing forces 9p and 6e are given by
!p= - tj{(n- 2)(n?- m)-aoe[n(n-l1)+ v(n- m2)]}
xAi, p' 3-'i cosmo,
.9 =_ - ~ 2_ M[ 2 - ma - (n- 1)(1 - V)]f=1
XAmnpnl-3i sinmO.
pQ do
Qdp 3
FIG. 4. Equilibrium of the elementary slice with respect (1) tothe tangential axis wt giving 2p, (2) to the radial axis wp giv-ing 9.,, and (3) to the axial axis wz giving q.
1337 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
(27a)
(27b)
Substituting the shearing forces in the equilibriumEq. (25), we find for the external load,
FIG. 5. Configurations of thickness 3C04 and loads F0 giving thecorrection of the third-order spherical aberration.
As a first approach to the best form to give to theprimary mirror, we have to search to elastically com-pensate for (i) the third-order spherical aberration A04
and (ii) the fifth-order elliptical astigmatism A24, whichare the two most important aberrations (fifth- andseventh-order spherical aberrations can also be totallyeliminated in a next stage).
(i) For the third-order spherical aberration case,we have m = 0 and n = 4. Considering only the simplestsolutions a,, we retain that case for which the externalload q is zero. This solution, which corresponds to acentral force F0 in reaction pushing against the periph-ery of the mirror, is obtained by finding the roots ofthe trinomial coefficient of A104 of Eq. (28). For q=Owe find a1 = 8/(3 + v) and a2 = 2, which leads to thicknessof the form
-C 4 = (l/pB/ 32
+') - I/p 2 )1 3 (29)
This thickness is zero at the mirror's edge and in-finite at its center since the paraxial curvature is in-variant with respect to flexure (Fig. 5), In practice thecentral thickness can be reduced to a finite value in ac-cordance with an angular resolution criterion or using acriterion like that of Mar6chal 38 concerning a given val-ue of the normalized intensity at the diffraction focus,
(ii) For the fifth-order elliptical astigmatism casewe have m = 2 and n = 4, Considering, as previously,the case of a zero external load (q = 0), the roots of thecoefficient of A024 are a1 =3+ v and a 2 =0. The thick-ness is then of the form
FIG. 6. Configurations of thickness JC24 and loads F 2 cos 20giving the correction of the fifth-order elliptical astigmatism.
1338 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
3
FIG. 7. Interferograms at 632.8 nm of the elastic aspheriza-tion with respect to a plane, obtained (1) in the case of Fig. 5,(2) in the case of Fig. 6 with a thickness xk04 instead of 3c24 ,and (3) with superposition of (1) and (2).
(30)
and the astigmatism correction is obtained by exertingalong the periphery of the mirror a line load p definedby the shearing force 94 for p= 1 of the relation (27a).This load, of the form p =F2 cos 20, is represented inFig. 6.
The evolution of the profiles S 0 4 and JC24 being com-parable to within a few per cent, the compensation ofaberration (i) and (ii) can be effectuated by superposi-tion of the two preceding systems of forces by choosing,for example, the thickness profile SC04 as can be seenin the interferogram of Fig. 7.
V. A NUMERICAL EXAMPLE
Anf/i. 5 Schmidt reflecting telescope was constructedfor the FAUST rocket program1 s for the Centre Nationald'Etudes Spatiales and the Laboratoire d'AstronomieSpatiale du CNRS. The focal length is 180 mm and theangle of inclination of the primary mirror is q = 15°,resulting in a 30° beam deviation at the field center.Detector acessibility was facilitated by a plane mirrorwhich reflects the focal surface to the outside of the in-strument (Fig. 10), To minimize the central obstruc-tion, the pupil was located on this plane mirror. Thisarrangement requires anf/1 primary mirror.
The coefficients A,,, defining the surfaces of the pri-
TABLE I. Values of the A,,,,n coefficients for the primary mir-ror.
A04 = 0. 0151198 A 26 = - 0. 0006095
A24 =-0. 0010473 A 17= 0.0010467
A440 = 0.0000091 A 08 = 0. 0028435
AO= 0.0054565
Gerard Lemaitre
E24 = (l/p3+v - V13,
1-4 A1o'- rd 04 A24 A06 A26 A05 A44 A17
FIG. 8. Ray traces evolution forf/1.5 and 6 15' of the on-axis image when adding successive coefficients A.,,. With the sevencoefficients given in Table I for the primary mirror, the angular image sizes are Ax/f= 3. 8 10- and Ay/f= 5. 7 10-6 (i. e., 1arc sec).
mary and secondary mirrors have been calculated ac-cording to the preceding theory [Eqs. (8a, b) and (19a-g)] for a 15° primary inclination, and for a purelyspherical secondary mirror. The value of these coeffi-cients is given in Table I A computer program wasdeveloped to introduce each component Am,, separately.
One can see in Fig. 8 the evolution of the on-axis imagewith the introduction of successive Am..
For unit focal length the best focal surface is givenwith respect to T* by
*=l- 1++ P*2+1p* cosO*, with 4 = 15° . (31)
On this surface the image quality along a 20 50 semi-field is given in Fig. 9 for four angular sections.
The aberration compensation was done in situ by de-forming the primary mirror of the prototype telescope.The fifth-order astigmatism A 24 , resulting from theA04 p
4 correction, is corrected by the superposition of aflexure exerted at four points of a relatively rigid ring.The four forces associated with these points are decom-posed into two identical forces applied to the extremi-ties of the section Oy push in the negative C direction,
FIG. 9. Ray traces of four angular sections along the focalsurface. The spot diagrams are given for 1.25' and 2.5'from the axis. The image sizes in the field must be consid-ered as maximum values. For a given field, the optimizationray tracing process leads to better performance at the edge ofthe field.
1339 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976
i0 i1 i2 i3 i4 i5
FIG. 10. Reduction of the on-axis aberrations of anf/l. 5 all-reflective Schmidt telescope (f=180 nm) with a 30' beam de-viation. Starting from the paraxial image io (which is reducedby a factor 2 in the figure), one can see various stages the re-duction of spherical aberration (it, i3, i4) and astigmatism(i2 , 4)-
Gerard Lemaitre
and two identical forces applied to the extremities ofthe section Ox pull in the positive D direction. The ringis connected to the edge of the mirror by a collarettegiving a faint built-in moment, thus approximating afree support. The rigidity of this ring allows us to ob-tain a high continuity of displacement in cos2G at theedge of the mirror.
The various stages of the reduction of the axial imageare shown in the photograph of Fig. 10, The resolutionwas also tested by Cohendet39 by means of numerous USAFresolution targets placed in the focal surface of a col-limator. The on axis resolution was found to be atleast 250 line pairs/mm, Over a field of 2° radius itwas at worst 180 line pairs/mm.
VI. CONCLUSION
The optimization ray tracing process was not con-sidered in the last example. For a given field, thistechnique will give a significant gain in image quality.
The dioptric elasticity method developed is foundedon the weak deformations of thin plates. This hypothe-sis is in agreement with the amplitude of the aberrationsthat must be corrected for optical designs slower thanf/1. Wherever possible, this principle should be ap-plied to the figuring of aspherical surfaces in general.The use of this method requires, of course, a fairlyelaborate apparatus, but this is a small price to pay inreturn for great improvements in profile continuity,i. e., in detection and resolution. The quality of theresults confirms the possibility to extrapolate the newmethod to large space telescopes. Presently, an aper-ture of 60 cm can be obtained without new difficultieswith a titanium-glass component. Recent advances incopying technique (Dakin and Loewen)40 could also allowthe duplication for mirrors of 40 cm aperture.
ACKNOWLEDGMENTS
I wish to thank G. Court6s who greatly supported thiswork by making the facilities of the Laboratoire d'As-tronomie Spatiale available for the first practical trials,and G. Monnet who encouraged this project in manyways. M. Cohendet was responsible for the opticaltests of the first prototype and greatly aided with manyof the initial practical problems. J. Caplan translatedthis article into English.
'C. R. Burch, Proc. Phys. Soc. London 55, 6, 433 (1943).2 E. H. Linfoot, Recent Advances in Optics (Clarendon, Ox-
ford, England, 1955), p. 277.
3C. G. Wynne, J. Opt. Soc. Am. 59, 5, 572 (1969).4K. Schwarzschild, "Untersuchung zur geometrischen Optik,"Abhandlungen der KUnigl, Gessellschaft der Wissenschaftenzu GCttingen, Math. -Phys. Kl. 4, (1905-1906), Nos. 1, 2, 3.Other references in Wynne paper (Ref. 3).
5B. Schmidt, Zentral Zeit. f. Opt. Mech. 52, 2 (1931).6C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).TF. E. Ross, Astrophys. J. 92, 400 (1940).8J. G. Baker, Proc. Am. Philos. Soc. 82, 323 (1940).9R. D. Sigler, Appl. Opt, 14, 2302 (1975).10 G. Lemaitre, Astron. Astrophys. 44, 305 (1975).11 D. 0. Hendrix and W. H. Christie, Sci. Am. 8, (1939).12 M. Migeotte, Ciel Terre 64, 145 (1948).13 L. C. Epstein, Publ. Astron. Soc. Pacific 79, 132 (1967).1 4L. C. Epstein, Sky Telesc. 33, 204 (1967).15L. C. Epstein, Appl. Opt. 12, 4, 926 (1973).16 G. Courtes, "New Techniques in Space Astronomy, "' Sympo-
sium IAU N'41, pp. 273-301, Eds. Labuhn and LUst (1971).17 K. Henize, "The Role of Surveys in Space Astronomy, "' in
Optical Telescope Technology, NASA-SP-233 (U. S. Govern-ment Printing Office, Washington D. C. 1970).
'HG. -Monnet, R. Zaharia, and G. Lemaitre, "ProgrammeFAUST," Publication C.N.E. S., FAUST-PJ-02-70 (1970).
19 D. E. Oinen, "A Correcting Surface for Two Surface All Re-flecting Schmidt Lens, "' Publ. Eastman Kodak Company, 901Elmgrove Road, Rochester, N. Y. 1"650.
20D. Korsh, Appl. Opt. 13, 9, 2005 (1974).21G. Lemaltre, C. R. Acad. Sci. Ser. B 276, 145 (1973).22 M. Born and E. Wolf, Principles of Optics (Pergamon, New
York, 1959), p. 210.23 H. Chr6tien, in Calcul des Combinaisons Optiques (J. and
R. Sennac, Paris, 2958), p. 377.25A. Baranne, "Le T6lescope Ritchey-Chretien de 3, 60m."
(1932).26 R. Schorr, Mitt. Hamburg Sternw. Bergedorf 7, 42, 175
(1936).27R. Schorr, Z. Instrum. Kde, 56, 336 (1936). Another paper
(Ref. 28) gives details on the elasticity method that was usedby Schmidt.
28R. Schorr, Astron. Nachr. 258, 45 (1936). This paper wastranslated into English by N. U. Mayall (Ref. 29),
29N. U. Mayall, Publ. Astron. Soc. Pacific 58, 282 (1946).30A. Couder, C. R. Acad. Sci. Paris 210, 327 (1940).3"B. A. J. Clark, J. Astron. Soc. Victoria 6, 76 (1964).32 E. Everhart, Appl. Opt. 5, 713 (1966).33 F. Cooke, Appl. Opt. 11, 222 (1972).34 G. Lemaitre, Appl. Opt. 7, 1630 (1972).35G. Lemaitre, Astron. Astrophys. 44, 305 (1975).36G. Lemaltre, Nouv. Rev. Opt. 5, 361 (1974).37S. Timoshenko and S. Woinowsky-Krieger, "Th6orie des
