Centering the secondary mirror of the Tirgo telescope

Silvio Guidarelli

A procedure for centering the secondary mirror of a Cassegrainian telescope on its mechanical mount hasbeen devised, using the classical principle of observing the movement of a reflected image when the mirroris rotated. The centering is obtained by iteration, keeping the reflecting surface in contact with a circularcentered edge. The achievable accuracy, accounting for the change in the hyperboloid shape of the mirrorfrom a sphere, has been calculated; the theoretical values were checked using the procedure for centeringthe secondary mirror of the Tirgo infrared telescope.

1. ProcedureThis paper refers to a technical problem in the design

and construction of the Tirgo infrared telescope, nowbeing put into operation by the Italian Group for Cos-mic Physics (GIFCO) at Gornergrat. 1

The problem is how to position the optical axis of thesecondary mirror of the telescope, whose meridianprofile is a hyperbola.

To position the optical axis means that the lattermust coincide in tilt and transversally within prescribedtolerances, with a suitable mechanical axis that can bethe spindle of a precision lathe. This then allows themetallic plate, to which the mirror is fastened, to beadjusted. The task of centering is related only to thisplate, which in its turn is connected to the other me-chanical parts of the telescope. The responsibility ofthe final centering is thus transmitted to the accuracyof these mechanical parts.

The best way to achieve this is to extend the classicalprocedure called catadioptric centering,2 normally usedfor elements of high class lenses. In this case it shouldrather be called catoptric centering, since it deals withonly one reflecting surface.

This operation is guided by the image of a targetproduced by a beam after reflection on the mirror, afterreflection and refractions in the case of a lens. Theimage is then observed through an eyepiece or othersuitable optical device, relating it to a reticule.

When the mirror together with the spindle is rotated,the image will move. Acting micrometrically on the

The author is with Istituto Nazionale di Ottica, Largo EnricoFermi-6, 50125 Arcetri-Firenze, Italy.

Received 8 December 1979.0003-6935/80/152520-04$00.50/0.© 1980 Optical Society of America.

position of the mirror with respect to the spindle, themovements of the image are reduced by successive at-tempts.

As all influences on the ray path are stationary, exceptthat of the mirror, the image remains motionless onlywhen the mirror rotates around its axis of symmetry.Thus the accuracy reaches the optical limit. At thispoint the metallic plate fastened to the mirror is ma-chined to obtain a plane surface, normal to the opticalaxis of the mirror, as well as a small cylindrical surface,which gives the transversal position of the axis itself.

There are four degrees of freedom: two in rotationsince the one around the axis itself, which has to be lo-calized, has no influence; and two in translation, as theone along the same axis is of no interest here. As aconsequence the adjustments necessary to bring themirror into a centered position can present some diffi-culty.

To simplify the operation: in the classical procedureof catadioptric centering, one of the two spherical facesof the lens is kept in contact with a mechanical circularedge, i.e., intersection between a plane and a cylindricalsurface, both mandreled, so that centering is certain.Obviously the edge rotates with the lens. At this pointonly two degrees of freedom are left, namely, freedomto slide on the contact, to bring the curvature center ofthe other face onto the mechanical axis.

Transferring the procedure to the case of the hyper-bolic mirror, an analogous contact seems to be requiredto restrict the degrees of freedom to two, leaving onlythe adjustments, sliding on the contact, to bring thevertex of the hyperboloid onto the mechanical axis (Fig.1).

It does not matter whether, before bringing the vertexof the hyperboloid onto the mechanical axis, this contactcannot be completed. Indeed a circumference, such asthe mechanical edge, can belong to a generic surface of

2520 APPLIED OPTICS / Vol. 19, No. 15 / 1 August 1980

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rotational symmetry only if it is centered. Neverthelessin this case the deviations are very slight.

However, when the vertex of the hyperboloid isbrought onto the mechanical axis, thus attaining acomplete contact, the position of the mirror will becorrect also in tilt, owing to the contact itself. If themirror were spherical such a contact alone, in this casealways complete, would ensure centering. The surfacecould slide on itself with the same effect as if it werestationary. Thus the image would remain motionless.Instead, if the hyperbolic mirror is rotated around themechanical axis not passing through the vertex butnever losing the contact, the distance from the opticalaxis to the area of reflection changes. Therefore thedirections of the normals change inasmuch as the sur-face differs from a spherical one, and then the imagemoves.

Therefore adjustments to bring the vertex of hyper-boloid onto the mechanical axis are all the more un-certain the less the surface of the mirror deviates froma spherical surface. It is worthwhile to calculate theamount of these deviations so that the accuracy can beknown.

11. Asphericity of the Reflecting SurfaceThe normal equation of the hyperbola, the meridian

section of a reflecting surface, is known, so the sag canbe easily computed for a number of points at variousdistances y from the optical x axis. That axis is thetransverse one for the hyperbola, i.e., the symmetry forthe hyperboloid.

The tilts of the normals with respect to the x axis giverise to the directions of the reflected rays; they are thetilts of the tangents with respect to the y axis, namely,the first derivatives of x (y). Finally their variations canbe obtained as second derivatives.

As can be deduced from what has been said above, theabsolute values of the variations are not of direct in-terest, but the differences between these variations andthe corresponding ones of a spherical surface are in-teresting.

It seems suitable to choose as a reference sphericalsurface that one tangent to the hyperboloid along thecircumference of contact, as all the normals to the re-flecting surface along that contact define the center of

Fig. 1. Layout of centering: M = reflecting sur-face; V = vertex of the hyperboloid; Ct = center oftangential curvature; Cr = center of radial curvature;P = a point of contact on the mechanical circularedge; OP = radius of the mechanical circular edge;t = tangent in P; n = normal in P; H = lathe head;S = lathe spindle; L = lens of the autocollimator; E

= eyepiece; T = target; D = splitter; G = reticule.

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the sliding movement. Therefore the radius of thisspherical surface does not coincide with the curvatureradius of the hyperbola on the point of contact, i.e., theradial curvature of the hyperboloid, whose center liesoutside of the axis. This contrasts with the necessaryaxial symmetry, that is to say, with the necessary con-tact along the entire mechanical circular edge.

On the other hand the tangential curvature center ofthe hyperboloid, as a surface of rotational symmetry,is on-axis, and the radius of the reference sphericalsurface is made to coincide with the radius of tangentialcurvature. This can easily be deduced from the radiusof the mechanical circular edge, which is its projection:it is sufficient to calculate the tilt on the symmetry axisof the tangent to the hyperbola on the point of contact(Fig. 1).

The sag, the tilts of the normals and their variationswith respect to the circular section on the meridianplane of the reference spherical surface, are then com-pared with the corresponding values of the hyper-bola.

Ill. Numerical ValuesIn the case of the Tirgo secondary mirror the hyper-

bola has these characteristics: radius of the osculatrixcircumference at the vertex r = 1037.30 mm; and ec-centricity corresponding to the value of the conicalconstant 1 -e2 = -0.58737. Thus we have a = 1766.00mm and b = 1353.47 mm in the normal equation of thehyperbola. The diameter of the reflecting surface is 200mm, but during centering its use is restricted to a di-ameter of 190 mm on account of the mechanical circularedge of contact.

For various distances y from the axis, the sag of thehyperbola have been calculated as well as those of thereference circumference whose radius has been obtainedby the above procedure. The differences between thesag, that is to say, the deviations of the hyperbola fromthe circumference, are between zero for y = 95 mm,contact on the mechanical circular edge, and the max-imum value on the axis, that is <0.015 mm.

For more evidence on the asphericity of the reflectingsurface, the differences between the sag of other cir-cumferences and those of the hyperbola have also beencalculated. For the osculatrix circumference at the

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vertex the deviations go from zero on the axis to themaximum value on the border, that is <0.018 mm.

For a suitably chosen circumference including thevertex, the maximum absolute value of the deviationsis <0.003 mm, but this value can be even smaller if thecondition of the vertex is not required.

All the deviations mentioned are shown in Fig. 2; theyshow that there is not much difference between the re-flecting and spherical surfaces.

The differences between the tilts of the normals to thehyperbola and those to the reference circumference arethen calculated; they are shown in Fig. 3, noted as x' -x . The value is zero on the axis and on the contactpoint and reaches a maximum of 0.233 mrad (-48 sec)at y 55 mm.

Finally, the differences between the variations of thetilts are of interest regarding the effect on the mobilityof the image; they are shown in Fig. 3, noted as x- Xr.

The difference is null for the distance y 55 mm fromthe axis, and reaches 6.35 X 10-6 rad/mm on the axis,and 14.8 X 10-6 rad/mm on the border. The sign hasno importance with respect to the perception of an os-cillation of the image. The border however cannot beused, as mentioned above, on account of the mechanicalcircular edge.

It seems suitable to use the interval between y = 75mm and y = 95 mm, on which a 20-mm diam beam can

be reflected, with a theoretical angular resolution of 3.9X 10-5 rad, i.e., -8 sec for the He-Ne laser wavelengthand less for shorter wavelengths. The difference is x"- xr = 8.9 X 106 rad/mm, i.e., -1.8 sec/mm at y = 85mm, where the principal ray impinges.

An eventual residual 0.5-mm centering defect of thevertex of the hyperboloid, therefore a variation of y =1 mm during one turn, would produce an oscillation ofthe principal reflected ray of 17.8 X 10-6 rad, i.e., about3.7 sec.

To observe the image it is suitable to have a telescopewith 40X magnification and a pupil diameter able toaccept the slightly diverging beam reflected by themirror, corresponding to a beam of incident rays, forexample, parallel rays having 20-mm diam. Obviouslyfocusing should be possible. Such a magnification of-fers the observer's eye an oscillating image of -2.5 min,well distinguishable by a normal eye, all the more sowhen the perception of an oscillation reaches sharpervalues than does the perception of a separation.

IV. AccuracyThe accuracy is prescribed under two different as-

pects: in tilt and transversally. Following the adoptedprocedure the contact of the reflecting surface on thecentered mechanical circular edge should guarantee thatthe coincidence in tilt of the optical axis with the me-

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Fig. 2. Deviations of the hyperbola from certain circumferences: DOC = de-viations from the osculatrix circumference; DRC = deviations from the referencecircumference; DCV = deviations from a circumference through the vertex;

MDC = minima deviations from a suitable circumference.

2522 APPLIED OPTICS / Vol. 19, No. 15 / 1 August 1980

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chanical axis is achieved when the coincidence intranslation is also reached.

As a tolerance of 17 sec in tilt coincidence is pre-scribed, the contact of the reflecting surface on themechanical circular edge must occur without deficien-cies >0.015 mm. If some doubt should arise about ob-taining this accuracy, a check must be made.

It is sufficient that the principal ray of the beam,yielding the image guiding the centering, is brought toimpinge on the reflecting surface at y = 55 mm, that isto say on the area where the difference between thesecond derivatives is zero. An error of coincidence intranslation, therefore, does not cause an appreciablemovement of the image, whereas it could be caused bya deficiency of contact on the mechanical circularedge.

A deficiency of coincidence in tilt of -17 sec wouldcause an oscillation of the reflected ray >1 min; check-ing accuracy would thus be more than adequate evenwhen using a telescope with low magnification.

A 0.07-mm tolerance is prescribed for the coincidenceof the axis in translation; but it can seem unobtainableif we take into account that the limit of possibility ofperception, as we have seen above, is reached when thelack of centering of the vertex of the hyperboloid ismuch larger, i.e., 0.5 mm.

Moreover a beam diameter greater than the onechosen, that would impinge on useless areas of the re-flecting surface, would be required to give informationon oscillations of the reflected rays that are smaller than

3.7 sec corresponding to such a lack of centering. Butthe contradiction is apparent. We must in fact specifythat the 0.07-mm tolerance refers to a translation nor-mal to the axis, so that the variation of x' (plane y = xbeing the basis) is of interest, whereas in the procedurethat keeps the contact on the mechanical circular edgeit is the variation of the difference x' x' (the referencespherical surface being the basis) that is interesting.The ratio of x' to x' - xr is as weighty as the ratio of thelack of centering to the tolerance.

More precisely, comparison between the two varia-tions shows that the second procedure is the more rig-orous. On the other hand the sensibility of the opticaldevice that guides the centering, rather than the me-chanical method employed for the adjustments, deter-mines the accuracy that can be attained. Therefore theoperating procedure adopted seems adequate, and nomore complex procedure seems necessary.

These theoretical investigations made at the IstitutoNazionale di Ottica have guided the actual centering ofthe secondary mirror of the Tirgo telescope, carried outby P. Saraceno of the Laboratorio Plasma Spazio ofFrascati. They have also been checked to be quanti-tatively accurate.,

I thank my colleagues F. T. Arecchi, P. Saraceno, andF. Scandone for useful discussions.References1. 0. Citterio, Mem. Soc. Astron. Ital. 49, 57 (1978).2. F. Twyman, Prism and Lens Making (Hilger & Watts, London,

1952) p. 227.

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