Testing large hyperbolic secondary mirrors

Testing large Testing large hyperbolic secondary mirrors Robert E. E. Parks* 4149 E. E. Holmes Holmes Arizona 85711 85711 Tucson, Arizona Lian Lian Zhen Zhen Shaof Shaot Research Corp. Tucson Optical Research 210 S. 210 S. Plumer Tucson, Arizona 85719 85719

Abstract. Various hyperbolic secondary secondary mirrors mirrors are are Various methods methods of testing hyperbolic reviewed, and applicability to to testing testinglarge largesecondaries secondariesisisassessed. assessed. reviewed, and their applicability Traditional and a a combination combination of tests tests is is Traditional methods methods are are found found wanting, wanting, and suggested secondaries. suggested for for large, large, high high quality, astronomical secondaries. Subject Subject terms: terms: optical optical design; optical testing; aspherics; secondary mirrors; hyperbolic mirrors. Optical Engineering 27(12), 1057-1062 1988). 27(12), 1057 -1062 (December 1988).

CONTENTS 1. Introduction 1. 2. Review of traditional secondary mirror test methods Cartesian test 2.1. Cartesian Near stigmatic test 2.2. Near Telescope test 2.3. Telescope Lytle test 2.4. Lytle Hindle test 2.5. Hindle Modified Hindle Hindle test 2.6. Modified Test plate plate 2.7. Test Comments about the traditional tests tests 2.8. Comments Secondary mirror tests using refraction 3. Secondary Including glass homogeneity in the testing 4. Including Conclusions 5. Conclusions Appendix: a note on lens design software software 6. Appendix: References 7. References

1. INTRODUCTION INTRODUCTION 1. There is currently substantial interest interest in in building building very very large large There for ground ground-based astronomy,1-4 1 "4 and some of these telescopes for -based astronomy, projects are projects are now now funded funded and andunder underconstruction. construction.'5 The impetus for for these these projects projects has has come come from from new new technological technological apapproaches to fabricating and supporting very very large large primary primary proaches to fabricating and supporting mirrors or from the the use use of of multiple multiple smaller smaller telescopes telescopes mirrors or from single largelarge-aperture telescope. 6 mounted to function as aa single aperture telescope.6 For these these projects, large secondary mirrors are also needed. Little study, study, however, has has been been given given to to the the fabrication fabrication and and Little secondary is is still still small small testing of secondaries because a large secondary and thus secondaries secondaries are are compared to many existing primaries and not problem area. area. Large Large secondary secondary mirror mirror not perceived perceived as as a problem testing does does present present some some noteworthy noteworthy challenges, challenges, and and those those testing are are the the subject subject of this this paper. Various determining the optical Various tests tests presently presently used used for for determining the optical figure convex, hyperboloidal hyperboloidal secondary secondary mirrors figure of convex, mirrors are are re*The author *The author is is also also affiliated affiliated with with the the Optical Optical Sciences Sciences Center, University of Arizona, Tucson, AZ AZ 85721. 85721. Permanent address: Nanjing Nanjing Astronomical Astronomical Instrument Instrument Factory, Factory, Nanjing, Nanjing, Permanent address: China. Invited OD-104 12, 1988; 1988; revised revised manuscript manuscript received received Invited Paper OD -104 received received July July 12, Aug. 1988; accepted for publication Sept. 3, 3, 1988; 1988; received received by by Managing Managing Aug. 31, 1988; Editor 16, 1988. 1988. Editor Sept. Sept. 16, © 1988 1988 Society of Photo Photo-Optical -Optical Instrumentation Instrumentation Engineers. Engineers.

viewed Both the the design design of of the the tests tests and and practical practical viewed in in Sec. Sec. 2. Both aspects application are considered. It is shown shown that aspects of their their application are considered. some are not not easily easily implemented implemented for for large large secsecsome of these tests are ondaries. Section suggested by by Meinel Meinel and and Meinel7 Meinel7 Section 33 reviews tests suggested and Puryayev8 that overcome limitations limitations of the the traditional traditional and Puryayev8 that overcome tests properties of of the the rear rear surface surface tests by utilizing the refracting properties of the secondary. secondary. To function function satisfactorily, satisfactorily, these these tests tests dedepend on on glass glass of aa high high degree degree of homogeneity, homogeneity, but such such glass glass is in the is difficult and expensive to obtain in in large large sizes. Glass in the sizes required can can be be made made to to about aboutone one-tenth the desired desired sizes required -tenth the homogeneity, and the inhomogeneity inhomogeneity is is accounted accounted for, homogeneity, and if the there is aa sequence sequence of oftests tests that that permit permit the the determination determination of the the hyperbolic tests that that hyperbolicsurface surfacetoto the the required required accuracy. accuracy. The The tests account inhomogeneity are account for the the glass glass inhomogeneity are discussed discussed in in Sec. Sec. 4. 4. 2. REVIEW OF OF TRADITIONAL TRADITIONAL SECONDARY SECONDARY MIRROR TEST METHODS The traditional methods methods of of testing testing convex, convex, strictly strictlyhyper hyper-The traditional boloidal optical surfaces are are described described below. below. The The higher higher optical surfaces order secondary secondary surfaces surfaces required required for forRitchey Ritchey-Chretien -Chretien designs an obvious obvious extension extension of this signs are not considered but are an this work. 2.1. Cartesian test The Cartesian test, although although seldom seldom used used for for testing testing convex convex The Cartesian hyperboloids, is is important important from from aa historical historical and and pedagogical pedagogical perspective. During Descartes's study study of of stigmatic stigmatic imaging, imaging, perspective. During he showed that aa hyperboloidal hyperboloidal surface surface would would form form aa stigstighe showed matic image at one conjugate if the the other other conjugate conjugate was infinite n'/n whereeeisisthe theeccentricity eccentricityofofthe thehyper hyper-nite and n' /n = e,e,where boloid, n' isis the the index index of of the the glass, glass, and and nn isis the the index index of air.9 air. 9 This test shown shown in 1, in in This property property isis the the basis basis for for the the test in Fig. Fig. 1, which point source source on on-axis refracted by by the the hyperbolic hyperbolic which a point -axis is refracted A plane plane surface surface on on the rear of surface into a plane wavefront. A the secondary mirror reflects the the wavefront wavefront back back to to retrace retrace its its forming aa geometrically perfect null steps to the initial object, forming test. Because the Because the range range of available available glass, glass, and consequently consequently n', n', is limited, the the Cartesian Cartesian test test works works only only for for Cassegrain Cassegrain teletelescopes to 4, 4, somewhat somewhat smaller smaller scopes with magnifications around 3 to than is generally desirable. The near stigmatic stigmatic test (Sec. 2.2) 2.2) often overcomes that drawback. One One advantage advantage of the the CarCaroften overcomes that drawback. OPTICAL ENGINEERING / December 1057 OPTICAL ENGINEERING / December1988 1988/ /Vol. Vol.27 27No. No.1212/ / 1057

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PARKS, SHAO PARKS,

SPHERE HINDLE SPHERE HINDLE

an hyperboloid with an convex hyperboloid a convex Fig. Fig. 1. 1. Cartesian Cartesianstigmatic stigmatictest test for for a plane and surface is plane rear surface The rear eccentricity eccentricity ee == n'. The and the the index index of air is 1.00. as 1.00. taken as

more is more test is the test aspherization, the early aspherization, in early that in tesian test test is is that tesian the of the than to that of back than plane back the plane sensitive to to the the figure figure of the sensitive hyperboloid. Thus, while the asphere is being ground or polhyperboloid. loses one loses that one error that of error fringes of many fringes so many not so ished, ished, there are not track of the figure.

2.2. Near stigmatic test By definition, 10 I° ifif the the curve curve on on the the secondary is a "Cartesian conjugates finite conjugates for finite obtained for be obtained stigmatic imaging will be oval," stigmatic as well. well. For many many practical practical cases, the Cartesian oval is a close close obbe obapproximation approximation toto aa hyperboloid, hyperboloid, and and aa near null can be tained tained by by placing placing the the appropriate appropriate spherical spherical radius radius on the rear surface of the surface the secondary. secondary. The The near stigmatic stigmatic test is is similar to the the Cartesian Cartesian test test in in Fig. Fig. 1, but the plane surface is replaced third-order gives aathird Holleran 11 gives spherical one. by by a spherical one. Holleran" -order derivation of this test. radius is correct radius advantage that the advantage This This test test has has the that if the correct the rear surface and the index is is precisely known, polished on the correct have the correct must have surface must hyperboloid on the the hyperboloid on the the front front surface the at the formed at is formed image is eccentricity and vertex radius if a null image drawback is design conjugate conjugate distance. distance. The The practical drawback is that the design accuracy depends test accuracy depends on on the homogeneity homogeneity of the glass in the expensive and secondaries it will be expensive secondary, and for large secondaries secondary, and homogenesufficient homogenewith sufficient may may be impossible to obtain glass with sensiparticularly sensinot particularly is not Cartesian test, this test is ity. Like the Cartesian errors in tive to errors in the the refracting, hyperbolic surface. Telescope test 2.3. Telescope Another obvious obvious method method of secondary testing often used hissecondary in the telescope and test torically torically is is to install the secondary slope-measuring similar slopeor similar edge or A knife edge measuring against a bright star. A test in the image plane plane of the telescope easily reveals defects advantage that in the secondary. This test test has has the advantage that if the prisecondary. This mary has mary has not not been been perfectly perfectly figured figured itself, itself, there is an opporseconthe seconcorrecting the of correcting process of the process in the tunity to fix the error in tunity observing valuable observing uses valuable and uses tedious and is tedious method is This method dary. This dary. seconsuitable seconbecause suitable necessary because often necessary test isis often time, time, but the test the telescopes, the large telescopes, very large For very optics do dary dary test test optics do not exist. For lost observing observing time time makes makes this test uneconomic. 2.4. Lytle test the secondary in the and secondary primary and mates the primary similar test A A similar test that mates test. 12 Lytle test.12 the Lytle telescope is the laboratory rather rather than than in in the telescope laboratory somewhat inside secondary is Here, Here, the secondary is placed placed somewhat inside the the center center of of curvature curvature of of the the primary, primary, and the primary along with a refracting null null lens lens produces produces aa null null for for the the secondary. secondary. Again, errors ing seconin the seconin the primary can be partially balanced by those in seldom workable test Although this dary. dary. Although this is is a workable test on on paper, paper, it is seldom

sphere is Hindle sphere The Hindle hyperboloid. The convex hyperboloid. Hindle test for aa convex Fig. 2. Fig. 2. Hindle near focus of the hyperboloid and the test point the near conjugate with the surface is hyperboloid surface conjugate hyperboloid The conjugate focus. The far focus. the far with the is shown shown in with dashed lines.

primary-secondary long primarythe long used because of the secondary spacing spacing and and the need for a null lens.

2.5. Hindle test Perhaps the classic method of testing secondary mirrors is the geometrical property 13 Use Hindle test. test.13 Use isis made made of the geometrical property of of aa Hindle wave-convergingwave sphericalconverging perfectspherical produce aa perfect hyperboloid to produce hyperboloid is emanating from front front emanating from the the far focus when the hyperboloid is illuminated illuminated by by a diverging spherical wavefront from from the the near near center of placed with sphere is concave sphere focus. When is placed with its its center When a concave focus. curvature concentric with the far focus of the hyperboloid, the as shown itself to the original object plane, as wavefront retraces itself hyper-thehyper eitherthe oneither errorson figureerrors offigure absence of the absence In the 2. In in Fig. 2. in formed in is formed image is stigmatic image boloidal surface or the sphere, aa stigmatic and perfect simplicity and basic simplicity its basic to its addition to In addition the object plane. In sensitive to doubly sensitive being doubly of being advantage of the advantage has the test has null, this test errors in the convex, hyperbolic surface because of the double verify the possible to verify reflection from that surface. Also, itit isis possible conjugates conjugates of the test by direct measurement. the Unfortunately, the the problem problem with with the the Hindle Hindle test test is is the Unfortunately, the of the diameter of the diameter about 2/3 must be Hindle Hindle sphere sphere itself. itself. It must be about 2 the from the seconrays from as fast because rays primary mirror and twice as incidence. For large telenormal incidence. must be reflected at normal dary dary must project of aa project much of as much becomes as optic becomes test optic scopes, building the test as making the primary. According According to to Norman Norman Cole Cole of Arizona Arizona Technologies, Technologies, Inc., Inc., Hindle chordal Hindle or chordal subdiameter or do subdiameter to do possible to is possible Tucson, it is Tucson, tests these tests but these optics, but tests of the secondary using smaller test optics, are insensitive to cone and azimuthal errors, respectively. 2.6. Modified Hindle test test Hindle test the Hindle makes the telescope makes geometry of the telescope When When the the geometry obscuration central obscuration the central when the (or when fast (or large or fast sphere very sphere very large derivative produced by the Hindle test cannot be tolerated), aa derivative be may be Simpson et by Simpson of the Hindle et al., al., may described by Hindle test, described of 14 In used. used.14 In this this case case aa meniscus, transparent spherical optic is concave The concave hyperboloid. The convex hyperboloid. the convex to the quite near to placed placed quite surface of this element serves as the Hindle sphere while light through the refracted through hyperboloid isis refracted the hyperboloid of the from the near focus of Otherwise, the test meniscus meniscus element, element, as illustrated in Fig. Fig. 3. Otherwise, curvature The curvature test. The Hindle test. original Hindle is exactly the same as the original

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TESTING HYPERBOLIC SECONDARY MIRRORS MIRRORS TESTING LARGE HYPERBOLIC

Fig. transmitting meniscus meniscus Hindle Hindle Fig. 3. 3. Modified Modified Hindle Hindle test test using aa transmitting sphere. meniscus sphere is conjugate conjugate sphere. The The reflecting reflecting surface surface of of the meniscus with the the near near focus focus of the the hyperboloid. The The test test point is close close to to the the far far focus.

of the convex convex side determined by lens lens side of the meniscus must be determined design design analysis analysis to to minimize minimize the the residual spherical aberration because because this this isis not not aa geometrically geometrically perfect perfect null null test. test. In many practical less costly costly to to impleimplepractical cases, cases, this this form form of the the test is less ment than than the the original Hindle test because of the smaller optic involved; also, the test does not obscure any of the secondary. For secondaries, this For very very large large telescopes telescopes and and their their secondaries, this test may sphere may suffer suffer from from the the lack lack of availability availability of a meniscus meniscus sphere of sufficient sufficient size and quality. quality. Calibration Calibration of of the the meniscus meniscus size and element also becomes composite wave wave-becomes a problem, although aa composite front 15 front may may be be obtained obtained in in transmission transmission off off aa reflecting reflecting sphere sphere's and and errors errors in in the the meniscus meniscus element may be removed from the secondary mirror test results. secondary 2.7. Test plate A concave hyperbolic hyperbolic test or A final final method method of of test test is is to use aa concave proof plate As plate to to determine determine the the figure of the convex surface. As long secondary or plate is transparent, transparent, long as as either either the the secondary or the the test plate this is an an acceptable acceptable test. this is test. It isis seldom seldom used used when when just one one secondary is secondary is made, made, but it proved invaluable in the fabrication of the six identical secondaries for the Multiple Multiple Mirror Mirror TeleTelescope. 16 The interpret without without special special scope.16 Thetest test isis quick, quick, easy easy to interpret alignment procedures, curves on all alignment procedures, and and ensures ensures identical identical curves all secondaries. This disadvantage that one of the two two optical optical This test test has has the the disadvantage that one surfaces However, the surfaces required required of of the the test test optics optics isis aspheric. aspheric. However, test for 17 illustrated in Fig. for that that surface, surface, the theSilvertooth Silvertoothtest test" Fig. 4, is very straightforward. straightforward. As is very As can can be be seen seen from from the the figure, figure, a spherical mirror 50% larger in diamspherical mirror is is required required that that is is about 50% eter than the secondary or the test plate. Another concern concern about about using using aa test test plate plate isis the the possibility possibility of of damaging plate by by mutual mutual damaging either either the the secondary secondary or or the the test plate contact. largely eliminated eliminated by contact. This This problem problem can can be be largely by boring boring a central plate and and blowing blowing clean, air central hole hole in in the the test plate clean, dry dry air through through the the hole hole to to act as a hydrostatic support for the secondary. support is is then then uniform uniform azimuthally azimuthally and and easily easily dary. The support calculable viewed through the back calculable radially. radially. The fringes are viewed of the the test test plate plate (Zerodur, (Zerodur, for example, is transparent enough for this itself isis supported supported pneupneuthis purpose), purpose), and the test plate itself matically over 5. matically over aa sheet sheet of plate plate glass, as shown in Fig. 5. One plate for for large One other other practical practical drawback drawback of of the the test test plate large secondaries isis the enough back from secondaries the difficulty difficulty of getting far enough from the setup to see see the the whole whole interference interference pattern pattern at at once once the test setup with with no no distortion. distortion. To To help help alleviate alleviate this this problem, problem, the the rear rear of of the plate should should have the test test plate have an an aplanatic aplanatic curve, curve, in in this this case case aa convex sphere of about twice the curvature of the secondary. secondary. convex sphere The The aplanatic aplanatic surface surface guarantees guarantees that that the the interference interference pattern

Fig. 4. for concave concave hyperboloidal Fig. 4. The The Silvertooth Silvertooth test for hyperboloidal surfaces. surfaces. The The test point point is is conjugate conjugate with with the thenear near focus focus of of the the hyperboloid, hyperboloid, and and the sphere sphere is is conjugate conjugate with with the far focus. - CONVEX HYPERBOLOIDAL CONVEX HYPERBOLOIDAL TEST TEST PLATE PLATE WITH APLANATIC BACK BACK

CONVEX HYPERBOLOID HYPERBOLOID .

O -RING SEAL

AIR INLET

HARD POINTS

PLATE GLASS

HOLE IN TEST PLATE FOR AIR SUPPORT

VIEWING DIRECTION DIRECTION

Fig. convex Fig. 5. 5. Pneumatically Pneumatically supported supported test test plate setup for testing convex hyperboloids. Fringes are the rear rear surface of are viewed viewed through the of the the test test plate and the plate glass window. An aplanatic aplanatic back back is is shown on the window. An test plate. plate. test

between the secondary and test plate plate will will be be seen between the secondary and the the test seen at at normal incidence incidence over the entire entire surface surface ifif viewed viewed from normal from the the aplanatic conjugate. 18 aplanatic conjugate.'s Comments about 2.8. Comments about the the traditional tests should be be pointed pointed out out that thattwo twofundamentally fundamentally different different It should types of tests have have been been discussed discussed here. here. In the the case types case of of the the Cartesian test, the index index of of refraction refraction of of the the secondary secondary must Cartesian be completely homogeneous homogeneous for to work work without without be completely for the the test to all of of the the other other tests tests described, described, and and typified typified by the error. For all only the the hyperbolic hyperbolic surface surface isis tested, tested, so so the Hindle test, only the index index homogeneity (or structural uniformity) uniformity) of the the secondary secondary isis homogeneity (or structural The importance importance of of this this difference difference isis fundamental fundamental immaterial. The alternatives to the above tests are described in the rest of as alternatives Only cases cases in in which which the the secondary secondary is is transparent transparent this paper. Only and reasonably reasonably homogeneous homogeneous are considered below. below. For and are considered opaque or structured structured secondaries, an an alternative alternative test test is is desdesopaque cribed elsewhere in in this this issue.19 issue. 19 TESTS USING USING 3. SECONDARY MIRROR TESTS REFRACTION The secondary secondary mirror The mirror test test suggested by Meinel and Meinel7 Meinel suggested by Meinel and makes use of refraction at the the rear rear surface surface of makes of the the secondary secondary reflection off the hyperboloid, hyperboloid, as shown shown in Fig. Fig. 6. This This and reflection not aa perfect perfect null null but but has has the the advantage advantage that that itit isis most most test is not sensitive hyperboloid and so to sensitive to to figure figure errors errors in in the hyperboloid and less less so errors on the rear surface. In Meinel and and Meinel Meinel illustrate illustrate two of of four four In their paper, Meinel with aa plane optimized cases: an autocollimating test with plane back on OPTICAL / December 1988 / Vol. 2727 OPTICALENGINEERING ENGINEERING / December 1988 / Vol. No. 1059 No. 1212 / / 1059

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PARKS, SHAO PARKS,

R = -4731.1

RS = -49531.9

K= -1.643

n = 1.5442 t = 155

rear Fig. 6. 6. The Theautocollimating autocollimating Meinel Meinel test test using using refraction refraction at the rear Fig. the reflected wavefront from the convex surfaceto to largely largely null null the reflected convex surface

2508.3 4924.1

is 0.8 wave rms. Residual error is surface. Residual hyperboloidal hyperboloidal surface. TABLE I.I. Keck TABLE Keck telescope telescope Cassegrain Cassegrain secondary secondary parameters. Diameter Clear aperture Radius of curvature Radius Conic Conic constant

1446 1446 mm mm 1397 1397 mm mm

4731.06 mm mm (convex) 4731.06

and refracFig. 7. The The Meinel Meinel test test (test 3) using separate conjugates and Fig. 7. recancel the relargely cancel to largely secondary to the secondary of the surface of rear surface the rear at the tion at The test is shown surface. The hyperboloidal surface. the hyperboloidal from the flected wavefront wavefront from error, residual error, rms residual wave rms 0.7 wave has 0.7 and has configuration and pass configuration in a double pass 1A this 633 nm. double pass at 633 nm. The The test has 1/4 thiserror errorwith with the the optimum optimum surface. rear surface. the rear spherical radius on the

-1.642929

and a case with separate test conjugates and a secondary and the secondary possibilalso possibilsurface. There are also spherical curve spherical curve on on the the rear surface. and aa curved back and autocollimating test an autocollimating ities for an test with with a curved ities valid make aa valid back. To make plane back. separate conjugate test test with with aa plane separate conjugate 10 m comparison between between the the methods, methods, the Keck Keck 10 m Cassegrain Cassegrain comparison because telescope parameters parameters given given inin Table Table II will will be be used because telescope design is is frozen and the telescope is now under constructhat design tion. tions5 Using Using the the Keck Keck secondary secondary parameters, parameters, the the autocollimating autocollimating worst produces the secondary produces the secondary plane back on the test test with with a plane the worst uncomaberration uncomspherical aberration of spherical waves of 100 waves null, null, with almost 100 By use of very long radius (65 m) convex back, this pensated. By test isis The test error can be reduced by a factor of 36 (see Fig. 6). The error residual error still unsatisfactory, however, because because of the residual unsatisfactory, however, still surface. and and the the difficulty of testing the convex spherical surface. improves the Using separate separate conjugates conjugates and and a plane back improves Using by another another factor factor of 2, 2, but testing the large plane surface test by concave to surface to Allowing the easy. Allowing is is not easy. the rear rear surface to go go concave to aa 20,491 mm radius and using separate conjugates reduces the 20,491 an pass, an single pass, rms single jjtm rms residual aberration residual aberration toto about about 0.06 0.06 µm reduction. Aldata reduction. removed in amount amount small small enough enough to to be be removed in data may be so that itit may concave so is concave though the rear radius is long, itit is the center of curvature, and the absolute radius is not tested at the conof condifferent set of (with a different particularly critical. This This test test (with particularly critical. jugates) is shown in Fig. 7. hyper-the hyper for the designed for be designed also be stigmatic test A A near stigmatic test can also eccentricity the eccentricity Since the 1. Since boloid boloid similar similar to to that that shown shown in in Fig. Fig. 1. available glass, index of the available equal the index does not case does this case in in this not equal spherical residual spherical has residual but has null but perfect null not aa perfect Zerodur, the test is not peak-to-valley. jirn peak -to- valley. The design for the test aberration of 0.07 µm long-radius is optimized with a longradius (49,531 (49,531 mm) concave sphere that is surface that secondary (see on (see Fig. Fig. 8), 8), a surface the rear of the secondary on the relatively easy easy to to measure. measure. As As shown shown in in the the next next section, section, the the relatively the test of the exact figure of of this this surface surface isis not not critical critical to to the the test exact figure hyperboloid in the proposed test method. The test looking through the rear surface and the stigmatic indepensecondary provide two indepenthrough the front of the secondary test through alhyperboloid, althe hyperboloid, figure of the dent methods of testing testing the the figure methods of dent in errors in though though the the Meinel Meinel test will have more sensitivity to errors 1)] hyperboloid by by roughly roughly aa factor factor of of 33 [2n [2nversus versus2(n 2(n-- 1)] the hyperboloid

hyperbothe hyperboat the refraction at using refraction 1) using (test 1) Fig. 8. Near Near stigmatic test (test Fig. 8. Residual rear.Residual therear. on the surface on reflecting surface concave reflecting loidal surface and aa concave surface and 633 nm. at 633 error is 0.02 wave rms at error is

doubled again be doubled can be sensitivity can This sensitivity than the stigmatic test. This concave spherical mirror conjugate to the small concave by placing a small As aa double pass. As autocollimating the near focus and autocollimating the test in double avoid aa to avoid order to in order necessity in almost aa necessity this isis almost practical matter, this source or point source the point either the obscuration in the test due to either large obscuration interferometer is an interferometer mode, an pass mode, double pass the double In the test device. device. In test easily placed at the far conjugate. are tests are Meinel tests and Meinel stigmatic and Unfortunately, Unfortunately, both both the the stigmatic For glass. For the glass. homogeneity errors equally sensitive sensitive to to homogeneity errors in in the equally 1445 the 1445 inhomogeneity is Zerodur, Zerodur, the inhomogeneity is ±± 11llxlO" x 10 -66 for the telescope Keck telescope the Keck for the mm diameter diameter by 157 157 mm thick blank for mm PasaAstronomy, Pasain Astronomy, Research in (California Association Association for for Research (California seriously to seriously enough to communication), enough private communication), Calif., private dena, Calif., dena, affect the figure of the hyperboloid. This problem necessitates into the procedure and that a third test that can be incorporated into allows the errors from the rear surface, the bulk material, and independently. treated independently. and treated separated and the hyperboloid to be separated THE IN THE GLASS HOMOGENEITY IN 4. INCLUDING INCLUDING GLASS TESTING hyperboloid, the hyperboloid, to the due to errors due the errors separate the To unambiguously separate the of the needed of test isis needed material, aa test bulk material, the bulk and the the rear surface, and secondary secondary in in transmission. transmission. In addition, the sphere on the the rear rear same radius for the same be the ideally be would ideally secondary would surface of the secondary the for the surface for rear surface the rear on the radius on the radius Since the tests. Since three tests. all three vertex the vertex of the maintenance of the maintenance to the critical to stigmatic test stigmatic test is critical question the question hyperboloid, the the hyperboloid, of the constant of conic constant radius and conic radius

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MIRRORS HYPERBOLIC SECONDARY MIRRORS LARGE HYPERBOLIC TESTING LARGE

-6723.0 R == -6723.0 R -4731.1 R H == -4731.1 RH -1.643 = -1.643 K = K

R = -49531.9

n =1.5442 t = 155

t=100.3

4395.1

a Fig. Fig. 9. 9. Transmission Transmissiontest test(test (test2)2) of of aa convex convex hyperboloid hyperboloid with with a concave concave rear rear surface surface off off aa concave concave sphere. sphere. Residual Residual error error is 0.5 0.5 wave wave 633 nm. rms, double pass at 633

arises as as to to the the residual residual aberration aberration in in the the test through through the the rear in used in surface if the surface the radius radius required required for the stigmatic test is used in Fig. shown in In the example shown the test through the rear surface. In the error produced by going through the rear surface opti7, the worse times worse four times mized for the stigmatic only four is only stigmatic test radius is mized for may be error may residual error This residual optimum radius. the optimum with the than with than radius. This but itit severe enough to warrant a null lens near the test device, but is not a necessity. place a Turning Turning to to aa test test in in transmission, transmission, one one choice choice is is to place the through the looking through for looking stigmatic point for source at point point source at the stigmatic hyperbolic hyperbolic surface. surface. These rays must hit the sphere on the the rear and autocollimate and to autocollimate normal incidence surface at normal surface incidence for the test to produce produce aa stigmatic stigmatic real real image image at at the the center center of curvature curvature of the the the from the about 50 concave concave rear rear surface. surface. This This image image isis about 50 m from surface, farther than than is is desirable. desirable. To shorten the optical path, immediately behind spherical mirror convex spherical a convex mirror could be placed immediately indetest indesecondary, but this mirror would be difficult to test the secondary, pendently. An needed. An mirror is needed. spherical mirror concave spherical against a concave A A test against optimized optimized version version of of such such aa test test is is shown shown in in Fig. Fig. 9. This test fifth-order uncorrectedfifth ofuncorrected peak-to-valley, 1 |xm,peak about 1µm, has has about -to- valley, of -order doublet removed with spherical aberration spherical aberration that that could could be be removed with a doublet would test would this test refracting null lens of modest size. size. In practice, this spherical small spherical probably probably be be used used in in double pass by placing a small mirror at the near conjugate. sources of separate the necessary to The The three three tests tests necessary to separate the three three sources concave error in in the hyperbolic surface, the concave the hyperbolic secondary-the the secondary error are spherical surface, spherical surface, and and the the inhomogeneity inhomogeneityininthe the glass glass -are the now now defined. defined. In the strictest sense, as is shown in Eq. Eq. (4), the stigmatic test is not needed to solve for the hyperbolic surface conand so confoolproof and so foolproof test is so secondary, but the secondary, of the but the the test front the front ground into the being ground as the asphere is being venient venient to to use use as The unwise not surface of the the secondary that itit is is unwise not to use it. The secondary that surface of the with the check with consistency check results results from from this this test also act as a consistency stigmatic test simplify the other two two tests. tests. To simplify the following, following, the stigmatic will be referred looking into the hyperbolic surface as in Fig. 88 will as sphere as test sphere the test off the transmission off test inin transmission The test 1." The "test 1." to as "test to the test using the hyperbolic and the shown in Fig. 9 will be "test 2," and surface in reflection reflection as as in in Fig. Fig. 77 will be "test 3." surface in WB , WH , WE, define WH, us define let us how the procedure is used, let To show how the hyperboloid, the the hyperboloid, of the error of figure error surface figure the surface as the Ws as and Ws and inhomogeneity error in the the bulk material, and the figure error errors figure errors Positive figure respectively. Positive spherical surface, respectively. of the spherical as and as surface and hyperbolic surface the hyperbolic on the bumps on defined as bumps will will be defined

Fig. Fig. 10. 10.Coordinate Coordinatesystem system used usedto to separate separatewavefront wavefront errors due due to to surface, the the bulk material, and the hyperboloid. the spherical rear surface,

direcholes holes on on the the spherical spherical surface, surface, i.e., i.e., as errors in the the +z direcside of right-hand andright topand Thetop 10. The Fig. 10. in Fig. indicated in tion, as indicated -hand side the aperture looking at the hyperbolic surface are also defined errors these errors flip of these left-to-right beaaleft will be There will 10. There in in Fig. 10. -to -right flip When secondary. When the secondary. of the side of when looking from the spherical side left-tothe left 3, the and 3, tests 22 and in tests around, as in turned around, surface is the the surface is turned -toright right flip flip is is designated designated by by (( x). -x). the at the reflection at and reflection refraction and of refraction Accounting for the effects of 10 indicated in secondary as various surfaces surfaces of of the the secondary as indicated in Fig. Fig. 10 various model the (where two (where two dummy dummy surfaces surfaces are are used used to to model the effects effects of flips left/right flips inhomogeneity) and using the conventions for left/right inhomogeneity) wavefront the wavefront above, the and directions of surface errors outlined above, Wj, is 1, W1, error (or optical path difference) measured in test 1, 1)WH + 2dnWB 2dnWB + 2nWs(-x) Wi 2nWs( -x) , WI = -2(n --l)WH

(1) (1)

while the wavefront errors in tests 2 and 3 are 1)WS , 4(n -- l)Ws 4dnWB (-x) -x) ++ 4(n l)WH(-x) - -4(n --1)WH( W2 -x) + 4dnWB( W2 = W3

-x) ++ 4(n 1)WS ,, 4(n --1)Ws 4dnWB((-x) = = --4nWH(~x) 4nWH( -x) + 4dnWB

(2) (3)

where where nn is is the the index index of the the glass glass and and dn do is is the the inhomogeneity. inhomogeneity. is result is the result and 33 is taken, the If the difference between tests 22 and 4(n -- 1)1WH(-x) W2 = W3 -- W2 = [-4n + 4(n W3 = --4WH(-x) = 4WH( -x) ..

(4)

knowlexplicit knowlThe figure of the hyperboloid is given without explicit inhomogeneity. theinhomogeneity. orthe surface or spherical surface the spherical either the edge edge of either sensitive to as sensitive shows that Furthermore, Furthermore, the the result result shows that the test is as the the surface errors errors in the hyperboloid as is the Hindle test. between difference between thedifference thatthe noted that be noted will be addition, it will In In addition, tests 11 and 2 can be written as (-x) , 2W, 2W1±± W2 W2== 4Ws 4Ws(-x)

(5) (5)

"odd" aberrations that are "odd" those aberrations for those used for where where the the " ++"" is used left/ or left/ left/right regard to left with with regard /right flips flips (such (such as as coma in the x or aberrations. "even" aberrations. for "even" used for right right direction) direction) and and the the "-" " -" isis used homogeneity the homogeneity and the surface and spherical surface the spherical in the Thus, the error in WH can be found from since WH WB can can be be determined determined explicitly since WB tests 2 and 3. Code with Code modeling with rigorous modeling to rigorous up to This simple model holds up and 91107) and CA 91107) Pasadena, CA (Optical Research Associates, Pasadena, V (Optical errors20 wavefronterrors20 thewavefront ofthe representations of Zernike polynomial representations Furexample. Furthis example. in this errors in modeled errors the modeled of the 10% of within 10% to within CODE on CODE exactly on modeled exactly be modeled can be process can entire process the entire thermore, the / /December OPTICALENGINEERING ENGINEERING December1988 1988/ /Vol. Vol.27 27 No. No.12 12// 1061 OPTICAL

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PARKS, SHAO

V similar analysis analysis software software should should the accuracy be V or similar the accuracy be required. 5. CONCLUSIONS If itit isis required required to to fabricate fabricate aalarge largehyperbolic hyperbolic secondary secondary mirror out out of aa solid, solid, reasonably reasonably homogeneous, homogeneous, transparent transparent mirror is an an option option for for testing testing that that uses uses one one additional material, there is spherical mirror mirror of ofabout about the the same same dimensions dimensions as as the the concave, spherical secondary. This option is a sequence of tests that make use of the properties of the glass to null the refracting refracting properties null out out most most of of the the asphericity introduced the hyperbolic hyperbolic surface. surface. The The apapasphericity introduced by by the proach is selfself-checking proach checking for for vertex vertex radius radius and conic constant and is independent independent of homogeneity homogeneity and and and figure figure errors errors in in the bulk and on the the rear rear surface surface of of the the secondary, secondary, respectively respectively bulk and (provided the errors are are smoothly smoothly varying varying and and within within reasonreasonable magnitude). There exists a near stigmatic test for the hyperbolic surface surface secondary when when the the rear rear surface surface isispolished polishedasasaalong long-of the secondary Autocollimation and and aa stigmatic stigmatic image image ensure ensure radius sphere. Autocollimation for the the secondary. secondary. the correct vertex radius and conic constant for Because the beam of light light traverses traverses the the surface surface being being Because the test beam worked in refraction, the test may be carried worked in refraction, the carried out while while the the surface is fine ground if the surface is oiled or waxed, aiding in the rapidity with which which the surface surface may may be be aspherized. aspherized. There exists near null null tests tests (tests (tests 2 and and 3) 3) that that There exists aa pair pair of near indicate the error of the hyperboloidal surface without without one's one's indicate the hyperboloidal surface having to know explicitly the homogeneity of the glass or the exact figure of the the rear rear spherical spherical surface surface of of the the secondary. secondary. exact figure transmission test After the transmission test (test (test 2) has been performed once, itit need repeated; only reflecting test (test 3) is is rereneed not not be repeated; only the the reflecting test (test peated through through the the completion of the hyperboloid. If test test 33 is used used in double pass, itit is is twice twice as as sensitive sensitive to errors is itself itself to to the the same same errors in the hyperboloid as the telescope is errors. This high test sensitivity helps to to ensure ensure aa good figure the secondary secondary and and minimizes minimizes concern concern about about the figure figure on the adequacy of the the finished finished secondary secondary to to telescope telescope performance. performance. adequacy The same concave spherical spherical surface surfaceon on the the rear rear of the The same concave the secondary the rear rear surface surface does does secondary works works for for all all three three tests, tests, so the not reworked for subsequent subsequent tests. tests. The transmistransmisnot have have to to be reworked sion and and reflection tests are not perfect null tests but are sion are close enough to null that null optics are not needed and the remaining spherical aberration can be removed in data reduction. If a perfect null is desired, desired, small small corrector corrector elements elements can can be be dedeperfect null signed for use near the test device.

6. APPENDIX: a note on lens design software Since this this isis aa paper in a feature Since paper in feature issue issue on on lens lens design, design, it is appropriate to to note note that that all of the test designs illustrated here appropriate were were originally originally analyzed analyzed and and optimized optimized using using a $90 program program for the IBM IBM PC PC (or (or clone) clone) called called "Beam "Beam .2" from Stellar Stellar for the 2" from Software Software (Box (Box 10183, 10183, Berkeley, Berkeley, CA CA 94709). 94709). All All of the designs were were rechecked rechecked with with CODE CODE V V and and tweaked tweaked up up slightly slightly signs because CODE V is is less less tedious to use. However, itit seems CODE V seems to to us that the Beam 22 program is well suited to the small shop for the analysis analysis and and design design of test optics (as well well as as more more comcomplicated designs). Its one potential drawback is that it handles only conies. For general aspheres, aspheres, the the $300 $300 version, version, "Beam "Beam only conics. needed. This This costlier costlier version version also also does does off off-axis sys3," isis needed. -axis systems, tilted elements, tems, elements, folded folded systems, systems, etc. etc. Thus, Thus, for for aa relarelainvestment in in software, a small optics firm can can tively modest modest investment greatly increase increase its capability.

7. REFERENCES REFERENCES 1. -Y. Wong, 1. L. L. D. D. Barr, Barr,W. W.-Y. Wong, and and R. R. J.J.Harris, Harris,"Planning "Planningthe theNational National New New Technology (NNTT): II. structures," in in Advanced Advanced Technology Telescope Telescope (NNTT): II. Telescope structures," Technology Proc. SPIE SPIE 628, 628, TechnologyOptical OpticalTelescopes TelescopesIII, III,L.L. D. D. Barr, Barr, ed., Proc. 80-90 (1986). 80-90(1986). D. Enard, Enard,"The "TheESO ESOVery VeryLarge LargeTelescope Telescope project: project: present present status," status," in in 2. D. Advanced Technology ed., Proc. Proc. Advanced TechnologyOptical OpticalTelescopes TelescopesIII, III,L.L. D. D. Barr, ed., SPIE 628, 221 221-226 -226 (1986). K. Kodaira Kodairaand and S.S. Isobe, Isobe,"Progress "Progressreport reporton onthe thetechnical technical study study of of the the 3. K. Japanese telescope project," project,"ininAdvanced AdvancedTechnology Technology Optical OpticalTelescopes Telescopes III, L. D. -238 (1986). HI, D. Barr, Barr, ed., ed., Proc. Proc. SPIE SPIE 628, 628, 234 234-238 (1986). P. A. A.Strittmatter, Strittmatter, "Columbus "Columbus Project Project overview," overview," to to be be published published in 4. P. Proc. ESO ESO Conf. Conf. on on Very Very Large Large Telescopes Telescopes and and Their Their Instrumentation Instrumentation (Garching, FRG, FRG, 1988), 1988), European European Southern Southern Observatory, Observatory, Garching. Garching. Nelsonetet al., al.,eds., eds.,"Design "Designofofthe theKeck KeckObservatory Observatory and and Telescope," Telescope," 5. J.J.Nelson Keck Keck Observ. Observ. Rept. Rept. No. No. 90, 90, Keck Keck Observ. Observ. Project Project Office, Office, Pasadena Pasadena (1985). Beckers and and J.J. Williams, Williams, "The "The MMT MMT as as itit exists exists today," today," in in Optical Optical 6. J.J. Beckers and Infrared Telescopes Telescopes for 108-128, Natl. for the the 1990's, 1990's, pp. 108 -128, Kitt Kitt Peak Peak Natl. Observ. (1980). 7. A. A. B. B.Meinel Meineland andM. M.P.P.Meinel, Meinel,"Self "Self-null -null corrector correctortest testfor fortelescope telescope hyperbolic 22(4), 520 520-521 (1983). hyperbolic secondaries," secondaries," Appl. Appl. Opt. 22(4), -521 (1983). 8. D. D. T. T.Puryayev, Puryayev, "A "Aquality qualitycontrol controltechnique technique for for convex convex elliptical, elliptical, 8. parabolic Tech. parabolic and and hyperbolic hyperbolic surfaces surfaces of of simple simplelenses," lenses," Sov. Sov. J.J. Opt. Opt. Tech. -686 (1971). 38(11), 684 684-686(1971). 9. E. E. Hecht Hechtand andA. A.Zajac, Zajac,Optics, Optics,pp. pp.100 100-102, Addison-Wesley, Boston -102, Addison -Wesley, Boston (1976). 10. G. A. A. Boutry Boutry and and R. R. Auerbach, Auerbach, Instrumental Instrumental Optics, Optics, p. p. 25, 25, Hilger Hilger and and 10. G. Watts, London (1961). 11. R. T. T. Holleran, Holleran, "Third "Third-order wavefronts and related null Appl. 11. R. -order wavefronts and related null tests," Appl. Opt. 1244-1246(1966). Opt. 5(7), 1244 -1246 (1966). 12. J. Lytle, 12. Lytle, "A "Asuggested suggested procedure procedure for for testing testing large large Cassegrain Cassegrain optical optical systems," Rept. 43, 43, Optical Optical Sciences Sciences Ctr., Ctr., Univ. Univ. of ofArizona Arizona systems," Tech. Tech. Rept. (1969). 13. 13. J. H. H. Hindle, Hindle, "A "Anew newtest testfor forCassegrainian Cassegrainianand andGregorian Gregorian secondary secondary mirrors," Mon. Not. R. R. Astron. Astron. Soc. Soc. 91(5), 91(5),592 592-593 (1931). -593 (1931). 14. A. Simpson, Simpson, B. B. H. H. Oland, Oland,and andJ.J.Meckel, Meckel,"Testing "Testingconvex convex aspheric aspheric 14. F. A. lens surfaces with with a modified modified Hindle Hindle arrangement," arrangement," Opt. Opt. Eng. Eng. 13(3), 13(3), lens surfaces G101 -G109 (1974). G101-G!09(1974). 15. I.I. Friedman, "The "The testing testing of of meniscus meniscus refractive refractive corrector elements," in Contemporary Contemporary Methods Methods of of Optical Optical Manufacturing Manufacturing and and Testing, Testing, G. G. M. Sanger, ed., Proc. -170 (1983). Sanger, Proc. SPIE SPIE 433, 433, 165 165-170(1983). 16. 16. M. M. Ruda, Ruda, "MMT "MMT optical optical prediction prediction and andperformance," performance," MMT MMT Tech. Tech. Rept. Steward Observ., Univ. of of Arizona (1980). (1980). Rept. 2, Steward 17. W. W. Silvertooth, Silvertooth,"A "Amodification modificationofofthe theHindle Hindletest testfor forCassegrain Cassegrainseconsecon17. daries," J. Opt. Opt. Soc. Soc. Am. Am. 30(3), 30(3), 140 140 (1940). (1940). 18. D. D. 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Robert Robert E. E. Parks received BA BA and MA MA degrees degrees physics from from Ohio Ohio Wesleyan Wesleyan University University in physics and Williams Williams College, College, respectively. respectively. Since Since and he has has worked as as an optical engineer at then he Eastman Eastman Kodak Kodak Company and Itek Itek CorporaCorporabefore picking picking up up optical optical fabrication fabrication exextion before

perience perience at at Frank Frank Cooke, Cooke, Inc. Inc. In In 1976, 1976, he he to the the Optical Optical Sciences Sciences Center Center of the the went to University of of Arizona Arizona to to manage manage the optical optical shops. For For the the past past several several years years he he has has shops. acted as as a a project project manager manager at at the the Optical Optical acted Sciences Center his time time as as aa private private Sciences Center and and isis now now spending spending most of his consultant in optical fabrication fabrication and and testing testing while while still still keeping keeping ties ties consultant with the University University of of Arizona. Arizona.

Lian Zhen Lian Zhen Shao Shao was was born in in Shanghai, Shanghai, China, 1945. She She received received the the BS BS degree degree China, in 1945. in physics from Fudan Fudan University, University, Shanghai, Shanghai, 1968 and and the MS MS degree degree in in astronomical astronomical in 1968 optics from Academia optics Academia Sinica Sinica inin1982. 1982. From From 1982 to 1986 she the Nanjing Nanjing 1982 to 1986 she worked worked at at the Astronomical Instrument Instrument Factory Factory of of AcaAcaAstronomical demia Sinica demia Sinica as as an an engineer. engineer. From From 1986 1986 to 1987 she at the the Optical Optical Sciences Sciences 1987 she worked at Center Center of the the University University of ofArizona. Arizona. Her Her reresearch search interests interests include include lens lens design, design, data data and optical optical testing. testing. processing, and

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