theory of three-mirror telescopes with diffraction limited performance

19
This article was downloaded by: [The University of Manchester Library] On: 20 December 2014, At: 09:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Optica Acta: International Journal of Optics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmop19 Theory of Three-mirror Telescopes with Diffraction Limited Performance D. Iorio-Fili a , G. Misuri a & F. Scandone a a Istituto Nazionale di Ottica, Firenze, Italy Published online: 14 Nov 2010. To cite this article: D. Iorio-Fili , G. Misuri & F. Scandone (1980) Theory of Three-mirror Telescopes with Diffraction Limited Performance, Optica Acta: International Journal of Optics, 27:8, 1035-1052, DOI: 10.1080/713820382 To link to this article: http://dx.doi.org/10.1080/713820382 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/ terms-and-conditions

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Page 1: Theory of Three-mirror Telescopes with Diffraction Limited Performance

This article was downloaded by: [The University of Manchester Library]On: 20 December 2014, At: 09:16Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optica Acta: International Journal ofOpticsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmop19

Theory of Three-mirror Telescopeswith Diffraction Limited PerformanceD. Iorio-Fili a , G. Misuri a & F. Scandone aa Istituto Nazionale di Ottica, Firenze, ItalyPublished online: 14 Nov 2010.

To cite this article: D. Iorio-Fili , G. Misuri & F. Scandone (1980) Theory of Three-mirrorTelescopes with Diffraction Limited Performance, Optica Acta: International Journal of Optics,27:8, 1035-1052, DOI: 10.1080/713820382

To link to this article: http://dx.doi.org/10.1080/713820382

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoeveras to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified with primarysources of information. Taylor and Francis shall not be liable for any losses, actions,claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms& Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Theory of Three-mirror Telescopes with Diffraction Limited Performance

OPTICA ACTA, 1980, VOL. 27, NO. 8, 1035-1052

Theory of three-mirror telescopes with diffraction limitedperformance

D. IORIO-FILI, G . MISURI and F . SCANDONEIstituto Nazionale di Ottica, Firenze, Italy

(Received 20 July 1979)

Abstract . Attention has been recently focused on three-mirror telescopes . Thepurpose of the present paper is to explore the range of acceptable configurations interms of achievable apertures, fields and vignetting, as a function of the gaussianparameters . The achievable quality of the image over this range has been exploredby computing the residual maximal root mean square of the wave-front error fortypical cases, and the limits within which a diffraction limited performance can beexpected, are discussed .

1 . IntroductionIn recent years more attention has been paid by optical designers to three-mirror

telescopic systems [1-3] and extensive treatment has been given by Robb to third-order correction and higher order optimization . The attractiveness of such systems isdue to the following features :

(a) the system, being totally catoptric, can be used over a wide spectral range ;(b) the system can be made truly anastigmatic, with theoretically perfect

correction up to third-order approximation over a truly flat field, and it hasbeen proven that excellent correction can be achieved up to higher orders forsmall, but significant, field angles ;

(c) the gaussian and geometrical parameters of the corrected solutions (curva-tures, conic constants, etc .) fall within ranges of values which meet therequirements of manufacturing technology .

On the other hand, three-mirror systems are subject to significant shortcomings,such as :

(d) the constraints to be satisfied in order to obtain a truly vignetteless image aresevere ;

(e) even the minimum requirement, that the field centre or zero-field image,should be vignetteless, requires in general rather high relative apertures (lowN-numbers) and/or significantly high linear obscuration factors, a featurewhich causes a degradation of the ideal MTF ;

(f ) as a consequence of (d) and (e) the practically achievable fields are limited,even if higher fields could be well corrected from the aberrations point ofview ;

(g) finally, the position of the focal field, which necessarily falls internally to theoptical system (or maybe in front, upstream of the secondary system), can beunacceptable or undesirable in many practical cases .

In order to assess the importance of three-mirror systems for practicalapplications, it is the purpose of this paper to explore the extent of acceptable

0030-3909/80/2708 1035 S02-00 1' 1980 Taylor & Francis Ltd

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D. Iorio-Fili et al .

solutions in terms'of achievable apertures, fields and the related gaussian parameters .An eye will be kept on the quality obtainable for the image and on the general

feasibility of the system . To make the text easier to read, we shall relegate allgeometrical and gaussian derivations to the final section (§ 5), to which we willafterwards refer .

2 . Gaussian constraintsIn the following, reference is made to figure 1, which shows a generalized

diagram of a three mirror telescopic system, where we have considered the possiblerequirement of an entrance pupil situated at a distance dp upstream of the primarymirror . In fact it was such a requirement which originated this study [4] . In theordinary case of a telescope in which the entrance pupil coincides with the primarymirror, dp will be zero .

Figure 1 contains the symbols of all quantities of interest to us . The signs of thesequantities are to be considered positive if oriented as in the diagram . For instance,the distance e3 of the focal plane F3 from the primary mirror is positive if F3 is infront of the mirror, it is negative if F3 is located after the mirror, etc . Unless specifiedto the contrary, all linear quantities are referred to a unitary equivalent focal lengthof the system, so that we can set feq = 1 .

We consider the system as unequivocally specified by five gaussian parameters(the mirror-powers 01 ; 02 ; 03 ; and the two distances S 1 S2 =d1 ; S 2S3 =p=totallength of the system), and by the conic mirror constants E l , E2 , E3 . We do notconsider here possible higher order deformation coefficients .

Of the five gaussian degrees of liberty one will be used to obtain the requiredpower 4 eq = 1/feq (which we take = 1) and another to obtain a zero Petzval curvature,

E .P1

h p ,

I

i S 2

d p

L

Figure 1 . Three-mirror telescope, schematic diagram illustrating vignetting constraints .E.P.=entrance pupil; S1, S2 , S3 =primary, secondary, tertiary mirror ; F1, F12,

F3 =primary, intermediate and final focal plane ; Y1i Y12, Y3=dimensions of primary,intermediate and final semi-field; L = total longitudinal encumbrance ; p = length ofoptical system ; dp= distance of E .P. from primary mirror ; d 1 = distance betweenmirrors S1S2 ; 112=back focal distance of Cassegrain system S 1S 2 ; d2 =distancebetween mirrors S 1S3; e3 =distance of final focus F3 from primary mirror . Alldimensions considered positive if located as in diagram .)

YF

F3

d,-e3

5h s

1

h4

Y12iI

F12

F12

P

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Page 4: Theory of Three-mirror Telescopes with Diffraction Limited Performance

0 .40

0 .35

0.30

Theory of three-mirror telescopes

1037

necessary for a truly corrected flat field (see equation (5.1)) . The remaining threedegrees of freedom can be used to satisfy other conditions in order to meet certaindesign or performance requirements, such as a given position of the focal plane, theobtainment of a certain field with specified minimum vignetting requirements or thecontainment of the linear obscuration factor below an upper limit .

The mathematical treatment in § 5, shows that it is convenient to express thesethree degrees of freedom by three parameters which are :

rl = 1 - d 1 4 1 which expresses the'zero-field' linear obscuration factor, i .e . the oneobtained considering only rays parallel to the optical axis . This value we shallcall the `nominal ?J-value' which will be inferior to the effective ?Jeff -value,which considers the entire unvignetted field and is therefore field-dependent ;

p which is the total length of the system, and coincides with the second opticaldistance S 2S 3 ;

K=(1 + k) (1 +qk) where k=02/01 is the power-ratio of the secondary mirror tothe first .

The expressions for all the quantities specifying a given gaussian design as afunction of the three (?J, p, K) parameters are given in equations (5 .2) to (5 .6) .

Once a value for h has been chosen, each point of a (K, p) cartesian representationon a plane represents a different gaussian configuration of the system . K is negativefor a cassegrain system of the first two mirrors, K is zero for a true telescopic (zero-power) configuration of the first two mirrors, for which focus F 12 (figure 1) is locatedat infinity. A positive K would correspond to a gregorian solution (concavesecondary mirror) . Figure 2 represents such a plane, on which several areas can beseparated by curves representing certain limiting conditions .

P

1+,K-0 .12 -0 .10

-0.14

-0.15

-0.16

- (t-q) 2K = 4q

Figure 2. (p, K) plane representations of acceptable configurations .

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D. Iorio-Fili et al .

Curve (a) satisfies the equation :

d1 -e3 =0

(2 .1)

and separates the solutions (to the right) for which the focal plane falls after thesecondary mirror from those (to the left) for which the focal plane falls in front ofthesecondary mirror . In a practical design, even if the latter position of the focal planecan be considered, it would obviously mean `immediately' in front of the secondary,so that we consider curve (a) in the following as a limiting boundary . Curve (b)satisfies the equation

e3 = 0

(2 .2)

and separates the solutions (to the right) for which the focal plane falls after theprimary from those (to the left) for which the focal plane falls between the twomirrors . Again we consider curve (b) a practical limiting boundary because if thedesign asks for a focal plane located after the primary, it will again be `immediately' atthe back of the primary, because of the very severe limitations imposed on admissiblefield and vignetting conditions on focal plane positions located between primary andtertiary mirrors . If a given position of the focal field is required, one can trace anintermediate curve (a') satisfying the condition :

d1 -e3 =q, (2 .3)which gives all the configurations for which the focal plane is situated at distance qafter the secondary .

The area (p, K) of acceptable configurations is further limited by field andvignetting constraints .

Referring again to figure 1 we can see that if a true vignette-free image is requiredup to a given semifield y o , one must make sure that the incoming ray hP grazing theexternal boundary of the obscuration disc (often represented by the edge of thesecondary mirror) and oriented at a yo depression angle, will traverse the F3 focalplane at a height h5 greater than the size of the entire focal plane obstruction YF ,which must include mechanical supporting structures, photographic plates orelectronic receivers .

We can write this condition as

h5? YF=feq . tanyl,

(2 .4)

where 7 1 is the object-space semifield subtended by the obstruction . Therefore ylwill be in general greater than y o , although there are special situations, as for instancein reference [3], where the field yo is scanned by a small or unidimensional device, sothat in such cases one can assume 71 <yo .

A second vignetting constraint arises at the boundary of the central bore of theprimary mirror, In fact, figure 1 shows that in order to accept the full aperture of thebeam directed at the desired semifield y o, the primary mirror should not extendlower than h3 i in order to let the outward oriented ray hP proceed towards thecassegrainian focal plane Y 12i while the above mentioned lower ray h' requires thatthe primary mirror should extend at least down to height hi in order to be reflectedback towards the secondary . This `second vignetteless condition' reads therefore :

h1 >h3

(2.5)

and this condition is not fulfilled by all configurations .

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Theory of three-mirror telescopes

1039

Figure 1 also shows that the intersection of the extreme rays hi and hi at heighthZrepresenting the secondary mirror boundary, determines the effective dimension neff

of the obscuration disc . The explicit expressions for equations (2.4) and (2.5) and for11eff will be found in equations (5.13), (5 .12) ; (5 .11) and (5 .12a) . These equationsinvolve, besides the r1, p, K parameters, also the N stop number and the e .p .coordinate dp .

Turning back to figure 2, we can trace curves (c) and (d), satisfying the equalitysign in (2.5) and (2.4) respectively, for the y o , y l , and N values required by aparticular design .

We can now say that every point of the (p, K) plane falling within the curvilinearquadrangle A, B, C, D will satisfy the requirements . The quadrangle is limited bythe following segments :

AB portion of curve (c) satisfying equality (2.5)BC portion of curve (a') specifying a lower q-value limit for the position of the

focal plane, and generally contained between curve (a) and curve (b)CD vertical straight line Klim = -[( 1 -q)']/411 beyond which no solution existsDA portion of curve (d) satisfying equality (2.4) .

Curve (d) can fall to the left of curve (b), in which case no solution with a focal planeat the primary mirror or beyond is possible, or it may fall to the right, in which casesuch solutions exist .

3 . Vignetting constraintsWe can now proceed to explore the range of performance which can be expected

by a three-mirror telescope .We can ask ourselves two kinds of questions, i .e . :

(1) What is the widest range of aperture and field values possible from a gaussianpoint of view if we accept very liberal vignetting conditions? This could be thecase of an instrument which is not meant for precision photometric comparisonof objects at different points of the field. In such a case we also want to know towhat extent we can expect a good image for such apertures and fields .

(2) What is the achievable range of aperture and field when there is a strictrequirement for a vignetteless image and/or a very good one, i .e . a diffractionlimited quality of the image? Obviously, most requirements will fall in anintermediate region between these extreme cases . We shall try to answersynthetically to both types of questions, beginning with the first one .

To simplify matters we shall from now on consider only the case dp =0, by far themost common. An entrance pupil location in front of the primary mirror, besidesbeing an uncommon demand, necessarily has the effect of restricting the range ofpermissible solutions .

We will make the assumption that the most liberal vignetting condition we will beprepared to accept is the one in which only the centre of the field (Y o =0) is reallyvignette-free, (meaning by this that no reduction in useful aperture is introducedover and above the obscuration disc due to the secondary mirror), while vignettingsets in gradually as we consider points of the field for yo > 0 up to the required fullsemifield value y l .

From equation (5 .13) we see that for T o = 0 we get the limitationz

Ntany i ,< I- di -e3 ) 11

,

( 3 .1)l 1z

2

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D. Iorio-Fill et al .

while equation (2 .5) is always satisfied at zero-field . From equation (3.1) we can drawthe following conclusions :

(a) the vignetting conditions dictate limitations only on the product N. tan y,while these two factors can be combined at will, for the same gaussianconfiguration ;

(b) the maximum values for theN . tan y, product are obtained when d1- e3 = 0,that is for a focal field located near the secondary mirror, when we obtain

rN . tan y1 <

2.

( 3 .1 a)

Table 1 shows the maximum semifield y l , in degrees, obtainable for differentobscuration factors j and different stop-numbers N. We should keep in mind that y,must include any kind of obstruction, and therefore the useful semifield may besomewhat lower than y l . Discarding objectionable rj-values over 0 .50 and exception-ally high apertures N< 3 one can see that fields of the order of 2y 1 - 5 ° can beobtained (see, for instance, reference [1]) .

Table 1 . Maximum values of semifield for vignetted solutions (equation (3 .1 a)) .

We want to know the amount of vignetting which is introduced, under thepreceding assumption, at the maximum semifield Y1 given by (3 .1 a) .

To define vignetting, we turn again to figure 1 and will call hs(y 1 ) the height atwhich the lowest admitted ray traverses the focal plane ; therefore :

h'5(y1) =feq - tan y 1 = tan y 1 (because we have set feq = 1 )and hs(y 1 ) will be the height at which the full aperture ray at the same depressionangle y 1 traverses the focal plane, therefore the breadth of the vignetted lunula (figure3) will be tan y1 -h' (y l ), to be compared to the entire breadth of the ray-bundlestraversing the focal plane which, considering the presence of the obscuration area,we will conventionally set as 2[hs(Y1)-h`s(Y1)], so that we can define a `linearvignetting factor' v as

tanyl - hs(Y1)V=

2[hs(Y1) - h's(Y1)](3 .2)

The real area-vignetting factor used to measure the energy loss in the image, is ofcourse the ratio of the area of the intercepted lunula to the area of the annularaperture but, for many evaluation purposes, such as the deterioration of the MTF,

0

N\0.3 0 . 35 0. 4 0 .45 0.50 0 . 55

0.6

2 1 .2889° 1 . 7541 ° 22906° 2 . 8981° 3 . 5763° 4 . 3248°

5 . 1428°2. 5 1 . 0312° 1 . 4035° 1 .8328° 2 . 3192° 2 . 8624° 3 .4622°

4.1182°3 0.8594° 1 . 1695 ° 1 .5275° 1 . 9330° 2 . 3859° 2.8862°

3 .4336`4 0.6446° 0 . 8773° 1 .1458° 1 . 4500° 1 . 7899° 2. 1655°

2 .5766°5 0. 5156° 0 .7018 ° 0 .9167° 1 . 1601° 1 . 4321° 1 .7327°

2.0618°6 0.4297° 0 .5849° 0 .7639° 0 . 9668° 1 . 1935° 1 .4440°

1 .7184°10 0.2578° 0 .3509° 0.4584° 0 . 5801° 0 . 7162° 0.8665°

1 .0312°15 .0. 1789° 02340° 0.3056 ° 0 . 3867° 0 . 4775° 0.5777° 0 . 6875°

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Theory of three-mirror telescopes

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the linear factor is more meaningful . Equations (5 .19), (5.191a) and (5 .19 b) give thebreadth of the vignetted lunula and the linear vignetting factor v for the limitconditions (3 .1) and (3 .1 a) . At the same time we must make sure that no furthervignetting is introduced by equation (2..5) . For this we must request that :

2

N . tan yi 52/[(1 +n)l 12 -rIdl ] .

( 3 .3)

We have taken several rl-values, i .e. from 0. 35 to 0 .55 and for each of then we have .found, with the expressions in § 5, the (p, K) configurations which satisfy (3 .1 a) and(3.3), and we have then computed the linear -vignetting factor v .

We have found almost constant v-factors ranging from 0 .09 to 0.11 so that we cansay that the semifield declared in table 1 can be obtained with a linear vignettingfactor which starting from zero value for the field centre, attains a 10 per cent value atthe edge of the field. As can be seen from equations (5 .20), (5 .20 a) and (5.20 b),vignetting does not begin immediately when departing from the field centre, butstarts from a minimal ylo field given by equation (5 .20 a) . y1o varies, through thesame range of rl values, from 0 .75 to 0 . 85 of the full semifield y l .

It is interesting to find the analogous conditions obtainable at the less favourablecase of a focal plane located at the primary mirror, i .e . for. e3 =0.

The obtainable fields y l , and the resulting vignetting factors . are given byequations (5 .16 a) and (5 .19 b), and the results are reported in the graph in figure 4,where they can be compared to the more favourable ones for d 1 - e 3 = 0. Roughly., theachievable Ntany l values for e3 =0 vary from 0 .7S to 0 .80 of those obtainable ford 1 -e 3 =0, and of course, intermediate reduction factors would be found forintermediate positions of the focal plane . The linear vignetting factor v turns out tobe approximately 26 per cent compared with approximately 10 per cent found ford 1 -e 3 =0.

The expression to be used for a generic position d1 -e3 =q can be found inequation (5 .19) .

Figure 3 . Vignetting of ray-bundles traversing . the focal plane :P 1 =hs(0),P3 =h5(0),

P2 :=hs(y 1 ),P5 =h5(y1),

outer boundary of ray-bundle,internal boundary of ray-bundle,

P4 =feg tany 1 , boundary of focal area .

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D. Iorio-Fili et al .

0 .15

0 .10

0 .05

Ntan~

IiII)IIII1II T0 .50

0.600.34

0.40

Figure 4. Three-mirror telescope range of optimal Ntany 1 values for vignetted solution .

We can now turn our attention to the second kind of question set out at thebeginning of this section, that is, what is the achievable range of aperture and fieldsatisfying a strict requirement for a vignette-free image? These conditions have beendiscussed in § 2 and are expressed by conditions (2 .3) and (2.4) . The correspondingequations in terms of the gaussian parameters are found in equations (5 .12) and(5 .13) . We have also described how all the configurations satisfying the vignette-freeconditions are represented by points in the (p, K) plane falling within a quadrangle,as in figure 2, delimited by the curvilinear segments representing the limitingvignette-free conditions and the limiting position of the focal plane . With referenceto figure 2, if we request a given position of the focal plane, at distance q from thesecondary mirror, any point of the (p, K) plane belonging to the BC segment of thed1 - e3 =q curve will represent a system configuration satisfying the requirement andfulfilling the vignette-free conditions, the latter represented by curves (c) and (d) .But the (c) and (d) curves depend (see equations (5 .12) and (5.13)) on the chosenN . tan y o and the tan y ,/tan y o = T-ratio . As we increase the value of the requestedN . tan y o , the (d) limiting curve becomes lower, i .e . is satisfied by lower (p, K) values,Therefore the highest admissible vignette free N . tan y o value (for a given T-ratio)calls for the lowest possible (d) curve . The extreme case is when the upper vertex Afalls on the dl - e3 = q curve, thus reducing the quadrangle to a single point where thevertices A and B coincide . The equations which the gaussian parameters must satisfyto obtain these conditions are found in equation (5.15) . The special cases in which thefocal plane falls at the secondary mirror (q = 0) and in which it falls at the primarymirror (q=d1) are obtained satisfying equations (5 .15 a) and (5.15 b) respectively .

Conditions (5 .15) as well as (5 .15 a) and (5 .15 b) are satisfied by a uniqueconfiguration (p, K) . When such a configuration is found, one can compute, throughequations (12) or (13), paragraph 5, the corresponding highest possible vignette free

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value for N . tan yo obtainable for a given n and T-ratio. The latter is in the generalcase near to unity, except for the special case in which the focal area does not representa physical obstruction, so that T can be near to zero .

We have set T=1 .05, to provide a clearance for the support of the field area, andcalculated the (p, K) solutions for many q values, both for the q = 0 and the q = 1 cases,and then found the highest admissible N . tan yo . The results are represented by thetwo curves in figure 5 . Obviously, for 0 < q < d, one would obtain an intermediatecurve .

From these graphs one can obtain any desired combination of N and yo ,representing always the highest possible vignette-free combination of theseparameters .

The N . tan y o values satisfying the strict vignette-free conditions are seen to beabout 0 . 75 to 0 . 80 of the values obtainable by the more liberal conditions previouslyconsidered in this paragraph .

0~ 10

0605

00130

0.40

0.50

0.60

Figure 5. Three-mirror telescope range of optimal Ntany 1 values for vignette-freesolutions .

4 . Performance of corrected solutionsIn order to test the kind of optical performance which can be expected from

three-mirror telescope designs covering the range of gaussian parameters describedin the preceding paragraphs, we have tried to correct a number of configurations fordifferentNnumbers and total 2y o fields, but all having the same entrance aperture of1500 mm (diameter of the primary mirror) . The configurations can be groupedunder four categories, i .e . :

(a) focal plane position at secondary mirror (q=0) and satisfying the strictvignette-free conditions ;

N tan 10

1

Theory of three-mirror telescopes

1043

I

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1044

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CD 'I- '1- 0

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NO

o OLn c V-C:,

'CNOD0*O'

d- .~ ~O 0C C' 00

in

00)n M 'O

OO o O M tn 00~Nv)00etv)00

00 v')00 cnM 10 IF

OM o ir)toCO*O'LflM00N'OO d O n m 00moo C. 0~C' NO N En

~) o

r C

OM* NM .--iO-,Cr) 0̀00

d Nv)

O'00000'C NO C' oo p'

Oo'M-tnMN N --~ MC, tM'CNd'ON*OM 7' M "'"

e-'- N N i--~ N N

O*M 111111N

N

N00 O MN 00 O -tn 00 ~tOO~OOIn CI'O~O~'tnNLf) 'CO

2c IIN11N

O C' N NO`MN--~u')00NMNO'N-M ~O \0 00 '0 'O

toi- r- M } M N

MM 11 1 11 1

t) 'C N V) C C'N N N C'O\C'10O`n-*'Cd-MM*MNd''T-O N .--SON

N00

1

1

1

1

1

1M

Ov, 0'000C'NO'OMmC-n v) - * 00 t} 00N° I I

1 I

I

O'000'N,OONNNC)*00*N**

C' C' I I I I IN N

C a° fl'~'rY'Zt A WWWIWIWIW

N

C'OM)n MM O`NrM V)4 0`

U•

C

dN

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Page 12: Theory of Three-mirror Telescopes with Diffraction Limited Performance

Theory of three-mirror telescopes

1045

(b) focal plane position at primary mirror (q=d 1 ) and satisfying strict vignette-free conditions as in (a) ;

(c) focal plane position at secondary mirror (q=0) but admitting vignettingoutside of field centre ;

(d) focal plane position at primary mirror (q=d 1 ) but admitting vignettingas in (c) .

Table 2 contains all the relevant data concerning seven selected configurationsfalling under these categories .

For configurations under (a) and (b) a 1 . 05 = T-factor has been assumed . For allconfigurations the optimal correction of aberrations has been sought at the extremeyo semi-field and for the zonal aperture h 2 =530mm=0.707 of full aperture(hb =750mm) (see reference [5]) .

The correction procedure has been as follows :

(1) The conic coefficients E1 , E2 , E 3 , for the three mirrors solving the system oflinear equations which nullifies spherical aberration coma and astigmatismin the third-order approximation have been found .

(2) The amount of higher order aberrations present in the solution obtained in(1) was determined by ray-tracing and new coefficients El , E2 , E 3 have beenfound which reduce the spherical, comatic and astigmatic residuals of thetransversal aberrations below any significant figure for the chosen field andaperture. In all cases the values of these modified E coefficients becamestable up to the sixth significant figure after the second iteration of procedure(2) .

(3) By a procedure described in reference [5], we have reconstructed the wave-front error function w (h,4), h height at entrance aperture, to azimuth referredto tangential plane (0=0) up to all the sixth-order coefficients .

(4) The root-mean square value of w (h, ca) for the entire aperture has beencomputed integrating over ten values of h and 24 values of 0 . Finally theroot mean square of the order of one hundredth of a wavelength .asymmetric terms . These root-mean square values we have called co andwasYm respectively .

Table 3 reports the results obtained for the seven configurations proposed intable 2 .

The following general conclusions can be drawn :

(1) The conic constant of the third mirror E3 which assumes important valuesfor q = 0, is reduced to modest values for the q=d1 configurations . This is amanufacturing point in favour of e3 = 0 solutions .

(2) Up to a field of 1 ° (y o = 0 ° •5 ) all configurations are diffraction limited, with 1root mean square of the order of one hundredth of a wavelength .

(3) Even for a 2° field (y o =1 °) practically diffraction limited configuration exist,as can be seen in the last column where a N=3 . 9, y o =1° configuration isshown giving a A/cD=13, thus satisfying the Marechal criterion which setsthe diffraction limited performance around 1/ui=14 (reference [6]) .

(4) For fields greater than 2° the performance seems to deteriorate rapidly, butwith careful optimization of all parameters one can probably obtain adiffraction limited up to a 2 . 5° field, especially if the telescope is used aroundthe ly wavelength .

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Page 13: Theory of Three-mirror Telescopes with Diffraction Limited Performance

1046

KE

CCOCOE'CCoN

II

~~

OII o

la0

D. Iorio-Fili et al .

Co

O OO*oO X X O

C C%, in O C-C -M N

I

I

~n

o d o x x00

o O -+ ~O [- MC'000 CC -.ON

O- O N OO

N in

O 'O~O 00 C O OC'OOOOC

ONM

o Co1

IO O

N X X c- M~ Cp 0 ~ M OO o -

IO OX XOTC' 00•

o0 d 0 N O• 2M O O O M t

CoCo

C) N

x x

~O~~n• Md'O p 0 "n4

C OO dM C C O

pM .-.tnp0

O OX X

M N .N-~ M

~~ oa

E~`ti

Ij I~KK

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Page 14: Theory of Three-mirror Telescopes with Diffraction Limited Performance

Good correction can be obtained up to relatively strong apertures, that is forlow N values .

(6) The solution presented in the first column (see table 2) is very near to theRumsey class of solutions as the third mirror almost coincides with theprimary location. Other solutions present a situation in which p < d l , i .e . thetertiary mirror is situated between the primary and the secondary mirror, asconsidered in some of the papers in reference [1] .Finally it should be remarked that the configurations used in this paper forcorrection testing are far from those which assure the best merit functionvalue as used by Robb (references [3] and [8]) . This is due to the fact that inthis paper a survey has been made of solutions satisfying a number ofdictated constraints, accepting the resulting performance . Whenever theuser's requirements or the engineering constraints allow a defined range inwhich the gaussian constants can vary, then a search of the best meritfunction value within that range can be conducted along the lines describedby Robb in reference [3] .

(5)

(7)

feq01 ; 02 ; 0 3 ;

power of primary, secondary, tertiary mirror ;4'12

power of Cassegrain system S 1S 2 ;d 1

distance of S 2 from S 1 ;d2

distance of S 3 from S 1 ip

distance of S3 from S2 i .e . length of three-mirror system ;13

back focal distance S 3F 3 of system ;back focal distance of Cassegrain system S 1S2 ;distance of final focus F3 to primary mirror (positive if F3

before primary, negative if after primary) ;nominal, or zero-field, linear obscuration factor ;parameter which characterizes the system ;

K= (1 +k) (1 +rik) K is negative if the first two mirrors form a Cassegrainsystem, it is zero if the said Cassegrain system istelescopic . K has a maximum value beyond which no realsolution exists for k .

All the parameters defining the three mirror system can be expressed as afunction ofp andK. The requirement for a flat anastigmatic field is expressed by thePetzval condition :

l12e 3

q =1- d1W1k=4'2/01

Theory of three-mirror telescopes

1047

5 . Mathematical formulation5 .1 . Gaussian parameters defining the system

All linear quantities refer to unitary focal length . Quantities are positive when asrepresented in figure 1 .

equivalent focal length of the system ;

4'1+4'2+4'3 =0 (5 .1)

For every set of values (ij , p, K) a three-mirror system is totally defined as follows :k- - ( 1+rl)-J[(1-11) 2 +411K]

(5.2)2ri

'

~1(1 -n)+-/r(1-n)2+4pK]

> (5 .3)2pK

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Page 15: Theory of Three-mirror Telescopes with Diffraction Limited Performance

1048

D. Iorio-Fili et al .

from which one finds

l -qd1=

,1

and

d2=p - dl ; 112= q ; 13 = q -PO 12 ; e3=dl+q - (1+112)p .

(5 .5)012

The distance of the final focus F3 (figure 3) from the secondary mirror is given by

rat=kral ; q'3=-(1+k)11 ;

2=(1+gk)4 1

d1 - e3=(1+cb12)p - q .

From equations (5.2) and (5 .3) one finds also the limit upper value for IKI as :

IKI (1 _q)2 or (1 -q )2

4q

4p

(5 .4)

(5 .6)

(5 .7)

where the lower value of the expressions on the right side must be retained .It is too be noted that the k and 0, values expressed by (5 .2) and (5 .3) are obtained

solving two second degree equations, namely :

(1 +k) (1 +qk)-K=0,(5 .8)

pK0i - ( 1- q)(p1 -1=0 ,

which, for K values limited by (5 .7) give two solutions for each equation, obtained byusing the opposite sign for the radical . Combining these values one obtains fourgaussian configurations for each pair of (K, p) values. Solutions with a minus sign ofthe radical in (5.3) give stronger 0 1 power values and are therefore less attractive :solutions with a plus sign of the radical in (5.2) give weaker 42 powers which of courseare acceptable, and can be useful in many cases . We shall indicate when necessarysuch values of k by k.

5 .2 . Aperture and field equationsFigure 1 shows how the envelope of the bundle of rays through the system, for a

given object-field angle y o is determined by two extreme rays denoted by thesuperscripts s and i . These rays have opposite signs for y, but the followingequations are written for the absolute value of y o ; i .e . the effect of opposite signs isincorporated in the algebraic expressions .

The height of incidence h of the rays on the different surfaces are denoted by thesuperscript p for the entrance pupil, 1 and 2 for the primary and secondary mirror, 3for the central bore of the primary mirror and 4 for the tertiary mirror . Finallysubscript 5 indicates the incidence on the focal plane . The relevant quantities for ourstudy are h,, hi, hz, h3, h4 which describe the external boundaries of the mirrors andof the central bore of the primary, while h,, hi, h' , h5 are necessary to compute theeffective obscuration factor and the clearances of the bundle of rays at the criticalpoints, i .e . at the central bore and at the focal plane . h2 is the height where both thes- and the i-ray must intersect .

With these conventions the following equations hold (all linear dimensions arefor unitary focal length) :

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Page 16: Theory of Three-mirror Telescopes with Diffraction Limited Performance

Theory of three-mirror telescopes

1049

for s-rays at -yo angle :I

hs =P 2N'

hl =hp + dp tan yo1

= 2N + dp tan y o ,

h2 =11h1 + d1 tan y o =2N +

( rjdp + d l ) tan yo,

(5.9)

d

d

d

d

dh3

1 112 h2 + tango= 1-llz

2N+1-llz (rldp +dl)+

tan y,

h4= 1- p h2-' +p tan y o = 1 p+ 1- P (rldp +dl)+p tan y o ;

112

11

112 2N

l12

q

for i-rays at +y o angle :

h p̀ =h2+(dp -dl) tan yo=2N+(1 +j)dp tan 70 ,

hi=hsz -d1 tan y o =rlh1=2N+ndp tan yo,

2hz = rlh2 - ( 1 +tl)d l tan y o =

2N+(1 2dp -dl) tan y o ,

( 5 .10)

h5- 1- d1-e3 h2-dl-e3 tanyo

l12

rldl -e3 112

dl -e3\ 2

dl -e3=C1-~- + C1- I(tl dp -dl)-~ tanTo,1 1 2

2N

1 12

11J

from which the expression for the effective obscuration factor J eff can be derived

h'Jeff = hP = 11 + 2N(1 +q)dp tan yo ,

(5.11)p

for dp =0, J e ff is not equal to il, as could be assumed by simply writing dp = 0 in (5 .10),because the obscuration boundary coincides with the secondary mirror edge h2( - yo)which is not located at the e .p. plane . Therefore, for dp =0 one must substitute (5 . 10)by :

11eff =hs =11+2N tan y o .

(5 .11 a)1

In § 1 we have explained that the vignette-free requirement imposes .twoconditions, expressed as (1) and (3) . We can give here, using equations (5 .2) to (5.9),the explicit expressions of these conditions in terms of the gaussian parameters .

The first condition h' >h' (equation (2.5)) requires that :1

(1++1)l12-rldl<g22Ntanyo+dp

;

(5.12)

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Page 17: Theory of Three-mirror Telescopes with Diffraction Limited Performance

1050 D. Iorio-Fili et al .

the second condition hs ifeq . tan T, (equation (2.4)) requires that :Ctany l dl -e3 1

-1

z1tan g o + q

di-e3 +d'_<~1 2Ntanyo+dp

(5 .13)1

1 12

It is to be remarked that the right-hand members of (5 .12) and (5 .13) areidentical, and that the inequality signs of the equations are the same .

Referring to figure 2, we recall from § 3 that the highest admissible vignette freeN . tan y o value for a requested position q of the focal field and for a given tan y 1 /tan y o= T ratio, can be achieved by a (p, K) configuration for which the upper vertex A ofthe quadrangle falls on the d1 - e3 = q curve, thus reducing the quadrangle to a pointwhere the vertices A and B coincide . This happens equalling the left side members ofequations (5.12) and (5.13), and requesting that d1 -e3 =q at the same time .Remembering equation (5.6) we obtain the simultaneous conditions :

(1+rt)(112 -d1)(11z - q) -CT

-q) 112=0,///

515( .

)(1 +412)p -11=q,

which are satisfied by a unique (p, K) pair .For a focal plane located at the secondary mirror (q = 0) the conditions become

( 1 +x1)112 -d1 - T= O,

(1 +412)p -11= 0and for a focal plane located at the primary, (q=d1 ) we obtain :

(1+r1)(112-d1)2-T+17

d)112=0,

(5 .156

(1+412)p-d1 -rf=0 .

The (p, K) pair satisfying these equations can be found by assuming an initial(negative) value for K and finding the p value which through equations (5.2), (5 .3),(5 .4) and (5.5) provides values for d 1 and 1 12 which satisfy the first condition andthen verifying the second . The original K value is gradually varied until bothconditions are satisfied simultaneously .

Considering the more liberal vignetting conditions discussed in § 3 point (1) werecall that equation (3 .1) sets the limitation

zN .tanyl<Ci- dl-e3 q

(5 .16)1 12

2

for generic position of the focal field d1 -e3 =q we can also write

(5.15 a)

zN .tany l <, 1- q n

(5 .16 a)1 12

2

which gives the maximum possible N . tan yl for q=O . For these values of N . tany 1only a limited field is truly vignette-free, beyond which vignetting sets in as the fieldincreases up to the semifield limit y l .

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Page 18: Theory of Three-mirror Telescopes with Diffraction Limited Performance

The vignetting effect is illustrated in figure 3, and is represented linearly by thesagitta s of the vignetted lunula or by the `linear vignetting factor' v as defined inequation (3 .2) .

We can compute hs(y 1 ) and hs(y 1 ) using equations (5 .9) and (5 .10) with dueconsideration to the applicable signs for y1 and find, for a generic semifield y 1 :

so that for a generic semifield y1 we get :2

S= 1+(l-'

d l + q tany l - 1-q n ,llz)

q

112) 2N'

1 +(1 -112)d, +q

which for q= 0 becomes

and for q=d 1

Theory of three-mirror telescopes

2

lh`s(y1) = C1-q~ q -

(1- q)d1+q ltany 1l12 2N

hz

q

hs(y1)=C1-q q--[_[(1-

q)d1

+q] tany,112) 2N

112

11

V=

1-q112

11 N tan 7, -r1(1-r1)

2(1 q

-n)

Introducing the maximum N . tan y1 value given in (5.16 a) we getz

s=[(1 -qd1+q](1 -q n ,

llz

11

hz)2N

V= 1-q dl+q n ,l1z)

nj 2(1-n)

,=(,-d1 +1 1-dl d,2

11,

112 q

112) 2N

v=(1-d1 +1)d1

nl12 q

2(1-t1)

from equation (5 .18) one can see that the field is vignette-free only up to the y 1 valuewhich nullifies the right hand expressions, that is for

1

qz

112qtanylo=

/,

+I1- 4 d1 + 4 2N'1- -\

112 n

1051

(5 .17)

(5.18)

(5 .19)

(5.19 a)

(5.19 b)

(5 .20)

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Page 19: Theory of Three-mirror Telescopes with Diffraction Limited Performance

[1] SHACK, R . V., and MEINEL, A. B ., 1966, Paper contributed to the Fiftieth AnniversaryMeeting of the Optical Society of America, Washington. KORSH, D ., 1972, Appl .Optics, 11, 2986. GELLES, R ., 1973, Appl. Optics, 12, 935 .

[2] RuMSEY, N. J ., 1971, Proc. astr. Soc . Aust ., 2, 22 .[3] ROBB, P. N ., 1978, App!. Optics, 17, 2677 .[4] IORlo-FILL, D ., and SCANDONE, F., 1979, European Space Agency Report (unpublished) .[5] IoRlo-FILI, D ., and SCANDONE, F ., 1978, Colloquium on European Satellite Astrometry,

Padua .[6] BORN, M ., and WOLF, E ., 1975, Principles of Optics (Oxford: Pergamon Press), pp. 468-

469 .[7] WETHERELL, W . B ., 1972, Instrumentation in Astronomy, edited by L . Laniore and R . W.

Poindexter, Vol . 28 of SPIE Proceedings (Society of Photo-Optical InstrumentationEngineers).

[8] ROBB, P . N., 1976, J. opt . Soc. Am ., 66, 1037 .

1052 Theory of three-mirror telescopes

1

z(5 .20 a)tan 71,0 =

.

2N'

1- d1

z

tan 7 1 , 0 = 112

q (5 .20 b)d l

dl 2N1+ 1-- dl +-

11 z

11

References

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