New far infrared and millimetric telescopes fordifferential measurements with a large choppingangle in the sky

Marco De Petris, Massimo Gervasi, and Fabrizio Liberati

The optical design of two Cassegrain-type telescopes of 1200- and 2500-mm primary mirrors has been studiedby wobbling subreflectors to obtain differential measurements in wavelengths ranging from 300 Am to 2 mm of

the spectrum. Wide chopping angles in the sky have been reached in spite of the large apertures and high

secondary magnifications. Diffraction-limited performances are maintained by tilting the secondary mirroraround the primary focus. This work is included in the TIR (Telescopio InfraRosso) project.

1. Introduction

The optical design of two Cassegrain telescopes withprimary mirrors of 1200 and 2500 mm is described tostudy diffuse radiation in the FIR and millimetricregions. More specifically they are used to researchanisotropies in the temperature distribution of thecosmic background radiation (CBR) on angular scalesfrom 30 min of arc up to 3° in the wavelengths rangingfrom 300,gm to 2 mm and to study intergalactic molec-ular clouds at 300,gm and 1 mm. An accurate opticaldesign is needed to obtain extremely precise observa-tions. The optical performances of such instrumentsare limited by diffraction and aberrations typical of anoff-axis system.

Bolometers cooled up to 0.3 K with Winston cones ofdifferent diameter sizes will be placed as detectors inthe focal plane of these instruments.

An input aperture 45 mm in diameter is used tostudy anisotropies and to increase sensitivity; we shallcall these configurations flux collectors. The 1200-mm telescope will be used only for this kind of mea-surement at X = 1 and 2 mm matched with a photom-eter1 with a throughput A -. = 0.62 cm 2 sr and an half-power beamwidth (HPBW) = 12.80(f/4.5) (hereafter1200FC). Two configurations are intended for the

When this work was done Fabrizio Liberati was with SeleniaS.p.A., Electrooptics Department, I-00040 Pomezia, Italy; he is nowwith Ottico Meccanica Italiana, Gruppo Agusta, 79-81, I-00146Rome, Italy; the other authors are with La Sapienza University,Physics Department, I-00185 Rome, Italy.

25 April 1988.0003-6935/89/101785-08$02.00/0.© 1989 Optical Society of America.

2500-mm telescope. The first one is a flux collector,2500FC, with a photometer with four bands rangingfrom 300 Am to 2 mm. To make possible correlatedmeasurements, each spectral band observes the samefield of view. However, the study of molecular cloudsneeds a higher resolution. Therefore, a detector with asmaller aperture (diameter = 3 mm) or an array of 3 X 3of these at a 300-gm operation wavelength or one witha diameter of 9 mm at X = 1 mm (2500AR) will beplaced in the focal plane of the 2500-mm telescope.

The observations with these instruments will takeplace onboard a stratospherical balloon and at theAstronomical Observatory in Campo Imperatore (AQ-Abruzzo, Italy).

All these configurations are of the Cassegrain type,preferred for its compactness and simplicity. Thescanning of the sky is obtained by wobbling the sec-ondary element. This choice, which enlarges the inci-dence of geometrical aberrations, has been made forseveral reasons. As far as aberrations are concerned,the best choice would have been to modulate on thefocal plane by displacement of the photometer or wob-bling of a tertiary mirror. Due to the excessive micro-phonic noise induced, the first method has not beenconsidered. The second one is dangerous because awobbling element is supposed to modulate only the skyradiation. But a tilting third mirror near the Casse-grain focus will modulate not only the radiation fromthe primary and secondary mirrors but also the 300 Kbackground radiation, which will cancel the one ofinterest. To avoid this it is advisable to let the second-ary mirror tilt.

The 2500 telescope has a 0.5 primary focal ratio.This gives a very compact structure, useful for placingit on the platform of a balloon. The Cassegrainianfocal ratio is equal to 4, the beam size of 20 min of arc

15 May 1989 / Vol. 28, No. 10/ APPLIED OPTICS 1785

with a 45-mm aperture and of 1 min of arc with a 3-mmone. To move from the 2500FC to the 2500AR config-uration it is sufficient to change the subreflector, leav-ing the structure and modulation system untouched.

The 1200 telescope has a 0.4 primary focal ratio anda beam size of 10 with a 45-mm aperture.

These very fast primary mirrors, combined withvery powerful secondary ones, are usually not recom-mended because of their excessive aberration prob-lems. Telescopes using such a modulation systemhave reached angular separations much less than wehave. Moreover, for radio wavelengths, they are limit-ed by a higher diffraction so that geometrical optimiza-tion could be almost ignored. Larger separations areachieved using an optical system with a single tiltingplane element. For the angles we are interested in,this solution imposes a great reduction of the through-put, thus greatly affecting the sensitivity of the mea-surement. The key to an accurate performance is theright position of the wobbling subreflector. We reportthe relationship between the telescope's beam shiftangle and the tilting angle of the secondary mirror as afunction of optical parameters. The features of thesystem at the three operation wavelengths were ana-lyzed according to the optimized configurations.

11. Optimization Procedure

The instrumentation will work correctly if there is agood coupling of telescope and photometer. This onehas an angular response which describes its field ofview. From our point of view a good choice is to let thedetector see the border of the secondary mirror at suchan angle that its signal drops more than 10% of themaximum. In this way we reduce the importance ofthe modulated signal diffracted by the border of thesecondary mirror which would cause an offset to thesignal we are studying. For this reason we have differ-ent throughput for the photometer and telescope.The 1200 telescope has a focal ratio equal to 3 to bematched with af/4.5 photometer. The 2500 telescope,both the configurations, has a Cassegrain f/No. equalto 4 and will be used with a photometer with an oppor-tune f/No. depending on its angular response shape.

The best optical system has been achieved using theconfiguration in which the secondary mirror has itsmaximum inclination with respect to the optical axisfor each system. In fact in such a configuration theaberrations are at a maximum.

The possibility of spurious signals due to modula-tion has been eliminated by checking that the detectorcould pick up only sky radiation through primary andsecondary mirrors. In particular, we avoided the de-tector seeing through the subreflector beyond the bor-der of the primary mirror. This is the same as consid-ering an entrance pupil with a diameter smaller thanthe physical diameter of the main mirror wisely decen-tered, which we call an effective primary diameter.This is justified by the fact that the secondary mirror,tilting, lets the detector see different regions of theprimary mirror. A different heating and then a differ-ent emission on the main mirror surface can induce a

signal synchronous with the chopping frequency. Tomaintain the thermal gradients within observationlimits and avoid earth and sun radiation, a carefulshield design is required. 2' 3

The optimization has been carried out so as to obtainmaximum separation in the sky. This enabled a mod-ulation among three fields, in the 45-mm cone aperturecase and of 30 min of arc in the 3- and 9-mm coneaperture cases. Moreover we looked for diffraction-limited performances at the smallest wavelength ofobservation.

We intend to observe differential measurements ofintensity coming from directions separated by an angleAO. The minimum value for AO is equal to the ampli-tude of the field of view, corresponding to the dimen-sions d of the detector on the focal plane. Every objectimaged by an optical system can be considered ascomposed of an infinite number of Fourier terms, i.e.,of a sinusoidal sequence of dark and light of ascendingspatial frequency. The peak sensitivity of such anideal system is reached at the frequency Vmod corre-sponding to the amplitude of modulation. When con-sidering two-beam modulation the sensitivity is main-tained also at lower frequencies, which are seen as adrift. At higher frequencies the sensitivity dropsabruptly because the detector integrates all the radia-tion collected. Therefore, to judge an optical systemwe must use frequencies up to Vmax = 1/2d and only forthe component along the wobbling tangential direc-tion. Moreover it is important to estimate this at theinner edge of the field of view, which is next to the otherone. A good performance at higher frequencies and inthe transversal direction allows the observation of afield of view almost unpolluted by unwanted radiation.The minimum spatial frequency of interest is given bythe maximum sky field separation in this way: Vmin =1/F tanAOmax, where F is the effective focal length. Inour optical systems these frequencies are reported inTable I. For our purposes the sky temperature distri-bution could be approximated by the Vmod componentof the Fourier series as

T(v) = To +- cos 27r- ,2 Vmod /

(1)

where To is the cosmic background temperature (2.7K). The cosine temperature distribution is imaged asa cosine distribution image of the same frequency

Table I. Limiting Spatial Frequencies

Telescope: 1200 mmMinimum frequency for two fields of modulation 6.4 X 10-3 c/mmMinimum frequency for three fields of modulation 9.9 X 10-3 c/mmMaximum frequency for a 45-mm detector aperture 1.1 X 10-2 c/mm

Telescope: 2500-mm flux collectorMinimum frequency for two fields of modulation 5.1 X 103 c/mmMinimum frequency for three fields of modulation 8.3 X 10-3 c/mmMaximum frequency for a 45-mm detector aperture 1.1 X 10-2 c/mm

Telescope: 2500-mm arrayMinimum frequency (maximum sky separation) 1.1 X 10-2c/mmMaximum frequency for a 9-mm detector aperture 5.5 X 10-2 c/mmMaximum frequency for a 3-mm detector aperture 1.6 X 10-1 c/mm

1786 APPLIED OPTICS / Vol. 28, No. 10/ 15 May 1989

15

t0

0L0

iX=2 mm

A=I mm

- A=300 Am I I-~~~~~ - - -- -_ - -

100 200Distance subreflector vertex-Pivot (mm)

300

Fig. 1. Spot size in the focal plane as a function of the distancebetween the subreflector vertex and wobble axis with a FOV of 31.5min of arc (2500FC): solid line; radial dimension; dot-dashed line;

tangential dimension. The three horizontal dashed lines are theAiry radius for X = 300 jm, 1 mm, and 2 mm.

Vmod.4 The temperature contrast in the sky AT/T(a 10-5 for the CBR) reduces itself on the focal planeto the product of AT/T with the tangential componentof the modulation transfer function (MTF). There-fore, to achieve the maximum sensitivity, the value ofthe frequency Vmod must be much smaller than thelimiting frequency v = 1/Xf , where f# is the Casse-grain f/No. number, and so the MTF must be as closeas possible to 1, with respect to the limiting value givenby diffractions:

MTF(v) = 2 (0- cos sino),Or

(2)

where = arcos(Xvf#).

Ill. Best Wobbling Subreflector Axis Position

In the optimization procedure we considered as freeparameters the decentration of the tilted secondarymirror as well as curvatures, conic constants, and sepa-ration between the two mirrors. In this way we veri-fied that the best wobble axis position of the subreflec-tor is coincident with the focal point of the primary

FOCAL PLANE MAP

.. ' .4.iJ..

W1,ie

:.. .: _2

1200FC FOCAL PLANE MAP

,aaMt.

e'I '

"Si:.-

'8Uat

2500FC FOCAL PLANE MAP 2500AR

Attaa.

tilt = 0 degrees tilt = 0 degreestilt = 0 degrees

-OW

)I

j- ..)":4;1.

tilt = 4 degreestilt = 1.5 degrees

tilt = 2.5 degreesFig.2. Maps of the spots in the focal plane for the three configurations with the subreflector tilted and on-axis. In the first two cases (FC) (a)and (b) the spots are focused on a circle having the dimension of the 45-mm photometer aperture, while in the AR case (c) each spot is on the

axis of the cone of the array.

15 May 1989 / Vol. 28, No. 10/ APPLIED OPTICS 1787

. ~ ~ ~ ~ ~ ~ ~ . .. . .l

. I I I . . . I . . I . I . . .

A: Hi Of

i e Sl {stow

,4

: :W: :: ---. -..... .

I-, .. t.'-'. . -

' ' like _�'_

: -' .

> -r

`W:4

mirror. It was possible to achieve large wobble anglesin spite of the high secondary magnifications. Figure1 shows in the case of the 2500FC configuration a plotof the radial and tangential dimensions of the spot as afunction of the distance between the wobble axis andvertex of the subreflector at a fixed angle of view. Itshould be pointed out that we have smaller aberrationswhen the rotation radius is equal to the secondaryvertex-primary focus distance. 6 7 We have to stressthat there are narrow limits on the rotation radius onlyat 300 Am to be diffraction-limited.

This rotation system has better optical perfor-mances than the one around the subreflector vertex,but it yields two mechanical disadvantages. The firstis that to obtain the same beam separation in the skywe have to tilt the secondary mirror more. Indeed therelationship between the telescope's beam shift AOand the subreflector's tilting angle AcI is8

= [R(B1f BDF) p (BDf + BDF)] A. (3)

Here R and p are the distances from the secondarymirror vertex to the subreflector pivot axis and to theprimary focus, respectively; f and F are the primaryand Cassegrainian focal lengths, and BDf and BDF arethe corresponding beam deviation factors.9 In ourcase with R _ p Eq. (3) turns into

AO0 = [i BDF + BDF A (4)

with y as secondary magnification. Note the oppositeangle direction between the subreflector and field axis.On the contrary when the wobble axis is closed to thesecondary mirror vertex R << p, and so

AO = (BDF + BDf)1A4D. (5)

The difference between Eqs. (4) and (5) is in the termsBDf and BDF/y. Configurations with high secondarymagnification, as we obtained, allow this inequalityBDF/y < BDf, and so they have a larger subreflector'sangle shift at a settled angle of view. The seconddisadvantage is the increasing of the subreflector'smoment of inertia and consequently the necessarysupply to nutate the mirror.

Maps on the focal plane of different objects on theaxis and on the edge of the field of view for the three on-and off-axis configurations are illustrated in Figs. 2(a),(b), and (c). The coma aberration is minimized,whereas, for these large chop angles, astigmatism dom-inates the spots. The astigmatic image dimensions,chopping the secondary mirror around the so-calledcoma-free or neutral point (primary focus), are lowerthan the estimated ones.'0 In Figs.2 the defocus in theon-axis position due to the mirrors approaching causedby rotation is also evident. Because the telescopes aredevoted to differential measurements, integrating ra-diation inside an instantaneous field of view, imagequality is not essential. We made sure that the tan-gential dimension of the spot was narrower than theAiry disk. In this way it is possible to achieve a steep

slope of the geometric angular response along the mod-ulation direction. This avoids overlapping betweenthe wings, which is contamination between the differ-ent fields of view observed.

IV. Optimized Configurations

The optimized configurations stray from true Casse-grain ones because both mirrors are hyperboloids.This configuration, which eliminates coma and spheri-

Table II. Optical Parameters of 2500-mm Telescope

PrimariesPrimary diameterPrimary curvature radiusPrimary f/No.

Flux collector configurationEffective primary diameterSubreflector diameterSubreflector curvature radiusCassegrain f/No.Effective focal lengthDistance subreflector-primary mirrorDistance subreflector-Cassegrain focusDistance subreflector vertex-pivot axisMagnificationCone aperture diameterHalf power beamwidthMaximum sky field separationMaximum subreflector chop angleTelescope throughput

Array configurationEffective primary diameterSubreflector diameterSubreflector curvature radiusCassegrain f/No.Effective focal lengthDistance subreflector-primary mirrorDistance subreflector-Cassegrain focusDistance subreflector vertex-pivot axisMagnificationCone aperture diameter ACone aperture diameter BHalf-power beamwidth AHalf-power beamwidth BMaximum sky field separationMaximum subreflector chop angleTelescope throughput ATelescope throughput B

2500 mm2494.5 mm

0.5

2000 mm410 mm539.78 mm

4.058151 mm1019 mm1480 mm228.6 mm

8.145 mm19.2 min of arc1.0502.500.77 cm2 sr

2400 mm360 mm434.01 mm

3.979273 mm1059 mm1393 mm181.5 mm

7.943 mm9 mm1.15 min of arc3.45 min of arc

31.5 min of arc1.503.4 X 10-3 cm2 sr3.1 X 10-2 cm2 sr

Table IlIl. Optical Parameters of 1200-mm Telescope

Primary diameterPrimary curvature radiusPrimary f/No.Effective primary diameterSubreflector diameterSubreflector curvature radiusCassegrain f/No.Effective focal lengthDistance subreflector-primary mirrorDistance subreflector-Cassegrain focusDistance subreflector vertex-pivot axisMagnificationCone aperture diameterHalf power beamwidthMaximum sky field separationMaximum subreflector chop angleTelescope throughputPhotometer throughput

1200 mm1002 mm

0.417900 mm250 mm296.1 mm- 3.0

2645 mm380.4 mm620.2 mm114.3 mm

7.245 mm

1.002.360401.52 cm2 sr0.62 cm2 sr

1788 APPLIED OPTICS / Vol. 28, No. 10/ 15 May 1989

0

-c

80

60

40 l

10 11

20 - 1\ 11

n TV . _ | I Il I , / -

0 0.2 0.4a (degrees)

0.6 0.8

Fig. 3. Tangential angular responses of the 2500FC configurationwith the tilt = 0 and 2.50 of the secondary mirror: solid line,geometric angular response; dotted line, diffractive angular responseat X = 300,um; dashed line, diffractive angular response at X = 1 mm;

dot-dashed line, diffractive angular response at X = 2 mm.

cal aberrations leaving astigmatism, is called a tilted-aplanatic configuration." The optimization was car-ried out by the FORTRAN optical program ACCOS V.

In Tables II and III are assembled the geometricaland optical data of the three configurations 2500FC,2500AR, and 1200FC, respectively. In spite of thelarge chop angles of the subreflectors, the systemsremain diffraction-limited. We show in Figs. 3-5 theangular responses along the modulation direction inthe on- and off-axis configurations. There are angularresponses for the aberrated systems and with the dif-fraction contribution at 300 Eim, 1 mm, and 2 mm as acomparison. It is clear that the two telescopes as fluxcollectors are able to modulate between three or twofields of view, as expected. In the case of 2500AR themaximum allowed chop angle in the sky is really -30times the beamwidth.

Because of the peculiar range of wavelengths (300,um-2mm), we deemed it necessary to analyze the dif-fractive contribution on performances of, the opticalsystems. We also have considered the effect of thecentral obstruction even if the higher secondary-pri-mary diameter ratio e is 0.2 for the 1200FC configura-tion. Really the normalized intensity distribution forthe Fraunhofer pattern of an annular aperture with E =0.2 is too close to the Airy pattern.' 2 In Fig. 6 wepresent the point spread function profiles (PSFs) andthe radial energy distribution plots (REDs) of thespots in the focal plane of the 2500FC on- and off-axisconfiguration. The sagittal and tangential compo-nents of PSF in the tilted case are also shown separate-ly. Only for 300 ,gm the geometrical aberrations arecomparable to diffraction size, and the sheer diffrac-tive and diffractive-aberrated curves are distinguish-able. This is manifest in the 3-D plots of the PSF inFig. 7, where the sagittal astigmatic dimension prevailsover the diffractive Airy disk. Indeed at wavelengths>300)um the geometrical aberrations are negligible. Awavelength aberration map is shown in Fig. 8 for X =300,4m in on- and off-axis configurations. The defo-

10

so

40

20

0

0

Cb

19

6 (rrmin)

0 (aremin)

Fig. 4. Tangential angular responses of the 2500AR configurationwith tilt = 0° and 1.50 of the secondary mirror for the case of a 3-mmdiam aperture (a) and a 9-mm one (b): solid line, geometric angularresponse; dotted line, diffractive angular response at A = 300 ,um;

dashed line, diffractive angular response at X = 1 mm.

too

80

0

-tg

B0

40

20

0 _0 0.5 1 1.5 2

B (degrees)

Fig. 5. Tangential angular responses of the 1200FC configurationwith tilt = 0 and 40 of the secondary mirror: solid line, geometricangular response; dashed line, diffractive angular response at A = 1

mm; dot-dashed line, diffractive angular response at X = 2 mm.

cus for tilt = 00 of the subreflector and the astigmatismaberration for tilt = 2.50 is obvious. The ranges of theoptical path difference are symbolized in the figurecaption. We analyzed the performances of all configu-rations by their modulation transfer function at wave-lengths and spatial frequencies of interest. We reportin Figs. 9-11 the plots for the three configurations.Only for the 2500FC we show the geometrical compo-nents separately to point out the best values of MTF

15 May 1989 / Vol. 28, No. 10/ APPLIED OPTICS 1789

, - \-:. - - --

._ _I _ I \'

Re

r.I

5b

S1

-15.00

-4.000 0.000 4.000

(mm)

1s.0-7.500 0.000 7.500

(mm)

-40.00 -0. 00 0.000 20.00 40.00

(mm)

Fig. 6. Point spread function profiles and radial energy distribu-tion plots for the image spots of the 2500FC configuration. Theoperation wavelengths are: 300 ,um (a), 1 mm (b), and 2 mm (c):solid line, tilt = 0°, diffractive case; dotted line, sheer diffractive casewith obstruction; dashed line, tilt = 2.50, sagittal geometric compo-nent; dot-dashed line, tilt = 2.50, tangential geometric component.

for the tangential one at the inner edge of the titledbeam, as required.

V. ConclusionsThis study was done because telescopes with such

large apertures, high secondary magnifications, and

1.000

= 300 ym0.775

0.000tilt = 0 degrees tilt = 2.5 degrees

i.0°0 Fig. 7. Three-dimensional point spread function for the 2500FCconfiguration in the two cases of tilt = 00 and 2.5° for X = 300 Am and1 mm. The dimensions of the grids are 9.5 and 18 mm, respectively.

0.7000

0.0000

0.2500

0.000

0 .000

'.'1 |b--|stisFIX!.-.

| * @ @ @ * *

| * 9 s s * @

: : : : : : :| : : : : : ::.-@@@^b : : : : : :f ::::::K . * * s * -

| * . @ @ @ @

s * 9 @ * * @

: : : : : : :: : : : : : :* - @ @ - @ |:::::::::::::t . * *. t,,,.,,...

l 1:: ad r

itill* liii,It 1111,

III:

11'

tilt = 0 degrees

. I Ii

a:

tilt = 2.5 degrees

Fig. 8. Wavelength aberration map for the 2500FC configuration inthe cases of tilt = 00 and 2.50 at the center of the field of view at A =300 ,um: subreflector obstruction; 0, optical path difference inthe range between - /4 and +X/4, +1 for A/4 ' OPD < 3X/4, and-1

for -3A/4 < OPD ' -A/4.

wide chopping angles in the sky have off-axis unsatis-factory optical performances. In the peculiar regionof wavelengths from 300 m to 2 mm there are specificproblems of aberrations for visible telescopes and ofdiffraction for radio ones. The two optimized tele-scopes with 1200- and 2500-mm main mirrors will op-erate on stratospherical balloons and at the Astronom-ical Observatory in Campo Imperatore (AQ) with alt-azimuth mounts. The mirrors are realized inaluminum alloy (G Al Si 7 Ti UNI 7257-73 Avio). Thestructure is a light and stiff ribwork. The 1200-mmtelescope was launched on 20 Sept. 1988. In this way itwas possible to test the modulation system and moni-tor the thermal gradients of the main mirror surface athigh altitude. The next flight is foreseen for July1989. We expect the 2500-mm telescope to be com-pleted to fly in 1990. In Fig. 12 is shown a sketch of thegondola from two different points of view.

1790 APPLIED OPTICS / Vol. 28, No. 10/ 15 May 1989

-0.000

....I I I :. . .

1.000

0.7sO

0.5000 5

0.2000 0.4000 0.6000 O.W00 0.000 0.2000 0.4000 0.6000 0.8000

c/m c/m1.OC

0.8000

0.6000

0.4000

0.2000

0.000

0.7500

0.5000

0.2500

0.000

1.000

i ooo

0.4000

0.6000

0.00

D. 2000

0.000

0.000 5.0OOOE-02 0.1000 0.1500 0.2000 0.2500 0.000 2.5000E-02 5.OOOOE-02 7.5000E-02 0.1000 Q.125

c/mm c/mmr

Fig. 9. Modulation transfer function for the 2500FC configuration. (a) Geometric components: solid line, tilt = 0°; dashed line, tilt = 2.50,

sagittal component; dot-dashed line, tilt = 2.50, tangential component. The two edges of the beam are distinguished with i for the inner one

and o for the outer one. (b) Diffractive components for X = 300 ,um. Dotted line, shear diffractive component. The other symbols of the

curves are the same as (a). (c) Diffractive components for A = 1 mm. (d) Diffractive components for X = 2 mm. The symbols are the same as(a) and (b).

L.000

0.7soo

0.5000

0.2500

0.000

1.000

0 .000

0.6000

0.000

.00

0.000 0.200 0.4000 0.6000 .E000 0.000 5.0000E-02 0.1000 0.1500 0.2000 0.250

c/mm, c/tmm

Fig. 10. Modulation transfer function for the 2500AR configuration: (a) diffractive components for X = 300 pm; (b) diffractive components

for A = 1 mm. The symbols are the same as Fig. 9.

15 May 1989 / Vol. 28, No. 10/ APPLIED OPTICS 1791

0. _

0.000

_ 0.25 _

_0.000

1.000

I

L-i

0.000 5.00005E02 0.0000 0.100 0.2DO o.0000

c/m

L.OW0

0.6571

0.7143

0.57i4

0.4286

0.0657

0.14a9

0.000

0.3D05 0.3500

1.000

0.8336

O.GWs

0.5000

0.3333

0. t667

- 0.0500.000 3.0000-02 6.00005-02 S.0050-0 0.1200 O.1505 0.1800

c/mm

Fig. 11. Modulation transfer function for the 1200FC configura-tion: (a) diffractive components for A = 1 mm; (b) diffractive

components for A = 2 mm. The symbols are the same as Fig. 9.

We wish to thank the ElectroOptics Department ofSelenia S.p.A. (Pomezia) for making possible use of theFORTRAN optical program ACCOS V. In particular weare indebted to G. Benedetti Michelangeli for manystimulating discussions on optical problems and to F.Melchiorri, G. Dall'Oglio, P. de Bernardis, and S. Masifor helpful suggestions on IR experimental problems.We also acknowledge M. Perciballi for his technicaldrawings.

This study has been financially supported by PianoSpaziale Nazionale of CNR for the development ofTIR (Telescopio InfraRosso) project.

References

1. C. Mason, C. Ceccarelli, S. Masi, G. Dall'Oglio, G. Ferri, and S. J.E. Radford, "An Improved FIR Photometer for Atmosphericand Astronomical Studies," Infrared Phys. 26, 273 (1986).

2. L. Piccirillo, A. Moleti, and S. Masi, "Considerations on Balloon-Borne Far Infrared Telescopes," Infrared Phys. 27, 215 (1987).

3. P. de Bernardis, M. De Petris, M. Epifani, M. Gervasi, G. Guari-ni, and S. Masi, "A Balloon Borne Millimetric Telescope: Ex-perimental Study of the Thermal Gradients on the PrimaryMirror," Infrared Phys. 28, 243 (1988).

4. W. J. Smith, Modern Optical Engineering (McGraw-Hill, NewYork, 1966).

5. A. Cox, A System of Optical Design (Focal Press, London, 1964).

Fig. 12. Sketch of the gondola with the 2500-mm telescope, whichwill fly in 1990.

6. H. van de Stadt, "Optimum Location of the Wobble Axis ofSecondary Mirrors in Cassegrain-Type Telescopes," IEEETrans. Antennas Propag. AP-32, 1128 (1984).

7. H. van de Stadt and J. Verkerk, "Large Chopping SecondaryMirror for the 15-m Submillimeter James Clerk Maxwell Tele-scope," Appl. Opt. 26, 3446-3454 (1987).

8. S. J. E. Radford, "Observations at Millimeter Wavelengths ofSmall Angular Scale Isotropy in the Cosmic Background Radia-tion," Ph.D. Thesis, Washington, DC (1986).

9. Y. T. Lo, "On the Beam Deviation Factor of a Parabolic Reflec-tor," IRE Trans. Antennas Propag. AP-8, 347 (1960).

10. A. B. Meinel and M. P. Meinel, "Aberrations of an IR ChoppingSecondary," Appl. Opt. 23, 2675-2676 (1986).

11. M. Bottema and R. A. Woodruff, "Third Order Aberrations inCassegrain-Type Telescopes and Coma Correction in Servo-Stabilized Images," Appl. Opt. 10, 300-303 (1971).

12. M. Born and E. Wolf, Principles of Optics (Pergamon, London,1959).

1792 APPLIED OPTICS Vol. 28, No. 10/ 15 May 1989