Heat trap with flare as multimode antenna

John C. Mather, Marco Toral, and Hamid Hemmati

A heat trap, also called a Winston cone or compound parabolic concentrator, has been designed and built toserve as the beam defining optic for the far-infrared absolute spectrophotometer on the Cosmic BackgroundExplorer satellite. This quasi-optical multimode antenna is coupled to a flared entrance section shaped like atrumpet bell to reduce response to off-axis radiation. The antenna has very low sidelobes over a two-decadefrequency range from 1 to 100 cm-1. Fabrication is described, and stray light data are presented andcompared to theory. A novel optical figure test for concentrators is presented.

1. Introduction

This paper describes a trumpet shaped optical ele-ment designed to collect radiation from a 7° field ofview on the sky and direct it into a far-IR spectropho-tometer. The optical element is variously called ahorn, antenna, Winston cone, compound parabolicconcentrator (CPC), or heat trap. It will be part of aninstrument to measure the brightness of the cosmicbackground radiation that is generally thought to bethe remnant of the primeval explosion, the Big Bang,which was the earliest known phase of our expandinguniverse. The instrument, the Far-Infrared AbsoluteSpectrophotometer (FIRAS), is to be part of the Cos-mic Background Explorer (COBE) satellite, to belaunched by NASA in the late 1980s. The aim of theexperiment is to compare the spectrum of the 3 Kbackground with that of a blackbody with an accuracyof better than 0.1% of the peak spectral intensity.Further descriptions of the objectives of the project,the satellite, and its instrument complement are givenby Mather' and Mather and Kelsall.2

Since the background radiation is faint and objectsnear the earth are extremely bright by comparison, thedriving design consideration for the antenna is rejec-tion of known sources of stray radiation over a broadbandwidth. In addition, the absorption in the wallshould be small, and the antenna should be mechani-cally compatible with a blackbody calibrator that cov-ers the aperture. The horn and flare design given hereis an outgrowth of an earlier design developed byMather for use on a balloon-borne payload. 3 Thisearly design used a simple conical reflective opticalelement to define the beam and a second conical reflec-tor with a larger flare angle as a baffle against strayradiation. This design served its purpose, but its off-axis rejection ratio was limited by diffraction at the

When this work was done H. Hemmati was with Bendix FieldEngineering Corporation, Columbia, Maryland 21045; he is now atJet Propulsion Laboratory, Pasadena, California 91109. The otherauthors are with NASA Goddard Space Flight Center, Greenbelt,Maryland 20771.

Received 22 February 1986.

joint between the cones and at the large end of thebaffle cone.

I. FIRAS Antenna Design

The present design, illustrated in Fig. 1, consists ofthree elements: a flared shield; a compound parabolicconcentrator; and a compound elliptical concentrator.This configuration improves on Mather's early designin two ways. It substitutes a compound parabolicconcentrator (CPC) as described in Ref. 4 for thestraight sided cone, greatly improving the geometricaloptics beam pattern. Second, it substitutes a roundedflare for the straight sided conical baffle, eliminatingsharp discontinuities in the surface and greatly reduc-ing diffractive stray light effects. This idea was suc-cessfully tested by Woody and Richards in an im-proved balloon payload,5 and the stray lightperformance of the flux collector itself has been report-ed for broadband far-IR radiation.6

The concept of the flared conical horn has beentested in the microwave region, as reported by Math-er.7 That paper develops an elementary theory fordiffraction around the flared bell, which agreed wellwith the observations, despite the simplifying approxi-mations made. The theory is based on the geometricaltheory of diffraction (GTD), as originally developed byLevy and Keller. 8

The geometrical optics of the CPC or heat trap havebeen well studied.4 The CPC is a nonimaging devicewhich is nevertheless an ideal flux concentrator in twodimensions. In two dimensions, it is constructed fromtwo parabolas, as illustrated in Fig. 1. Each parabolahas its focal point at the opposite point of the outputaperture, and its axis is tilted to lie parallel to theextreme ray that is to be accepted into the aperture.In two dimensions, every ray that enters the apertureCH within the angular acceptance range between theaxes of the parabolas will pass through the apertureDG. In three dimensions the figure is rotated aboutthe symmetry axis (not the axes of the parabolas) andstill has most of the properties of the 2-D version.However, it has been shown that an ideal 3-D concen-trator is impossible,9 a result which is manifest here asa blurring of the sharp edge of the accepted beam. A

2826 APPLIED OPTICS / Vol. 25, No. 16 / 15 August 1986

Ac B

E D

K

M

Fig. 1. Horn antenna consisting of compound parabolic concentra-tor CDGH, flare ABCHJK, and compound elliptic concentratorEDGF. Curve CD is a parabola with focus G and axis GH, while

curve GH is its symmetric partner with focus D and axis DC.

ray trace for this particular heat trap was reported byMiller et al.10 The axis of the generating parabola isinclined 3.50 relative to the symmetry axis, and thehalf-power beam angle is very close to 3.50. For angles<30 off-axis, all the tested rays are accepted, but allrays more than 40 off-axis are rejected.

The radiation received by the CPC fills a wholehemisphere of solid angle and passes through a smallhole DG at its throat. For use in the spectrometer, thebeam is recollimated by a compound elliptic concen-trator (CEC), which is a modification of the CPC for afinite conjugate distance. Numerical design proce-dures for the general conic section concentrator aregiven by Eichhorn." In our case, the conjugate pupilis 25 cm away from the aperture. Dimensions of theCPC, CEC, and flare are given in Table I.

The flared portion of this horn is not a simple toroidbut is the union of two toroids having different curva-tures. This has two advantages over the simple toroidof the same diameter: lower sidelobes in the long-wavelength limit and a more compact mechanical ar-rangement. This last feature is advantageous for usewith a pivoted calibrator blackbody whose pivot axis isat N.

11. Antenna Pattern Theory

A quantitative prediction of the antenna pattern ofthis horn is offered only for the far off-axis patterns inthe long-wavelength limit. We have not attempted topredict the detailed near-axis pattern because it de-pends on factors we did not study or attempt to con-trol, namely, the coupling of the detector or detectorwaveguide to the horn. The horn can support manymodes of the electromagnetic field which can interferewith one another in a complex pattern when the radia-tion is coherent. Such effects are readily seen in thenear-axis measurements reported below.

The far off-axis pattern at long wavelengths is pre-dicted from the geometrical theory of diffraction(GTD). Briefly, the simple form of GTD used hereapproximates diffraction by assuming that radiationpropagates according to geometrical optics exceptwhen it encounters obstacles such as slits or bound-aries. At such locations, the diffracting object is ap-proximated by a similar canonical object whose dif-fraction properties have already been studied. The

Table . Dimenslons of CPC and CEC

Diameters AK 30.00 cmCH 13.65 cmDG 0.779 cmEF 3.98 cm

Lengths AK-CH 20.86 cmCH-DG 95.36 cmDG-EF 19.79 cmEF-focus 25.00 cm

HM 49.48 cmJN 6.04 cm

Angles CPC acceptance 3.50Parabola axis tilt 3.50Large radius flare (HMJ) 200Small radius flare (JNK) 700

far-field (ray-optics) diffraction coefficients of the ca-nonical object are then used to estimate new fieldamplitudes on new rays propagating freely.

The particular canonical problem used here is dif-fraction around a cylinder. The exact solution for acylinder, expressed as a series of Bessel functions, wastransformed using the Watson transform to show thatit can be equally well expanded in another series.8 Inthe new series, each mode represents energy whichpropagates around the cylinder but is attenuated ex-ponentially as it travels, and the loss is converted intofreely propagating energy leaving the cylinder. Theattenuation coefficients depend on polarization andthe curvature of the surface, and there are additionalnormalization factors called attachment coefficients.For a more detailed analysis and a physical interpreta-tion, see Mather.7

The simple formula given below is derived from anumber of assumptions, some of which are difficult tojustify except for the approximate agreement withmeasurement shown below. First, we do the computa-tion in a time-reversed picture in which the source ofradiation is at the horn throat DG. Second, we ignorethe detailed geometrical optics of the horn and theeffects of diffraction and mode propagation within itand assume that the radiation at the plane CH iscollimated with the nominal beamwidth of 7 and is ofuniform intensity across the aperture CH. Third, weassume the first-order GTD result for radiation propa-gating around a curved surface, even though there arediscontinuities in the curvature. Fourth, we neglectthe fact that the creeping surface wave modes haveeffective thicknesses which are comparable to the radi-us of the aperture LH. This implies that we cannotpredict the ripples in the sidelobe pattern that arisefrom interference between rays from opposite sides ofthe aperture. Fifth, we neglect the effects of theabrupt discontinuity at the outer edge of the flare AK,so we cannot expect accuracy in the backlobe. Sixth,we neglect the higher modes in the creeping wave ex-pansion and consider only the one with lowest attenua-tion. Seventh, we assume that as radiation begins topropagate around the curve of the flare (HJ or CB), it isalready at an angle 0 = 3.5° off-axis. This angle is the

15 August 1986 / Vol. 25, No. 16 / APPLIED OPTICS 2827

one adjustable parameter of the simple theory, and itsvalue was chosen to give a fair match to observations.Not coincidentally, it represents the edge of the geo-metrical optics beam.

These are the same assumptions made by Mather,7and they give the same result for angles 0 of < 01 = 200,the angle at which the two toroids are joined:

(o) [0.89896] X' 1/3a 13

g() 2.5434 Jr, sinO

* exp -{25.93121 (a )1/3 (-°°)ex -1 L2.5844 J x / (1)

where the upper set of coefficients refers to the H planeand the lower set to the E plane. In Eq. (1), g(0) is theantenna gain relative to an isotropic antenna, so thesolid angle integral of g(0) is 47r. The value a, is theradius of curvature HM of the flare, r, is the apertureradius LH, X is the wavelength, and 0 is the angle off-axis.

For values of 0 greater than 01, we substitute (ala2 )I/3

for a, , where a2 is the second radius of curvature JN.This produces a discontinuity in g(0) at 01 by a factor(a 2/al)1/3 , which is -1/2 in this case. We also substi-tute [aI/3(01 - Oo) + al/3 (0 - 0)] for al'/3(0 - 00) in theexponential.

For comparison with observations, which were al-ways made relative to the forward gain g(0), it wasnecessary to estimate the beam solid angle Q = X [g(0)/g(0)]dg from the data. From this we have g(0) = 4r/Oby definition of the normalization. Since we did notmeasure the entire pattern, we computed the followingestimate for 0:

0 {27rSf[gH(O)/g(0)I sinOd011 2

*J2rf[gE(0)/g(0)] sinOd011/2 , (2)

where gH and gE are the gains measured in the H and Eplanes, respectively.

IV. Fabrication

The horn and flare were fabricated on a numericallycontrolled lathe. It was impractical to manufacturethem from a single aluminum block, as originally de-sired, so they were made in two parts of approximatelyequal length, joined at flanges held with bolts. Thejoint design was made so that the joining force wasconcentrated at the optical surface, and concentricitywas assured by mating cylindrical parts. The sectionswere joined before polishing, and the resulting bulge atthe joint was polished off. Tests with a profilometershowed that the surface discontinuity was typically 10,um. The joint was barely detectable by eye.

Polishing was done by hand using motor-driven ro-tating cloth buffing wheels charged with a polishingcompound, while the horn turned on a lathe. Thewheels were mounted on a long shaft to reach deep intothe horn. Because the polisher could not turn at alarge angle to the machining grooves, there are stillvisible signs of the machining irregularities, particu-larly in the small diameter portions. Nevertheless, thesurface seems to be quite shiny. According to machin-

c�c

(A)

SHUTTERLENS l

LAMP I

/DIFFUSEF

20cm

ILII I CAMERASLIT

SLIT

I/

cc CEC -25cm

I FILM

(B)

Fig. 2. (a) Arrangement for testing figure accuracy of CPC;(b) arrangement for testing figure accuracy of CED.

ists' comparison standards, the level of polish corre-sponds to a surface roughness of 0.2-,um rms.

V. Figure Verification

The figure accuracy of the horn was verified in twoways, mechanical and optical. The mechanical testused a computer-controlled mechanical sensor mount-ed on the end of a long wand. The system measuredthe interior diameter of the horn at intervals of 0.508cm, but the flexibility and vibrations of the wand limit-ed the accuracy of the measurement to -0.003 cm.Typical diameter errors are +0.01 cm, and typicalslope errors on scales of 2 cm are 1 mrad.

The optical tests of the figure error were done photo-graphically. The arrangement shown in Fig. 2 wasused to reduce the effective dimensionality of the hornfrom three to two, so that the sharp angular cutoffproperties of a CPC and CEC could be tested. Anarrow (1-mm) slit was placed across the CPC or CECin the plane of the axis. It allows a three-parameterfamily of rays to pass, which are imaged onto the 2-Dfilm. The rays which stay in the plane of the horn axisare focused onto a line across the center of the image onthe film. The boundary of the image should be per-fectly sharp where it intersects this line in accordancewith the edge ray principle of the concentrator. More-over, all the rays which come to this line are reflectedonly once or not at all. Thus interpretation is simple.There is no such guarantee for the rest of the image,which shows various zones of different intensities asshown in Fig. 3. The angular spread of the image edgeis twice the slope error of the reflecting surface, suit-ably averaged. Measurements on the film imageshowed that our CPC has slope errors of the order of 1mrad rms. Several different portions of the horn wereselected for study by restrictive apertures at the slit.Use of small apertures showed that the image edge wassharper, implying that the surface is relatively smoothon scales of several centimeters.

VI. Antenna Pattern Measurements and Interpretation

The antenna pattern of the horn was measured inseveral ways: a microwave antenna range using 31.4,

2828 APPLIED OPTICS / Vol. 25, No. 16 / 15 August 1986

I

0

30

Id03

Fig. 3. Photo taken in arrangement shown in Fig. 2(A) With 3.5-cmaperture lens centered on-axis. Slit is horizontal.

-100 -80 -60 -40 -20 0 20 40 60 80 100

ANGLE FROM AXIS, DEGREES

Fig. 5. Measured and theoretical antenna patterns at 31.4 GHz,E plane.

-40 -20 0 20 40 60

ANGLE FROM AXIS, DEGREES

-20

I)T

m

UJ

On

2

:

-30

-40

-60 -40 -20 0 20 40

ANGLE FROM AXIS, DEGREES

Fig. 4. Measured and theoretical antenna patterns at 31.4 GHz,H plane.

80F uI oo -80

Fig. 6. Measured theoretical antenna patterns at 90 GHz,E plane.

53, and 90 GHz sources; a far-IR (118-ptm wavelength)laser used directly and after expansion by a lens; andhot (10000C) and cold (77 K) blackbodies used both inthe near field and collimated by a parabolic mirror.

The microwave antenna range used is located out-doors at the Goddard Space Flight Center. The trans-mitter and receiver are both located on towers 1.5 and 6m, respectively, above the ground separated by 17 m.The transmitter is a coherent source, such as a Gunndiode, and the receiver is a Scientific-Atlanta model1750 heterodyne system with a logarithmic gain com-pression circuit which permits a dynamic range of 80dB to be shown on one output chart. It was calibratedwith attenuators.

The microwave receiver was coupled to the hornunder test by a small CPC, similar to the CEC anddirectly in contact with it, which focused most of theradiation onto the waveguide input to the receiver.However, no special transitions were made between

the guide and CPC to enhance efficiency. Therefore,no simple statement is possible about the particularmode excitation patterns in the horn. The horn itselfsupports multiple modes, as its etendue is 1.5 cm2 sr,somewhat larger than the square of the wavelengtheven at 31.4 GHz, the lowest test frequency.

Antenna patterns from tests at 31.4 and 90 GHz aregiven in Figs. 4-6. They show as expected that the H-plane pattern at levels below -30 dB is much sharperthan the E-plane pattern by a factor of -2 in angle.Theoretical predictions from the geometrical theory ofdiffraction (GTD), as described above, are also shown.The agreement is good considering the number of ap-proximations in the theory.

Measurements at 118 -,gm wavelength were madewith an Apollo far-IR waveguide laser operated in thedischarge chopped mode. (The pump laser inputpower was modulated electrically.) The output beamhad a diameter of only 1 cm, but this is not expected to

15 August 1986 / Vol. 25, No. 16 / APPLIED OPTICS 2829

Cl)

Uala

0C')

60 80 100

-60

-10

-20

,, - 30aa

Z0 40

> - 50

CC - 60-570

-290

0 5 10 15 20 25 30 35

ANGLE FROM AXIS, DEGREES

Fig. 7. (a) Antenna pattern measured at 118 /im witttest beam; (b) antenna pattern at -10 pAm; (c) anten

visible wavelengths.

greatly affect the far sidelobes. The measurmade indoors in a laboratory room, so EccoVHP foam material was placed at strategic 1the room where stray reflections might be:until additional Eccosorb produced no chetected signal. The signal was detected wiGolay cell or a helium-cooled bolometer in cwith a phase sensitive amplifier. The resuas curve (A) in Fig. 7.

Some attempts were made to explore Idence of the pattern on beam position insince the 1-cm beam fills only a few of the hiThe laser beam was also expanded to more taperture of the horn by the use of a divergthe laser, and the angular response patterntraced. The result was that the central ma:wider because of source divergence, but tlshapes were similar at angles more than 1It is expected that the pattern at 118 gmdependent on surface finish and particulatnation, so no special attempts were madebetter accuracy.

Shorter wavelength measurements werechopped 10000C and 77 K blackbodies. Iethylene detector filters were used to reiwavelength response, so that both blackbeffective wavelengths of 10 tim and very silobe patterns. Near-axis patterns were meesmall well-collimated sources, but large are;sources were required for adequate sensitivoff-axis angles. The curves were joined at200 off-axis for curve (B) in Fig. 7. The metshow as expected that the flare provides aeffect at an angle of 50° off-axis. Accordmetrical optics, all rays at angles greater theprevented from entering the CPC section athey scatter on the flare itself. Then they nagain on the CPC walls to reach the detect(

Photographic measurements were also nPolaroid film and varying time exposures, i

described previously. No special attempt at accuratephotometry was made. The photographic data, curve(C) in Fig. 7, show much stronger response at angles of5-10° off-axis than could be seen at IR wavelengths.On the other hand, wide angle sidelobes (300) are quitecomparable to the 10-kum results. Our interpretationis that different portions of the horn have differentdegrees of surface roughness because of the polishingtechniques and that the small diameter portions aredistinctly rougher. Rays a few degrees off-axis canpenetrate far into the CPC, bouncing 10 to 20 timesbefore being rejected, and have many opportunities toscatter light into the acceptance angles of the CPC.

VIl. Summary and Conclusions

We have shown that the quasi-optical multimode40 45 50 horn antenna works well over an extremely broad spec-

tral range. At long wavelengths, the geometrical the-ha 1-cm diam ory of diffraction describes the far off-axis sidelobes,

na pattern at which are well attenuated by the flared bell at the largeend of the horn. At short wavelengths, the geometri-

'ementwas cal optics beam pattern is observed to agree with raysorb (TM) trace predictions, and scattered radiation is stronglyocations in dependent on wavelength. A new method of testingimportant, the optical figure was presented.mnge in de-th either a We greatly appreciate the excellent manufacturingDnjunction work done for us by the Tinker Spinning Co. in Balti-It is shown more, MD and the GSFC shop. B. Borchardt of the

National Bureau of Standards in Gaithersburg, MDthe depen- was in charge of the mechanical figure measurements,lthe horn, and C. Fleetwood at GSFC assisted in the photograph-corn modes. ic tests. We also thank the reviewers for helpful ques-han fill the tions. This work was supported by the National Aero-Jing lens at nautics and Space Administration as part of the COBEI was again project.dimum wasie sidelobe)0 off-axis. Referencesis strongly 1. J. C. Mather, "The Cosmic Background Explorer (COBE)," Opt.e contami- Eng. 21, 769 (1982).to achieve 2. J. C. Mather and T. Kelsall, "The Cosmic Background Explorer

Satellite," Phys. Scr. 21, 669 (1980).

made with 3. J. C. Mather, "Far Infrared Spectrometry of the Cosmic Back-Black poly- ground Radiation," Ph.D. Thesis, U. California, Berkeley (Xe-duce short rox University Microfilms, Ann Arbor, MI, 1974).4. W. T. Welford and R. Winston, The Optics of Nonimagingoodies gave Concentrators (Academic, New York, 1978).milar side- 5. D. P. Woody and P. L. Richards, "Spectrum of the Cosmicsured with Background Radiation," Phys. Rev. Lett. 42, 925 (1979).

a (100-cm2

) 6. D. P. Woody, "An Observation of the Submillimeter Cosmic-ity at large Background Spectrum," Ph.D. Thesis, U. California, Berkeleyan angle of (Xerox University Microfilms, Ann Arbor, MI, 1974).sured data 7. J. C. Mather, "Broadband Flared Horn with Low Sidelobes,"shadowing IEEE Trans. Antennas Propag. AP-29, 967 (1974).ling to eo- 8. B. R. Levy and J. B. Keller, "Diffraction by a Smooth Object,"ng 57ogeo Commun. Pure Appl. Math. 12, 159 (1959).an a are 9. W. T. Welford and R. Winston, "On the Problem of Ideal Fluxat all unless Concentrators," J. Opt. Soc. Am. 68, 531 (1978).Lust scatter 10. M. S. Miller, W. L. Eichhorn, and J. C. Mather, "Sky Input Horn)r. for a Far-Infrared Interferometer," Opt. Lett. 7, 210 (1982).iade, using 11. W. L. Eichhorn, "Generalized Conic Concentrators," Appl. Opt.n the setup 21, 3887 (1983).

2830 APPLIED OPTICS / Vol. 25, No. 16 / 15 August 1986