On the Fabrication and Evaluation ofan Integrating Hemiellipsoid

R. P. Heinisch, F. J. Bradac, and D. B. Perlick

This report describes the fabrication of an integrating hemiellipsoid and the associated analytical andempirical evaluation of it. The focusing properties of the instrument are seen to be of high quality, andquantitative information is presented regarding the selection of the size of a detector to be used.

IntroductionIn 1899 Paschen' developed a hemispherical collector

to measure surface reflectance. Coblentz described asimilar instrument in 1913.2 The use of a hemispherein a reflectometer has been recently reported by Birke-bak and co-workers,8 ' 4 Kozyrev and Vershininj andJanssen and Torborg.6

Integrating hemispheres, such as those used by theabove authors, consist of a concave hemisphere with aspecular reflecting metal coating and areused to approxi-mate an ellipsoidal mirror. Instead of having two dis-tinct foci, the center of the sphere is used as a singleapproximation for the two. The specimen is located ashort distance one side of the center of curvature withthe detector at a conjugate position. Radiant energyis projected onto the specimen through an aperture inthe surface of the hemisphere. The reflected energy iscollected by the hemisphere and focused on the detector.

The purpose of this report is to describe the fabrica-tion of a hemiellipsoid and its associated analytical andexperimental evaluation. A similar instrument hasbeen constructed from glass.' However, it was expen-sive, difficult to fabricate, and the optical quality of themirror was not evaluated. Advantages of an integrat-ing hemiellipsoid over a hemisphere in a reflectancedevice Vre discussed by Neu7 and by Dunn and co-workers8 for a truncated section of an ellipsoid. Jans-sen and Torborg5 also have commented on the inade-quacy in using a hemisphere to approximate a hemi-ellipsoid.

F. J. Bradac is with Research Service, 985 Englewood Avenue,St. Paul, Minnesota 55114; the other authors are with Honey-well, Inc., Systems & Research Division, 2345 Walnut Street,St. Paul, Minnesota 55113.

Received 30 June 1969.

Nomenclaturea, semimajor radius of hemiellipsoid,b, semiminor radius of hemiellipsoid,g, defined by Eq. (4),h, defined by Eq. (5),r, distance between two points in geometric ray trace,x, coordinate in rectangular Cartesian system,y, coordinate in rectangular Cartesian system,z, coordinate in rectangular Cartesian system,X, wavelength.

Subscripts1, refers to source point,2, refers to point on hemiellipsoid,3, refers to location surface normal strikes xy plane,4, refers to image point.

Fabrication

The hemiellipsoid was constructed from two quartersections of an ellipsoid with a semimajor axis of 12.70cm and a semiminor axis of 12.19 cm. As a result ofthese dimensions, there is 3.556 between each focal pointand the center of the ellipse.

Rather than attempting to machine the quarter sec-tions of the ellipsoid directly, we chose to machine amale mold and then cast the surface of the ellipse withan aluminum filled epoxy. A convex male mold of onehalf of a ellipsoid (Fig. 1) was machined on a lathe bycarefully adjusting the cutting tool to x and y coordi-nates computed on a digital computer. The y coordi-nate of the tool location was computed accounting forthe tool radius, at 2 5.4 -A increments of x values. Theconvex mold was polished to a high luster to give the re-sulting casting a smooth finish.

The quarter ellipsoids were prepared by filling theannulus between an aluminum outer casting and theconvex male mold with epoxy. A very light coat ofsilicone vacuum grease was used as a release agent be-tween the male mold and cast epoxy. The quarter-

February 1970 / Vol. 9, No. 2 / APPLIED OPTICS 483

X

Fig. 1. Ellipsoid male mold.

SEMI-MAJORAXIS

Fig. 2. Hemiellipsoid.

Evaluation

Both laboratory and computational experiments wereused to evaluate the focusing properties of the ellipsoi-dal mirror. Although the mirror was developed for useat 10.6 /,u tests were performed using both visible lightand ir radiation. If the mirror is found adequate forvisible light, it would be more than adequate for the irsince the surface quality is less critical at longer wave-lengths.

Laboratory Tests

A fixture was made to locate accurately the focalplanes of the mirror as shown in Fig. 3. The focal planesare defined as the area in a horizontal plane immediatelysurrounding and passing through the focal points. Athin opal glass plate was placed in the fixture at onefocal plane to serve as a diffuse source. A Plexiglasblock with two 6-V light bulbs was placed on top of thediffuser plate for illumination. The diffuser plate andPlexiglas block were masked to form a circular lightsource in the focal plane of 1-cm diam, the 1-cm dimen-sion being thought to be typical for practical applica-tion. The image of that circular pattern was formed atthe second focal position. Visually the image appearedabout equal to the source in size and shape. The imagewas recorded on photographic paper.

Figure 4 is a photographic view showing both thesource and the image. The source was photographedby placing photographic paper in direct contact with it

ellipsoid was obtained by cutting the casting to includean additional 2.54-cm overlap along the semimajor axis.This gave the final hemiellipsoid an overlap as shown inFig. 2. The castings were cast with an additional 2.54-cm additional overlap along the semimajor axis so theresulting hemiellipsoid has overlap as illustrated in Fig.2. Joining the two quarters of the ellipsoid as demon-strated in Fig. 2 results in the desired hemiellipsoid.There exists a very fine crack or line created by joiningthe two quarters of the hemiellipsoid, but this line repre-sents only about 0.004% of the total surface area.Therefore, the line does not introduce an appreciablemeasurement error.

The area associated with the entrance slit required forsample illumination was about 0.19% of the total sur-face area. The slit in the hemiellipsoid contributed alarger error than the line of joining discussed above, how-ever, the magnitude of the error is quite small.

It was found necessary to evacuate the mold whilecasting the epoxy to remove entrained air bubbles onthe mold surface. After many tests, a complex pump-ing schedule was developed by which Devcon F, an 80%aluminum filled epoxy, could be cast with a minimum ofentrained bubbles.

Aluminum was evaporated on the inner surface of theellipsoid to provide high reflectance. The hemiellip-soidal mirror visually appeared to have excellent focus-ing properties in the visible, but quantitative evalua-tion was deemed necessary.

Fig. 3. Image quality determination fixture mounted on hemi-ellipsoidal reflector.

r4 7.112 cm X

X I+I. 1 _o

SOURCE IMAGE

Fig. 4. Experimental determination of ellipsoidal reflectorquality.

484 APPLIED OPTICS / Vol. 9, No. 2 / February 1970

SMOOTH SUBSTRATE1.0 -

0.9

W

I-U

I-w

0.8

0.7REPRESENTATIVE ROUGHSUBSTRATE

0.6 -

0.5

0.4

0.3

0.21 2 3 4 5 6 7 io

X (W)

Fig. 5. Reflection of aluminized epoxy specimens.

for a 1-2-sec exposure. This image is representative ofthe distortion of the source by the ellipsoid in the visiblerange of wavelengths. The distortion was less in thedirection perpendicular to the major axis than parallelto it. As shown in Fig. 4, no radiation was detectedoutside of a 2-cm square centered at the focal point forexposure times of less than 24 sec.

The reflectance of the ellipsoidal mirror surface wasevaluated by preparing several 2.54-cm diam specimensin the same manner as the ellipsoidal mirror. These sam-ples were aluminized in the same chamber concurrentlywith the ellipsoid. The specimens were chosen basedon the substrate surface quality. The spectral reflec-tance of two specimens measured from 1 A to 10 A isshown in Fig. 5.

One of these samples had a very smooth finish similarto the surface finish of the ellipsoid. The second speci-men was porous with qualitative appearance muchpoorer than the worst section of the ellipsoidal mirror.The reflectance is degraded by the poorer substrate sur-face finish. The data such as those of Fig. 5 are neces-sary to correct the measurement to ensure results ofhigh accuracy.

AnalysisThe focusing properties of the hemiellipsoid were

derived using geometric optics and then the mathemati-cal model was programmed on a digital computer. Be-fore discussing the details of the current analysis, otherefforts of a like nature will be summarized.

Brandenberg 9 conducted a geometric ray trace toevaluate the focusing properties of hemispherical andhemiellipsoidal collector mirrors. Analytical expres-sions for the aberrations of a hemisphere and an ellipsoidand the resulting distortions are presented by Branden-berg. However, his equations describing the charac-teristics of the hemiellipse were not derived explicitly.The equations were obtained via a transformation of co-ordinates on the resultant equations describing the

hemisphere. Brandenberg's numerical data on imageproduction were generated only for the boundary of thesource. There are few physical situations of practicalinterest where the source is defined by the boundary of,say, a circle. The ellipsoid studied by Brandenberg hada semiminor axis of 10 cm and a semimajor axis of 10.2cm. Twenty-seven equally spaced points on the sur-face of the ellipsoid were selected to be struck by a ray.This relatively small number of points severely limitsthe accuracy of the data. Preselection of the directionof the rays is likewise unsatisfactory. Proof of thishypothesis will be given in a later section.

By recourse to vector calculus, a geometrical raytrace was performed on the hemiellipsoid. The hemi-ellipsoid collector surface is described by

(X2/a2) + [(y2 + z2)/b2] 1, (1)

where a = 12.70 cm and b = 12.19 cm in a rectangularCartesian coordinate system as shown in Fig. 6.

Consider a ray which originates from the source point(xi, yi) and is reflected by the hemiellipsoid at an arbi-trary location (X2, Y2, z2). The reflected ray intersects thexy plane at the image point (X4, Y4). The normal to thecollector surface intersects the xy plane at the point (x3,y3). It can be shown that

(2)

(3)

X3 = 0,

Y3 = Y2(1 - a2/b2).

Define, for convenience,

g = a2/b2 (4)

x

Fig. 6. General coordinate system for ellipsoid.

(X1, Y1)

(0, Y3 )

(x4 , y4 )

Fig. 7. Two-dimensional coordinate system.

February 1970 / Vol. 9, No. 2 / APPLI ED OPTICS 485

-

i l l l --

,

Z

Fig. 8. Image of source boundary for rays emitting from a 45°cone about the surface normal of the source.

long. The image of an entire circular source radiatingin a 10-deg vertical cone is shown in Fig. 9, therebydemonstrating the high quality of the focusing for radia-tion reflected at small angles of inclination to the nor-mal. About 10,000 rays are traced. Finally, the entirecircular pattern radiating in all directions is shown inFig. 10. Over 25,000 rays were used to generate thephotograph of Fig. 10. The image spreads to about12 times the source size, but the flux density in theouter portion appears to be less than that contained in a1-cm diam circle (which corresponds to the size of thesource). The inclusion of rays emitted at low anglesfrom the source causes these aberrations.

Quantitative information was also obtained describ-ing the particular ellipse investigated. At the detector,47% of the total energy was outside of a circle of 1-cmdiam, while 11% of the total energy was outside of acircle of 1.5 cm in diameter. Only 0.7% of the totalenergy was outside of a circle of 1.5 cm in diameter.

and

h = r23'/(7 I'!- - V132) = 7-34/7-13. (5)

By restricting the source and image to be in the equa-torial plane of the hemiellipsoid (i.e., z = 0), only thegeometrical pattern in the xy plane need be considered.This pattern is shown in Fig. 7. Using geometricalrelationships, we can obtain the following expressionsrelating the source and the image points:

4 x1h,

Y4 = Y2(1 - g)(1 + h) - hy,,

(6)

(7)

where

r122

= ri2 + r22 - 2r, X r = X12 + y12 + x,2 + y2' + Z2

2

- 2x1 X2 - 2y1y2,

r132 = Y22 ( - )2 + X1

2 + yl - 2yjY2(1 - 9)

r122 - r132 = X2(X2 - 2x) + z2 + y.2[y2(2 - g) - 2yl],

r232= x22 + z22 + g2y22.

Equations (6) and (7) were programmed on a digitalcomputer. A diffuse circular source 1-cm in diameterradiated from one focal point in the hemiellipsoid.Rays were permitted to emit from either the boundaryof or the entire source. The location and the directionof emission were chosen using a random number ('onteCarlo scheme) generator. Each ray so selected reflectsfrom the hemiellipsoid and is imaged in the second focalplane. Each discrete point thusly imaged in the secondfocal plane was displayed on a large oscilloscope used asan output device of the digital computer. The datawere recorded on a photograph of the oscilloscope dis-play.

The results of the analysis are presented in Figs. 8-10.Figure 8 shows a photographic image of the boundary ofa 1-cm circular source for about 8000 rays emitted in a450 cone about the surface normal. For reference pur-poses the grid lines shown on the photographs are 2 cm

Fig. 9. Image of entire source emitting from a 100 vertical coneabout the surface normal.

Fig. 10. Image of the entire source emitting in all directions.

486 APPLIED OPTICS / Vol. 9, No. 2 / February 1970

Only 0.7% of the total energy lies outside of a 2-cm cir-cle. It should be noted that this information is predi-cated on the assumption that the source is a diffuseemitter. The quantitative information described aboveis deemed essential in the selection of a detector for usein such an instrument.

The authors would like to express their thanks toJens K. Anderson and Roger N. Schmidt for their con-tributions. The direction of Dwayne Hinton, NASA/Langley and the editorial assistance of E. M. Sparroware also gratefully acknowledged.

This work was performed as a subsidiary phase ofNASA/Langley Research Center contract NAS 1-8447.

References1. Von F. Paschen, Gesammtsitzung 27, 405 (1899).2. W. W. Coblentz, Nat. Bur. Stand. Bull. 9, 283 (1913).3. R. C. Birkebak and J. P. Hartnett, Trans. ASME 80, 373

(1958).4. R. C. Birkebak, E. Ma. Sparrow, E. R. G. Eckert, and J. W.

Ramsey, J. Heat Transfer 86C, 193 (1964).5. B. P. Kozyrev and 0. E. Vershinin, Opt. Spektrosk. 6, 542

(1959) [Opt. Spectrosc. 6, 345 (1959)].6. J. E. Janssen and R. H. Torborg, in "Measurement of Thermal

Radiation Properties of Solids," J. C. Richmond, Ed., NASASP-31, 1963.

7. J. T. Neu, "Design, Fabrication, and Performance of anEllipsoidal Spectroreflectometer," NASA CR-73193, March1968.

8. S. T. Dunn, J. C. Richmond, and J. A. Wiebelt, J. Res. Nat.Bur. Stand. 70C, 75 (1966).

9. W. M. Brandenberg, J. Opt. Soc. Amer. 54, 1235 (1964).

Submillimeter Waves International Symposium

New York City, 31 March-2 April 1970

The submillimeter wave region of the spectrum, extending from far infrared to millimeter wavelengths, still remains

relatively unexplored and unused. Progress in this area has been limited by lack of interchange of ideas and

techniques between those who approach this range from the microwave region of the spectrum and those who

view it from the optic region. The purpose of this Symposium is to bring the optical and microwave groups

together. Following survey papers on the application of submillimeter waves to physical experiments, astronomical

observations, and communication systems, various ideas, techniques, materials, and devices will be explored which

are valid for the submillimeter wave region. Since activity specifically at submillimeter wavelengths is limited,

papers discussing work at millimeter or infrared wavelengths are welcome if the results show promise of future sub-

millimeter wave use. The technical areas covered in the Symposium will include long wavelength lasers, nonlinear

effects in semiconductors, thermal and quantum detectors, and parametric interactions. Implications for compo-

nents (sources, amplifiers, detectors, etc.) and application to systems (radar, communications, etc.) will be con-

sidered. Papers are solicited in the following areas: long wavelength lasers (e.g., discharge lasers, three-level

masers, novel pumping techniques); nonlinear and nonreciprocal interactions (e.g., lumped and distributed para-

metric effects in solids, gases, and liquids, ferrimagnetic and antiferromagnetic materials); semiconductor sources

(e.g., LSA Gunn effect, avalanche transit time devices); transmission techniques (e.g., lenses, beam waveguides,

quasi-optic techniques, dielectric guides, and integrated circuits, atmospheric propagation, and dispersion); detectors

and amplifiers (e.g., thermal detectors at room and cryogenic temperatures, pyroelectric effect, Josephson effect,

quantum detectors, point contact and Schottky-barrier detectors, parametric amplifiers); and systems techniques

(e.g., communication systems, radiometry, image conversion, radar, material and plasma diagnostics, spectroscopic

techniques, radioastronomy). In addition, papers on other novel techniques in the far infrared to millimeter wave-

'length region are welcomed. The Symposium is conducted with the co-sponsorship of the Air Force Office of

Scientific Research, the Office of Naval Research, the Army Research Office, the Institute of Electrical and Elec-

tronics Engineers Group on Microwave Theory and Techniques, and the Optical Society of America. The Proceed-

ings of the Symposium on Submillimeter Waves will be published by the Polytechnic Press as Volume 20 of the

MRI Symposia Series with members of cooperating societies entitled to a special discount. Address all corre-

spondence to Polytechnic Institute of Brooklyn, MRI Symposium Committee, 333 Jay Street, Brooklyn, N. Y. 11201;

Attn: Jerome Fox, Executive Secretary.

February 1970 / Vol. 9, No. 2 / APPLIED OPTICS 487

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488 APPLIED OPTICS / Vol. 9, No. 2 / February 1970