JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 52, NUMBER 3

Study of the Effect of Large Aperture on the Performance of anEbert Spectrometer

LAWRENCE R. MEGILL AND LEANN DROPPLEMANU. S. Department of Commerce, National Bureau of Standards, Boulder, Colo.

(Received July 27, 1961)

A description of the effect of large aperture upon the resolution limit of an Ebert spectrometer is discussed.The calculations were made by ray tracing on a high-speed computer. Variations of the aberration pattern atthe exit slit are noted for several parameter changes which may be useful in design considerations.

THE Ebert or Fastie-Ebert spectrometer has beenT enjoying increasing popularity since its revival

by Fastie."2 The basic spectrometer arrangement isshown in Fig. 1. The rays from the entrance slit go toone half of the spherical mirror, are collimated, sent tothe grating, and the dispersed radiation is then imagedon the exit slit. Only one-half the mirror is used for eachreflection so that, for a given aperture, twice the mirrordiameter is required as would normally be calculated.This system is a special case of the Czerny-Turnersystem which utilizes two spherical mirrors. The Czerny-Turner arrangement becomes an Ebert when the centersof curvature of the two mirrors are coincident. However,the single mirror makes adjustments much easier toaccomplish and results in a simple, reliable instrument.In most cases the relative cost between mirrors andgratings also justifies the use of the larger mirrors.

The image-formation quality of the system is excel-lent if curved slits are used. The aberrations due to theuse of the spherical mirror are partially corrected in thesecond reflection. There is no correction for sphericalaberration nor is astigmatism reduced; however, thegeometry of the curved slits renders the astigmatismharmless.

For many applications in the infrared, the resolutionsof spectrometers is limited, not by the instrument, butby the energy available at the detector. The temptationunder these conditions is to increase the aperture of thesystem. The Ebert optical system, however, is limitedfor use with large apertures by the curvature of the wavefront at the grating. The question which must be decided

in the design of large-aperture instruments is whetherthe resolution limit due to the instrument is greater thanthat due to lack of energy.

In order to obtain a parameter which is of use in thedesign of spectrometers, consider the two extreme wavefronts s and s' in Fig. 2, such that the angle between thetwo fronts is the same as the maximum angular differ-ence between any two points. If both of these fronts hadthe same wavelength they would have an angular dif-ference 0= (cosi/cosO) i upon diffraction from thegrating, where i is the angle of the incident wave frontto the grating surface normal and is the angle of theexit wave-front normal. In the Ebert spectrometeriO so that for apertures smaller than about f/4 wemay write 0 8U.

Now consider a perfectly plane wave front with morethan one wave length. The wavelength spread whichcorresponds to the angle 30 = i is XA = 2d cosi or 3i=2d cosO3O, where the approximation i 0 has again beenmade. The reciprocal of the factor 2d cosi is the dis-persion d/dX = D, so that one has the relationshipX = i/D from which the resolution limit due to the

curvature of the wave front may be calculated for agiven instrument.

In order to study the behavior of the wave front withvarying aperture, a ray-tracing routine has been devel-oped for use on a digital computer. The parameteri (or 0) can be determined from the angles between the

various rays.A complete analysis of the effects of the curved wave

front would be extremely difficult. The general nature

Fio. 1. A scheimatic drawing of the spectrometer howing theparameters used in the calculations.

1 W. G. Fastie, J. Opt. Soc. Am. 42, 641, (1952).2 W. G. Fastie, J. Opt. Soc. Am. 42, 647, (1952).

S,

I

//A/II

IaI ll

FIG. 2. A schematic draw-ing of the wave-front shapedefining the parameter .

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PERFORMANCE OF EBERT SPECTROMETER

FIG. 3. A partial drawing showing the 41 raystraced in the calculations.

of the wave front is as shown in Fig. 2. Although theextent of the curvature is exaggerated, the shape of thewave front is approximately that of an S in the planeperpendicular to the axis of the grating.',2

The calculations were carried out in the followingmanner. From a point source a bundle of 41 rays wasconsidered. The rays were chosen as shown in Fig. 3.Five cones of rays were considered, with eight raysequally spaced around the cone being calculated. Theinternal angles of the cones were chosen so that thesolid-angle increment included between two successivecones was constant. The internal angle of the largestcone was computed from the aperture of the systembeing considered. The forty-first ray traced was thecentral ray of the system.

These rays were traced by using the exact analyticalexpressions for the reflections off the mirror. The cal-culations were carried out by an electronic computer.

The results of the value i are plotted in Fig. 4 as afunction of f number to f/20. The term f number ishere defined as the ratio of the focal length F to theradius of the larger mirror d in Fig. 1, as only half themirror is being used on each pass. The performance ofthe instrument degrades very rapidly as the aperture isincreased. It may be that in some cases this degradationis permissible for energy-limited cases. In these casesthe curve of Fig. 4 can serve as a useful engineeringguide.

qualities of the system exclusive of the grating may beconsidered. Examples of the aberration pattern ob-tained at the exit slit are shown in Fig. 5. The largeaberrations are along the length of the slit and so do notdegrade the performance of the instrument. Because ofthe symmetry of the system, all points on the curvedslits behave identically with the aberration patternalways oriented along the slit.

The change Az in the maximum width of the aber-ration pattern as the system aperture is varied is cal-culated in order to compare the effect of the wave-frontcurvature in limiting the resolution of the instrument.In order to normalize the quantity we have plotted, inFig. 4, Az/F, where F is the focal length of the instru-ment. This parameter plays essentially the same role incalculation of the instrumental effect as does the param-eter i. It is seen that the quantity Az/F is consistentlysmaller than i. Since both are upper limits, however,one cannot be completely certain of this interpretation.The effect in limiting the resolution is not calculablefrom the aberrations since we cannot be assured a priorithat after diffraction all rays will represent equal energy.

In many instruments it is necessary for practicalconsiderations to place the grating in front of the slitplane. The effect of varying the quantity Q (Fig. 1) foran f/4 system was investigated. The value z/F is

100

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0.5

0.1

0.05

IMAGE FORMATION

The extension of the ray-tracing technique beyondthe grating must be undertaken with care since it isdifficult to take diffraction accurately into account.However, we have extended the ray trace in order tostudy the image-forming properties of the Ebert system.This has previously been done experimentally byFastie2,"2

In order to carry out calculations for the rays extend-ing beyond the grating, the grating is replaced in thecalculations by a plane mirror. Now the image-forming

0.01

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FIG. 4. This plot shows the parameters bi andAZ/F plotted as a function of f number.

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259March 1962

L. R. MEGILL AND L. DROPPLEMAN

20 -20 -10 0 10 20 -20 -10 0 10 20

ARBITRARY SCALE

FIG. 5. This figure shows the shape of the spot at the exit slit in an f/4 system. The three plots show the effect of moving the gratingforward. The scales are arbitrary since the value AZ depends on the focal length used. For a 1-meter focal length the scale is in microns.

plotted in Fig. 6 as a function of the fractional changesin Q. The aberration pattern narrows slightly as thegrating is moved in front of the slit plane. A similarinvestigation carried out at f/8 and f/6 gives a verysimilar curve.

Another deviation from the ideal case which may bemade for practical reasons is that of increasing theradius of the slit circle to values larger than required toinclude all the light on one side of the mirror. The effectof this modification on an f/4 system for percentage

0.8

0.4X/

0 001 00 O^ i 00N 0

Q-QoQ

FIG. 6. This plot shows the variation of AZ/F for an f/4 systemwhen the grating is moved forward.

increases in the slit radius to a value twice that of theideal was investigated. The results of this calculationare shown in Fig. 7. This curve was also checked at f/8and f/6, with similar results.

The computer program has also been designed tocheck for vignetting on the collimating mirror, thecondensing mirror, and the grating, and for obstructionof the beams by the grating. These checks have beenfound useful in the design of specific instruments.

CONCLUSION

Upper limits for the effects of wave curvature andaberration in limiting the resolution of an Ebert spec-

0

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1.0 1.2 14Ppo

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Fig. 7. This figure shows the variation of AZ/IF as the slits aremoved out to a larger diameter. The plot is for one /4 system,

260

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Vol. 52

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PERFORMANCE OF EBERT SPECTROMETER

trometer have been calculated. A series of curves ispublished which establishes engineering guide linesuseful in the construction of spectrometers.

ACKNOWLEDGMENTS

The authors are indebted to Miss Pauline Jamnickfor help in the early stages of the problem and to Mr.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Arthur Shafer's helpful conversations concerning theproblem of ray tracing after diffraction.

Special notice is due Dr. Charles Barth of the JetPropulsion Laboratories, who, in the use of the programfor a specific design problem, noted the diminishing ofthe diffraction-pattern width as the grating is movedforward.

VOLUME 52, NUMBER 3 MARCH, 1962

Optical Properties of Evaporated Lead Monoxide FilmsA. E. ENNos

Research Laboratory, Associated Electrical Industries, Aldermaston Court, Aldermaston, Berkshire, England(Received June 6, 1961)

The optical properties of evaporated lead monoxide were investigated in relation to their use as high-refractive-index films for multilayer optical filters. Refractive index of the films was measured through-out the visible wavelengths by a spectrophotometric method. nD= 2.55 and vi (reciprocal dispersion) =8.8.The optical absorption coefficient, appreciable in the visible range when oxide is evaporated in high vacuum,could be kept below k=0.004 by evaporating in low-pressure oxygen. Strong absorption appeared in theviolet and ultraviolet. Cementing properties and atmospheric effects on the films were also investigated.

1. INTRODUCTION

MANY materials have been investigated in thin-film form to assess how suitable they might be

in multiple-film optical elements. Experience has shoA nthat only a few of these can be prepared with repro-ducible and stable optical properties while also possess-ing the hardness and resistance to atmospheric condi-tions that make them really useful. However, there aremany others possessing certain unique properties whichfit them for special applications. Lead monoxide, withits high refractive index and dispersion, is one of these.

A high-refractive-index material finds many uses inmultiple-film filters. For example, pairs of high-refrac-tive-index and low-refractive-index films one-quarterwavelength in optical thickness can be combined toform dielectric mirrors of high efficiency, and it isdesirable to use films having as wide a divergence ofrefractive index as possible so as to obtain high re-flectance with the minimum number of layers. Formost purposes, zinc sulphide (n= 2.37) and cryolite(n= 1.35) are used, owing to their transparency in thevisible, their stability, and their ease of preparation byvacuum evaporation. Materials having a higher re-fractive index than zinc sulfide either exhibit strongselective absorption in the visible (e.g., antimony tri-sulfide) or require special preparation techniques whichpreclude their use in multiple layers (e.g., titaniumdioxide). Bismuth oxide (n= 2.45) is an exception, butthe films have to be formed by reactive sputtering toachieve transparency. It has been found, however, thatlead monoxide films having small absorption may bevery simply prepared by vacuum evaporation and,while these do not possess ideal properties in other

respects, their high refractive index and dispersionjustify their inclusion in the list of possible materialsavailable to the designer of multilayer filters.

2. PREPARATION OF FILMS

Lead monoxide, PbO, exists in two crystalline forms,litharge and massicot. Litharge is the stable form atroom temperature, and transforms to massicot whenheated above 300C. It melts at 8880 C. The monoxideis readily reduced to lead by reaction with metals, suchas tungsten, molybdenum, or tantalum, or with carbon.It also reacts with silica to form glassy lead silicate. Itis thus imperative to use a platinum container for theevaporation. The lead monoxide begins to evaporatein vacuum at a temperature close to its melting point,and condenses on a cold substrate to form a film whichelectron-diffraction examination has shown to be com-posed of polycrystalline massicot.

Films were prepared by evaporating "Specpure"lead oxide (obtained from Johnson Matthey, London)from a directly heated platinum boat made from 0.001-in. foil. When evaporation is carried out in high vacuum(i.e., 5X1O5 mm Hg pressure), the films exhibit asmall absorption of approximately constant valuethroughout the visible spectrum (see below), in addi-tion to stronger absorption in the violet. The formeris most probably due to a very small proportion ofmetallic lead derived from decomposition of the oxide.If, however, oxygen is leaked into the evaporationchamber so as to raise the pressure to 3X 10-4 mm Hg,and evaporation carried out under these conditions,the absorption is materially reduced without affectingthe other properties of the film very greatly.

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