JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Fresnel Zone Plate for Optical Image Formation Using ExtremeUltraviolet and Soft X Radiation

ALBERT V. BAEZSmithsonian Astrophysical Observatory, Cambridge, Massachusetts

(Received April 25, 1960)

A new type of Fresnel zone plate has been constructed whichcan focus ultraviolet radiation of any wavelength down to the softx-ray region. It consists of a set of thin circular gold bands madeself supporting by radial struts, leaving the transparent zonesempty. Experimental tests at 6700, 4358, and 2537 A showed thatthe theoretical minimum angular resolution obeys the Rayleighcriterion, sinomin= 1.22X/D. The diameter of the zone plate isD=0.26 cm and contains 19 opaque zones, the narrowest of whichmeasured about 20 u across. The zone plate was better than theoptimum pinhole in resolution by a factor of about 6 and in speedby a factor of 40. The zone plate produced pictures that compared

favorably with those made with a lens of similar focal length andaperture. The lens was about 20 times faster than the zone plateat 4358 A, but at 1000 A the zone plate would have been far fasterthan the lens. Focusing tests are contemplated at 1000 A and at100 A where lenses and mirrors, the conventional image-formingdevices, may fail. The angular resolution at 2537 A was close tothe theoretical value of 1.2)X 10-4 rad and held over a field of atleast 1.75 X 10-2 rad, which is 2.0 times the angle subtended by thesun's disk at the earth. A zone plate telescope, operating in thesoft x-ray or extreme ultraviolet region, far above the earth'satmosphere in an orbiting satellite, now seems possible.

INTRODUCTION

THIS paper describes experiments in the use of anew type of Fresnel zone plate for image forma-

tion with visible light and ultraviolet radiation. Testswith visible light and ultraviolet down to 2537 A aredescribed below, followed by a description of our plansto extend these tests down to about 100 A. There isgood reason to believe that the same zone plate willoperate over a large range of wavelengths from the softx-ray region through the infrared because its trans-parent zones are completely open. Therefore, it is op-portune to speculate on the possibility of using the zoneplate to focus even other radiations and particles. First,however, let us consider the motives that promptedus to construct this zone plate for soft x-ray and extremeultraviolet radiation.

Early research in the means of focusing light wasmotivated in part by interest in the construction ofmicroscopes and telescopes. These instruments were,after all, the earliest "space probes" into both the inter-molecular and extragalactic worlds. Later, the propertiesof the electromagnetic spectrum outside of the visibleregion prompted natural extensions of image-formingdevices into the near infrared and the ultraviolet.

Today similar motives exist for the construction ofmicroscopes and telescopes to work in the soft x-rayand extreme ultraviolet (hereafter referred to as euv)region. While no definite bounds exist for the euvregion we shall limit our study to wavelengths from 10 Ato about 1000 A. Although the developments in x-raymicroscopy and microradiography have already beenresponsible for two international congresses,', 2 the newinterest in telescopes that can operate in the soft x-ray

I Symposium on X-Ray Microscopy and Microradiography,Cambridge, England, August 16-21, 1956. The proceedings ofthis conference were published in the following work: V. E.Cosslett, A. Engstrom, and H. H. Pattee, Jr., X-ray Microscopyand Microradiography (Academic Press, Inc., New York, 1957).

2 Second International Symposium on X-Ray Microscopy andX-Ray Microanalysis, Stockholm, Sweden, June 15-17, 1959.

and euv regions stems chiefly from discoveries inastrophysics.

Rocket experiments have already demonstrated thatsolar, stellar, and interstellar sources of euv exist,3 butthe earth's atmosphere prevents most of the radiationsshorter than 3000 A from reaching its surface. Theadvent of rocket and satellite-borne experiments justi-fies serious consideration of telescopes and spectrographssensitive to wavelengths between 1 and 3000 A.

We distinguish between instruments, such as lensesand mirrors, whose main function is to form opticalimages, and those such as prisms and gratings, whosepurpose is to produce dispersion. Our interest for thepresent is primarily in image formation. The classicalmeans of focusing visible light for image formationinvolve reflection (mirrors) and refraction (lenses).Proper choice of materials allows the use of reflectorsand refractors in the ultraviolet from 3000 A to about1000 A. Between 1000 and 10 A refraction is out ofthe question because most materials are opaque toradiation in this region. Special types of glass4 have beendeveloped which can transmit about 70% at 2200 Athrough a thickness of 1 mm. But this is exceptionalsince the transmission of most types of glass drops es-sentially to zero below 3000 A. Although transmissionof wavelengths below 10 A does occur, the index ofrefraction is so close to unity that refraction isimpractical.

The transmission of several materials is shown in Fig.1.5 Except for thin films of aluminum, tin, indium, andbismuth, as shown, 6 the materials of which lenses mightbe made cannot transmit below 1000 A. Below thiswavelength, then, lenses are apparently not worthconsidering.

I R. Jastrow, J. Geophys. Research 64, 1647 (1959).4 L. R. Koller, Ultraviolet Radiation (John Wiley & Sons, Inc.,

New York, 1952), Chap. 5, p. 146.6 F. L. Whipple and R. J. Davis, Astron. J. 65, 285 (1960).6 W. C. Walker, 0. P. Rustgi, and G. L. Weissler, J. Opt. Soc.

Am. 49, 471 (1959).

405

VOLUME 51, NUMBER 4 APRIL, 1961

ALBERT V. BAEZ

CRYSTALS (2 MM. THICK)METALS (0.1 ' THICK)

FLUORIDE CRYSTALLINE QUARTZFUSED QUARTZ

/ CALCIUM // FLUORIDE /

M|NU / /, 1 / */~*CORUNDUMUMINUM t .-

\ I / STANDARD BAND PASS FILTER

NiSO4 COMPLEX

TINA. INDIUM

IUTH .i

400 800 1200

TRANSMISSIVITY

1600 2000 2400 2800

(I/lo) OF VARIOUS TRANSPARENT SUBSTANCES

3200 3600

FIG. 1. The transmissivity ofseveral materials in the ultra-violet. The crystalline materialsthat might serve for lensescease to show any appreciabletransmission below 1000 A. Thethin metaljfilms could serve asfilters.

X (A)

How about mirrors? Anastigmatic image formationwith a single mirror requires good reflectivity at normalor near-normal incidence. Recently Hass7 and his co-workers have developed coatings of evaporated metalfilms which, when deposited on glass or other materials,produce high reflectivities down to fairly short wave-lengths. Figure 28 summarizes the characteristics ofsome of the best coatings. The data were gathered frommany sources and are intended to show general trendsrather than authoritative values for which the readeris referred to the original literature, 7 9 especially forvalues of the reflectivity of aluminum overcoated withmagnesium fluoride at wavelengths below 800 A. Atthe present writing it seems safe to say that for wave-lengths shorter than 585 A the Fresnel zone plate mayfind application as a focusing device, although one mustnot exclude the possibility of using very large mirrorsat normal incidence with very low reflectivities.

For angles of incidence close to 90° it becomes con-venient to speak of the "grazing incidence angle,"that is, the complement of the usual angle of incidence,which is the angle subtended by the incident ray and thenormal to the reflecting surface. For x rays, the phe-nomenon of total reflection at grazing incidence is wellknown. 0"'" The reflectivity is 100% because the indexof refraction for most materials is less than unity. Point-to-point grazing-incidence systems for forming imageshave been built by Kirkpatrick and his students,12-14

7 G. Hass and R. Tousey, J. Opt. Soc. Am. 49, 593 (1959).8 G. R. Sabine, Phys. Rev. 55, 1064 (1939).9 P. H. Berning, G. Hass, and R. P. Madden, J. Opt. Soc. Am.

50, 586, (1960).10 L. M. Rieser, Jr., J. Opt. Soc. Am. 47, 987 (1957)." A. H. Compton and S. K. Allison, X-Rays in Theory and Ex-

perimient (D. Van Nostrand Company, Inc., Princeton, NewJersey, 1935), 2nd ed., Chap. IV, p. 305 et seq.

12 P. Kirkpatrick and A. V. Baez, J. Opt. Soc. Am. 38, 766(1948).

13 P. Kirkpatrick and H. I. Pattec, Jr., "X-ray microscopy,"Ilandbuch der Physik edited by S. Flugge (Springer-Verlag, Berlin,1957), Vol. XXX.

14V. E. Cosslett, A. Engstrom, and H. H. Pattee, Jr., X-rayMicroscopy and Mficroradiography (Academic Press, Inc., NewYork, 1957).

but reflection from a single curved mirror at grazingincidence is highly astigmatic and can be corrected onlyby the difficult technique of crossed mirrors. Systemsof this type have not received much attention for wave-lengths as long as 100 A but they may have to beconsidered.

If we restrict ourselves to the lesss formidable pro-cess of image formation by normal incidence reflectionat a single surface, we can probably conclude that a regionexists, somewhere between 10 and 1000 A, where imageformation by reflection or refraction is either difficultor impossible.

ZONE PLATE FOR FOCUSING EXTREMEULTRAVIOLET AND SOFT X RADIATION

Diffraction offers still another way of bending lightto produce focusing. Several such methods have beensuggested'", 6 but even the simplest, the Fresnel zoneplate'"" has not received serious use in any region ofthe spectrum, let alone that between 10 and 1000 A.Several writers have noted that Fresnel zone platesmight be used for image formation in this difficultregion, but no successful attempts to build a zone platefor x rays or euv have come to this author's attention.

This paper describes the construction and use of aFresnel zone plate whose opaque bands are made ofthin gold and whose open bands are completely trans-parent to radiation between 10 and 1000 A, becausethey are empty.

A Fresnel zone plate consists of alternately opaqueand transparent bands bounded by circles of radii

r,=rni; n=1,2,3-- . (1)

Its principal focal length f obeys the simple relation

AX= r, 2 , (2)

15 A. V. Baez, J. Opt. Soc. Am. 42, 756 (1952).16 D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).17 0. E. Myers, Jr., Am. J. Phys. 19, 359 (1951).18 J. Strong, Concepts of Classical Optics (W. H. Freeman &

Company, San Francisco, California, 1958).

80

60

40 ALU

20

BISN0 -.-- J

0

. . . . . . . . . . . . . . . . . .

406 Vol. 51

FRESNEL ZONE PLATE

FIG. 2. Normal incidence reflectiv-ities of various materials in the ultra-violet. Aluminum overcoated withmagnesium fluoride gives useful reflec-tivity even below 1000 A. Below 400 Athere is a dearth of experimental in-formation on reflectivity.

REFLECTIVITIES OF EVAPORATED METAL FILM

where X is the wavelength, and r1 is the radius of thecentral circle.

In considering the focusing of x rays for microscopywe see that the product fX can be disturbingly small.For example, in an x-ray microscope one might considerf== 1 cm and X= 1 A. This requires r1

2 =fX= 10-1 cm2 orr 1= 10-4 cm or 1 . By Eq. (1) we find that the widthof the nth zone is

so= [(a+ 1)§-Negro, (3)

which, for large n is approximated by

Sn= rl/2nl. ~~(4)

If, for example, n= 25 and r1= 10-4 cm, then sn= 10-4/2(25)- cm= 10-5 cm or one tenth of a micron. Since thebest photographic emulsions (e.g., Eastman 649 spectro-scopic plates) have a resolution of about a micron, azone plate made to these specifications looks impossibleindeed. The product f X needs to be considerably largerbefore zone plates look feasible. Fortunately, the currentinterest in the use of x rays and euv in microscopy andastrophysics can lead to larger values of both f and X.

Consider X first. Wavelengths much longer than 1 Aare important for different reasons. In x-ray microscopy,for example, there has been a trend toward wavelengthsapproaching 100 A because microscopic objects of bio-logical interest have low atomic numbers and theirdifferential absorption is therefore greater in this re-gion. 9 On the other hand, the wavelengths from 10 to100 A, which are very long x rays, are considered veryshort compared with the wavelengths of ultravioletradiation, and are of potential interest in astrophysics.Also, the lines around 150 and 300 A emitted by thesun's corona could be of interest as sources for telescopicimage formation.2 0

Now consider the focal length f. In microscopes wewant f to be small, but in a telescope a long focal lengthis an advantage; in fact, a focal length of 100 cm wouldbe practical and one of 1000 cm, though somewhat large,

19 B. L. Henke, Proceedings Seventh Annual Conference onIndustrial Applications of X-Ray Analysis, University of Denver,Denver, Colorado, 1958.

10 C. Dejager, Ann. Geophys. II, 1 (1955).

is not completely beyond possibility if we could buildthe right kind of platform. The reader must bear inmind that the ultimate platform for telescopes operatingin the soft x-ray and extreme ultraviolet regions must bea satellite orbiting above the earth's atmosphere.

To choose figures of the right order of magnitude,consider f= 100 cm and X= 100 A. Using n= 25 we getsn=r 1 /2n=10- 8 cm or 10yl. This value is about 10times the resolving power of the finest-grained photo-graphic films and makes the construction of a zone platefor soft x rays seem feasible.

However, a zone plate made by exposing a photo-graphic film would be useless in the region between 10and 100 A because the base on which the emulsionrests and even the emulsion layer itself are practicallyopaque to these wavelengths.

FIG. 3. An enlarged photograph of the self-supported gold zoneplate used in these experiments. The diameter of the outer circleis 0.2596-40.0002 cm. The central circle has a diameter of0.042640.0002 cm. The thickness of the gold is estimated as 10 ,u.The white bands representing the transparent regions are com-pletely open and hence will transmit electromagnetic radiation ofall wavelengths.

407April 1961

ALBERT V. 3AZV.

FiG. 4. The layout of the apparatus on the optical bench. The light source consisted of a tungsten filament lamp and a condensinglens system. For the experiments at 2537 A the source was a 4-w General Electric germicidal lamp, No. GT4/1. The filters are de-scribed in the text. The zone plate was mounted at the open end of a bellows extension. The camera was an Exakta VX Ila. EastmanKodak Microfile Contrast Copy film was used throughout, processed in D 19. Object and image distances are labeled p and q, respectively.

Recently we have succeeded in producing a zone platewhose opaque elements are thin concentric bands ofgold made self-supporting by the use of thin radialstruts (see Fig. 3). The streaks in the photograph areimperfections in the form of very thin filamentlikepieces of gold. The outer diameter of the zone plate is0.259640.0002 cm. The central circle has a diameterof 0.042640.0002 cm. The bands are bounded by circleswhose radii obey the relation r = 0.021301 , it= 1, 2.. 38.For a zone plate with a large number of rings the moreprecise formula for r would be needed,

r,,= (nXl)-E1+ (X\/4f)]4. (5)

The narrowest band was designed to measure 0.0017 cmin width. Actually it varied between 0.0010 and 0.0020cm. The zone plate was made for us by the BuckbeeMears Company of St. Paul, Minnesota. It was madeby a combination of etching and electrodepositioncapable of high resolution. As far as we know this isthe first zone plate with transparent regions that arecompletely open (except for the supporting radialstruts). The images formed are very sharp, comparingfavorably with those made by lenses of similar focallength and aperture (see Fig. 9). We may suggest oneexplanation for the good quality obtained with thesezone plates as compared with that of the photographic-ally based zone plates. The phase condition that requiresan increment of exactly one wavelength between suc-cessive transparent bands must be difficult to meet witha film base many microns thick that is not held to opticalflatness. This condition is probably less difficult to meetwith the new zone plate that has no film base at all.

Since the transparent regions of the zone plate arecompletely clear they should transmit radiation of allwavelengths. The gold bands are practically opaque toall radiations between 10 and 1000 A; therefore, thezone plate should be able to focus soft x rays and euvin this region. Of course, it can also operate in the visibleand infrared regions. It should also work with particleshaving wavelike characteristics of the proper wave-length, but our present interest is in the soft x-ray andeuv region.

Our first zone plate, with which all the tests reported

in this paper were made, was designed to have a focallength of about 400 cm at 100 A. At 4000 A its focallength is 10 cm; it thus lent itself very conveniently totests with visible light. All the tests described belowwith the exception of those at 2537 A were made withvisible light. We plan to continue tests at 100 and 1000 Awith newly acquired sources.

RESOLUTION

Theoretically, the smallest angular separation min

between two monochromatic point sources at infinitythat can just be resolved when imaged by a zone plate,can be shown7 to obey the Rayleigh criterion for a lens:

sinOmin= 1.22 (X/D). (6)

Here X is the wavelength and D the diameter of theoutermost circle of the zone plate. To test this relationwe used fine mesh screens as objects, illuminating themby transmission as shown in Fig. 4. In Eq. (6) we couldvary X, the wavelength being used, but not D, as allour zone plates had the same diameter, approximately0.26 cm. A zone plate is a highly chromatic device,having a focal length that is inversely proportional tothe wavelength. We used filters to monochromatize theradiation or relied on the strength of a spectral line,as in the case of the 2537 A source, to produce approxi-mately monochromatic radiation. The object distancep, and the image distance q (Fig. 4), were chosen toexhibit the improvement in resolution that accompaniesthe use of shorter wavelengths. We operated in a regionclose to the limit of resolution. The parameters werepurposely chosen to exhibit the improvement in resolu-tion from red light to ultraviolet. For example, in Fig. 5three pictures of the same object mesh made with thezone plate are shown superimposed on one another toexhibit the ratio of the sizes in which the originalpictures appeared when taken with red (6700 A), blue(4358 A) and ultraviolet (2537 A) light. The increasein image size results from the fact that the focal lengthof the zone plate increases as the wavelength decreases,while the object distance p remains fixed at 47 cm.This increase in image size alone can bring about animprovement in signal-to-background ratio, but notice

408 V7ol. 51

FRESNEL ZONE PLATE

that an improvement in resolution has also taken place.The ultraviolet picture is not only the largest, it is thebest resolved because it was made at the shortestwavelength.

To exhibit this improvement in resolution apart fromthe change in magnification, the pictures of Fig. 5 wereenlarged photographically so that the distance betweensquares remained constant. The improved resolutionbecomes clearer. In Fig. 6 (left), p equaled 47 cm andq was adjusted to 7.2 cm, a value that produced thebest focus experimentally for a wavelength of 6700 A.A gelatin filter was used with a pass band of about400 A. The object mesh had 4 lines per mm. The anglesubtended at the zone plate by two successive centersof open mesh was 0.00053 rad, which is 1.68 6

min forthat wavelength.

The center picture of Fig. 6 was made at p=47 cm,q= 12.5 cm, at a wavelength of 4358 A. A Baird inter-ference filter with a total pass band of about 100 A wasused. Here the angular separation between adjacentcenters is 2.59 0

min. An improvement in resolution isapparent.

Figure 6 (right) shows an image of the same meshtaken at 2537 A. The value of p remained fixed at 47 cm,q was 27 cm and the angular separation between objectcenters was now 4.44 min. The source was a GeneralElectric germicidal lamp number GT4/1. The manu-facturer states that this lamp yields 60% of its energyat 2537 A. A filter was used to remove the visible radia-tion but the monochromatic effect at 2537 A is due

FIG. 5. Pictures of a mesh with four lines per mm taken withthe zone plate at three different wavelengths. Working out fromthe center we see pictures taken at 6700, 4358, and 2537 A, insizes proportional to the original image sizes obtained with a fixedobject-to-zone plate distance of 47 cm. The shorter wavelengthresults not only in a longer focal length and hence a larger imagesize but also in inproved resolution. See also Fig. 6.

FIG. 6. The pictures of Fig. 5 are here enlarged to produceequal image sizes. From left to right, they were made at 6700,4358, and 2537 A. The angles subtended at the zone plate by twoadjacent midpoints of the open area of the object mesh were1.68 6min, 2.59 0,mn, and 4.44 0mi., where 0mi is the theoreticalminimum angle of resolution computed at the respective wave-lengths. The improvement in resolution with shorter wavelengthis apparent.

only to the relative intensity of this line. This demon-strates a point about zone plate focusing that may beuseful in ultraviolet and x-ray astronomy: A relativelyintense and isolated spectral line may preclude the useof filters. These pictures lead us to believe that theresolution will continue to improve in a predictableway as we go down first to 1000 A and later to 100 Awith our zone plate.

COMPARISON OF ZONE PLATE WITHPINHOLE AND LENS

Next we compared the image-forming qualities of ourzone plate with those of a pinhole. The pinhole size wascalculated to give the optimum resolution for the chosenvalues of , p, and q. The optimum diameter for such apinhole is given by the expressions

d= 20.9pqX/(p+q)}. (7)

This may also be written d= 2(0.9fX)1 which, interest-ingly enough, is very close to the diameter of the inner-most circle of a zone plate that gives good focus at thesame distances and for the same wavelength. In otherwords, the pinhole behaves like a zone plate with asingle circular opening.

Figure 7 is a picture of a mesh with 4 lines per mm(not visible because unresolved), framed by the linesof a coarser screen with 0.7 lines per mm (poorly re-solved), made with the pinhole at a wavelength of4358 A. With the same screens as an object and all otherfactors the same, a picture was taken with the zone plateinstead of the pinhole, with the results shown in Fig. 8.The 4 line/mm screen is clearly resolved. The angularseparation at p= 26.7 cm, between adjacent centers of

21 J. E. Mack and M. J. Martin, Te Photographic Process(McGraw-Hill Book Company, Inc., New York, 1939).

409April 1961

ALBERT V. BAEZ

FIG. 7. An object consisting of a coarse grid, one rectangle ofwhich measures 1.43 mm by 1.72 mm within which there is a finemesh with four lines per mm, photographed through a pinholewhose aperture was chosen to produce the optimum resolutionfor the wavelength and distances involved. Only the coarse gridis resolved. Compare this with Fig. 8.

the smaller screen pattern was 5.3X 10-4 rad or 4.5 Omin,

as computed for the zone plate. Theoretically, the im-provement in resolution of the zone plate over that ofthe pinhole is a factor ;Bs of 6.23 for our zone plate of38 rings. The actual improvement as demonstrated inFigs. 7 and 8 is quite apparent.

The zone plate has another advantage over a pinholein that the light flux reaching the image is greater withthe zone plate, so that much less exposure time is re-quired if all other factors have been left unchanged. Inour notation, n stands for the number of circles in thezone plate pattern. Hence z is the number of openzones or bands. The expected improvement in exposureshould be nt since all the zones have equal areas. Forour zone plate, n= 19. The actual exposure time usedfor the pinhole was 40 times greater than that requiredfor the zone plate. Since the densities of the plates werecompared only by eye, the experimental values do notseem out of line.

Next we compared the resolution of our zone platewith that of a lens of similar focal length, of apertureequivalent to that of the zone plate. For both the lensand the zone plate f= 10.2 cm and D= 0.26 cm. Figure9 (right) shows the image of a screen with 4 lines permm photographed with the zone plate. Adjacentcenters of the object screen subtended an angle equalto 2.6 Otnin. Other constants were p=47 cm, q= 13 cm,and X=4358 A. The lens had a simple plano-convexform. If a highly corrected camera lens had been chosenthe comparison might have favored the lens, but sinceit was stopped down to f/18 and the field covered wasnot very large, it was not subjected to undue demandsas an image-forming device. We can conclude that thezone 1plate image compares favorably with that of thelens.

The zone plate was about 20 times slower than the

lens for the following reason: In the vector diagramrepresenting the amplitude of the waves that passthrough successive Fresnel zones, the continually chang-ing phase produces a spiral (Fig. 10). In Fresnel'streatment of the contributions of successive circularzones, the amplitude of the contribution due to onezone is represented by the vector AB. The length ofthe arc ABC, starting at A and ending at C, would beproportional to the amplitude produced by two zonesof a lens (since the lens has the property of orientingall the infinitesimal vectors in the same direction). Theratio of these amplitudes is approximately wr. The ratioof intensities would be 7r

2 . This argument would holdfor a perfect zone plate, that is, one in which the widthof an opaque band never exceeded the mathematicallycomputed value of Sn.

With the real zone plate, the widths of all the goldbands are greater than the theoretical value of ,thereby reducing the intensity of the light gathered atthe focus. A perfect zone plate introduces the factor 7r2

and inaccuracies in manufacture introduce a factor ofabout 2, making the ratio approximately 27r2, which isabout 20. This means that a single zone plate is not avery efficient gatherer of light when compared with alens. However, with sufficiently intense sources it canproduce images with good resolution. Certainly theresolution and light-gathering power are much betterthan those of a pinhole. We conclude that the zoneplate is much better than a pinhole both in resolutionand light-gathering power; it is comparable to a lens inresolution but about 20 times slower than a lens ofsimilar aperture when used with visible light.

To overcome the intensity limitation the method ofsuperposition of images could be used. It has beenshown22 that ni pictures of the same object each exposed

FIG. 8. The object of Fig. 7 photographed through a zone plate,with all other factors fixed except the exposure. The exposuretime required for the pinhole picture was about 40 times greaterthan that uised for the zone plate p~icture. This timne the finemeshed screen is clearly resolved.

2 S. Kirby, Pubis. Astron. Soc. Pacific 71, No. 334 (1959).

410 Vol. 51

FRESNEL ZONE PLATE

for a time t (where t would yield an underexposed nega-tive) give negatives which when superimposed producean image comparable in exposure to that of a singlepicture exposed for a time nW. Instead of photographicrecording some form of electronic readout might beused to integrate the signals from X zone plates andproduce an exposure in seconds that ordinarily wouldhave required nt seconds. Because noise increases alongwith the signal, the advantages in resolution are notsimply multiplied X times. Nevertheless, a gain doesresult from this technique and since zone plates mayeventually be produced fairly cheaply, it is conceivablethat the signals received simultaneously from manyzone plates could be integrated advantageously.

B

w_-2

I

CONCLUSION

In summary, then, we have produced a Fresnel zoneplate consisting of a set of gold bands supported byradial struts, leaving alternate bands almost completelyempty, thereby making possible the transmission andfocusing of soft x rays and extreme ultraviolet radiations.We have tested it in visible light and in the ultravioletonly as far as 2537 A. All tests indicate that the experi-mental resolution and transmissivity are what one wouldexpect from simple theoretical considerations. Thismeans that from the point of view of resolution andexposure the zone plate is a great improvement over apinhole. It compares favorably in resolution, but notin speed, with a simple lens, if we make the comparisonin the visible region. Below 1000 A, however, the speedof the zone plate would be thousands or millions oftimes greater than that of a lens made of any knownmaterial. All the experimental evidence now in existenceseems to indicate that there is a region in the euv where

FIG. 9. Comparison of a lens and a zone plate as image formingdevices near the limit of resolution. On the left is a picture of afine mesh screen taken with a zone plate. On the right is a pictureof the same screen taken with a plano-convex lens of similaraperture and focal length. All other factors were kept the sameexcept exposure which was 20 times greater for the zone plate.The wavelength used was 4358 A. If the exposure comparison hadbeen made at 1000 A the results would have been much morefavorable for the zone plate.

C

K~~~~~

I

I-,

A

FIG. 10. The vector AB represents the resultant of the infini-tesimal vectors contributed by the subzones of one open Fresnelzone in a zone plate. The length of the arc ABC represents theamplitude of the contributions due to two successive Fresnelzones in a lens. The continuously changing thickness of a lenscauses the infinitesimal vectors to line up since they all have thesame phase. The ratio of the length of the arc ABC to the lengthAB is approximately 7r.

the zone plate could serve as the only practical focusingdevice, because the reflectivity and the transmissivityof known materials would make the use of lenses ormirrors at normal incidence either difficult or impossible.

The lines of future research now seem clear. First,experiments to focus soft x rays and euv radiationbetween 10 and 1000 A should be made since in thisregion the conventional methods of image formationseem likely to break down. Next, as the zone plate isa highly chromatic device, sources of essentially mono-chromatic radiation must be found for it. Severalschemes suggest themselves at once. For example,limiting stops placed strategically can absorb the radia-tion of one wavelength while almost all else proceedsessentially unaffected. If we could precede the circularzone plate by similarly self-supporting diffraction grat-ings, new types of euv spectrographs with little or noastigmatism might become feasible.

The uses to which these zone plates may be put in theastronomical telescopes of the future will be determinedby the problems that need to be explored in astrophysics.Because of the great intensity available from the sunsome early solar experiments would probably be inorder. The isolated lines at around 150 and 300 A inthe spectrum of the sun's corona20 28 seem ideal for an

23R. Giacconi and B. Rossi, J. Geophys. Research 65, 773(1960).

411April 1961

M

ALBERT V. BAEZ

experiment with a zone plate telescope "tuned" toone of these lines. Next in intensity-and in possibleinterest-might be the moon and the planets. The pres-ent limitations on speed indicate that we must developmeans of greatly increasing the effective speed of azone plate device before it can be useful in gatheringeuv and x rays from the stars.22

JOURNAL OF TIHE OPTICAL SOCIETY OF AMERICA

ACKNOWLEDGMENTS

I wish to thank Dr. Robert J. Davis for encouragingthis research, Andras M. Bardos for doing most of theexperimental work with the diligent help of PeterHofmann, and the Buckee Mears Company of St. Paulfor the delicate work which produced the zone platewithout which this research would have been impossible.

VOLUME 51, NUMBER 4 APRIL, 1961

Diffraction Images of Circular Self-Radiant DisksLUTHER W. SMTH AND HAROLD OSTERBERG

Research Center, A inerican Optical Company, Soultbridge, Massachusetts(Received November 25, 1960)

The diffraction image formed by an Airy-type objective when the object is a circular, uniformly self-radiantdisk is found to exhibit the phenomenon of a secondary Airy disk when the radius of the object is near 1.2Airy units. The time-averaged irradiance in the diffraction image is presented for radii up to 4 Airy units.A separate expression for the irradiance at the geometrical radius in the image shows that as the objectbecomes very large the value of this irradiance approaches one-half that at the center of the image.

INTRODUCTION

A THEORETICAL investigations 2 of diffractionimages of nonself-radiant, disk-shaped particles

viewed under full Kdhler illumination has provided afunction which can be used to describe the diffractionimage formed by an Airy-type objective when theobject is an extended, uniformly self-radiant circulardisk. This function determines the time-averagedirradiance3 in the diffraction image versus radialdistance from the center of the image. Within theapproximations of the theory, details of the diffractionimages are-found to depend only upon the ratio of thesize of the geometrical image to the size of the Airyunit ra in the image space. This Airy unit is defined bythe equation 27rnpmra= 3.8317X, where 3.8317 is approxi-mately the first nonzero root of Ji(x); pm is sinOm; Gmis the maximum angular opening of the objective withrespect to its image space; it is the refractive index ofthe image space (taken as unity here); and X iswavelength.

A special solution for computing in a more convenientmanner the irradiance in the image at points cor-responding to the geometrical edge of any disk hasbeen added.

This work complements the investigation by H.Nagaoka,4 who obtained curves of irradiance in tele-scopic images when the radius of the source is manytimes the limit of resolution.

I H. Osterberg and L. Smith, J. Opt. Soc. Am. 50, 362 (1960).2 L. Smith, J. Opt. Soc. Am. 50, 369 (1960).

Formerly called energy density.4 H. Nagaoka, Phil. Mag. Ser. 5, 45, 1 (1898).

EXPRESSION FOR IRRADIANCE

Nagaoka's expression for the distribution of ir-radiance H(L) in the plane of the sharply focusedimage of a self-radiant disk can be written in the form

X 2 r

M2H(L)/pm2= f f [J12 fl)/co 2]tdtd0, (1)

in whicha= (L2+ t2-2Lt coso) 1,

03-3.8317,

where L is distance in Airy units from the center ofthe diffraction image, K is the radius (in Airy units)of the geometrical image' of the self-radiant disk andM is the magnification ratio of the objective.

Equation (1) can be derived from modern theoriesof image formation by requiring that the self-radiantdisk shall radiate uniformly in a Lambertian mannerand that the objective shall be of the Airy type, i.e.,shall image a small dipole radiator as a distribution ofamplitude and phase with the form J1 (ax)/(ax).

Integration of Eq. (1) can be carried out as shownby Osterberg and Smith,6 for the double integral inthe right-hand member is just the function 1md whicharises in the theory of images of nonself-radiant disk-shaped particles viewed with full K6hler illumination.The behavior of M2H(L)/pm 2=Iud for radii up to 4Airy units is shown in Fig. 1, which is a plot of values

I K is also the radius of the self-radiant disk in Airy units withrespect to the object space. However, the Airy unit with respectto the object space is 1/ MI times the Airy unit r with respectto the image space.

6 See p. 365 of work cited in reference 1.

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