Diffractive-Refractive Optics for Focusing Hard X-Rays Beams

Jaromir Hdy*a, Nikolai Artemiev*', Andres Freund**', John P. Quintana***'

ajnstitute of Physics ASCR, Praha ; bESJJ France;CAPS Argonne National Laboratory, USA

ABSTRACT

The sagittal focusing of x-ray beam diffracted on symmetrically cut crystals with a longitudinal parabolic groove on theirdiffraction surfaces has been proved experimentally and the results have been already published. This kind of focusing isbased on the refraction phenomena occurring during Bragg x-ray diffraction. In this paper our new developments in thisfield are reported. First, it was shown experimentally that in some cases a channel-cut crystal monochromator withlongitudinal parabolic grooves can be replaced by a single crystal with a round hole drilled parallel to diffracting planes.This substantially simplifies the manufacturing of such a focusing monochromator. Second, it has been experimentallyproved that the refraction effect, on which the focusing is based, may be substantially enhanced by cutting the longitudinalparabolic groove into the surface of an asymmetrically cut crystal (or drilling a hole whose axis is tilted with respect to thediffracting planes). A very simple formula describing the focusing properties for this case is derived. Finally, the results ofthe first experiment on the meridional focusing of x-ray beam diffracted on a crystal with a transversal groove on its surfaceare reported. Some experimental results are compared with the results of ray-tracing simulations, which were developed forthis purpose.

Keywords: x-ray crystal monochromators, focusing hard x-rays.

1. INTRODUCTION

It is well known, that in Bragg diffraction of x-ray beam on a perfect crystal the angles between diffractingcrystallographic planes and the incident or diffracted beams are always somewhat larger than the angle °B (Bragg angle)which is calculated from Bragg 's law. This is the consequence of refraction. If a crystal is cut such that its surface is parallelwith diffracting planes (symmetrical diffraction) then the angles between the diffracting planes and the impinging ordiffracted beam are equal, both beams and the normal to the diffracting planes lie in one plane (coplanar case) and thesituation is similar to the reflection on mirror. If the surface is not parallel to the diffracting planes then we can observe asmall deviation from the mirror-like behaviour. This means that the angle of incidence on the diffracting planes differsslightly from the angle of reflection and/or both beams and the normal generally do not lie in one plane (non-coplanar case).The situation for two extreme cases, inclined and asymmetrical diffraction, is shown in Figs. 1 and 2. The deviation frommirror-like behaviour is represented by an angle 6, whose value is of course much smaller than shown in the picture. Theinclined diffraction is obviously non-coplanar.

These phenomena, here called diffractive-refractive phenomena follow from the dynamical theory of diffraction andmay be utilised for the focusing of synchrotron radiation. The corresponding device plays a twofold role: it functions as amonochromator and a focusing lens. In this paper the previous works are briefly described and then some new results arepresented.

* hrdy@fzu.cz; artemiev@fzu.cz ;phone +4202 66052148; fax +4202 8581448; Institute of Physics, Academy of Sciences of theCzech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic; ** freund@esrf.fr ;ESRF, B.P.220, 38043 Grenoble, France;jpq@northwestern.edu; APS, Argonne National Laboratory, Argonne, IL 60439, USA

Invited Paper

X-Ray Mirrors, Crystals, and Multilayers, Andreas K. Freund, Tetsuya Ishikawa, Ali M. Khounsary,Editors, Proceedings of SPIE Vol. 4501 (2001) © 2001 SPIE · 0277-786X/01/$15.0088

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polychromatic

Fig. 1. Inclined diffraction

polychromatic

N

Fig. 2. Asymmetric diffraction

2. SAGITTAL FOCUSING USING BRAGG DIFFRACTION ON LONGITUDINAL GROOVE ORINSIDE A HOLE

2.1 Longitudinal parabolic groove produced into the surface of symmetrically cut crystalsIt was suggested by Hrdy1 that an x-ray beam diffracted on a symmetrically cut crystal with parabolic longitudinal grooveproduced on its surface is sagittaly focused. The wall of the groove is inclined with respect to the diffracting planes

Fig. 3. Sagittal focusing onlongitudinal parabolic groove

Focus

Fig. 4. Geometry of beams in inclineddiffraction

such that the diffraction is an inclined one (Fig. 1) and thus each point on the wall creates certain sagittal deviation & Theparabolic shape is necessary to create the focus (Fig. 3). As known from the dynamical theory of diffraction a beam ofcertain wavelength A is accepted within certain angular interval and is diffracted within certain angular interval. For theinclined diffraction this situation is in detail seen in Fig. 4. The beams 1 and 2 represent the borders of accepted anddiffracted region for certain A. It is seen, that the diffracted beam creates a fan in sagittal direction. It may be shown thateven a single parallel but polychromatic incident beam creates such a fan when diffracted (Fig. 1). This fan causes a

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smearing of the focus. The longitudinal groove also increases the size of the diffracted beam in meridional direction (thehorizontal beam with "zero" vertical size will have the vertical size 2tcosO after diffraction on the groove of depth t) .Boththese effects may be eliminated by using crystals with grooves in dispersive (+,-,-,+) arrangement (Fig. 5).On this picture itis schematically shown that the vertical broadening of the beam leaving the second crystal is completely cancelled by thesecond pair of crystals. This arrangement also automatically keeps the position of exit beam fixed. In principle a simple

Fig. 5. Fourcrystal (+,-,-,+) arrangement Fig. 6. Channel-cut crystal with

double crystal (+,-) arrangement may also be used if the depth of the groove is small in comparison with the vertical size ofthe SR beam. If the depth of the groove is large then the increase of the vertical size of the diffracted beam may be partiallyreduced by the more complicated shape of diffracting surface as described by Hrdy1. The (+,-) arrangement, however, doesnot cancel the smearing.

The angular deviation of the central ray may be calculated according to the following approximate formula1:

o= Ktanf3,

where for Silicon K= 1.256x103 d[nm] X[nm], and 3 is the angle between the surface of the crystal and diffracting planes(angle of inclination).

The shape of the parabolic groove is given by

y [mm] = a [mm'J (x[mm])2,

where a may be determined from the number N of diffraction events on the grooves, source-to-lens distance S [mm] andlens-to-focus distancef [mm] as follows:

a=(S+J)/2NKJS.

As K is proportional to X, the focusing distance decreases with the increase of X. This means that the focus of thefundamental harmonic is located on the broad background of higher harmonics. Thus the contamination of the focus byhigher harmonics is small. We have showed2 that the above approximation is good up to about f3= 86°.

The first experiment on such kind of focusing was performed in NSLS3. By means of two channel cut Si(1 11) crystalswith identical parabolic grooves (y = 1.01x2) produced into each of four diffracting surfaces (Fig. 6) we were able to focus15 keY radiation at 5.7 mfrom the crystals. The focus is seen on Fig.7. The two lateral weak spots are caused by the lateralparts of the beam diffracted on the flat part of the crystal surface (the size of the beam was somewhat larger than the widthof the groove, which was about 2 mm). The size of the focus was 0.29 mm and the size of corresponding unfocused beam atthe place of the focus was 2.7 mm. (The size of the demagnified image of the source was 0.03 mm). The experimentshowed that this kind of focusing works well but the quality of the surface of the grooves has influence on the sharpness ofthe focus. This influence, however, is much less than in the case of x-ray mirror.

parabolical grooves

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Fig. 7. Focus of 15 keV radiation created at 5.7 m from the focusingmonochromator. The weak lateral spots are caused by the lateral partsof the beam diffracted on the flat part of the crystal surface.

2.2 Diffraction inside a hole produced into a crystal parallel to diffracting planes

It was shown by Artemiev, Hrdy et al.4 that in some cases the parabolic groove may be approximated by a groove withcylindrical shape and such a channel-cut grooved crystal may be replaced by a single crystal with a hole drilled parallel todiffracting planes. The beam is then diffracted twice (or even more times) inside the hole. The diameter D of the hole issimply

D[mm]=1/a

Such a device may suffer from an aberration and less horizontal acceptance but its production is very simple and cheap.The first test experiment was performed on BM5 beamline in ESRF4. The focusing distance at this beamline was 2 m,

the source-to-focusing monochromator distance was 40 m. For this geometry and the wavelength A =0.154 nm the diameterD of the hole was 1mm. We used two such crystals arranged both in dispersive and nondispersive position. The crystalswere arranged such that the plane of diffraction was horizontal to utilise the small sagittal component of the demagnifiedimage of the source, which is projected into the focus. The focusing effect was seen in both arrangements, the better resultwas, however, obtained in nondispersive arrangement, which is probably connected

Fig. 8. The snap shot of the 8keV radiation diffracted inside the 1 mm holesin two Si(1 11) crystals in the dispersive setting. On the left side there is an imageof the third harmonic diffracted in the hole of the first crystal and penetrating thesecond crystal. This roughly shows what would be the vertical size of the beamif there would be no focusing effect.

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with the circular shape of the diffracting surface. In both arrangements the beam filled the whole profile of the holes andconsequently the diffraction took place also from those parts of the holes which differ substantially from the theoreticalparabolic shape. Nevertheless, the intensity in the focus was about 5 times higher than the intensity diffracted from the flatpart of the crystals. The ray tracing simulation (developed especially for the study of the diffraction — refraction phenomenaby Artemiev5 shows that in this case the realistic increase of the intensity in the focus when the whole hole is filled byradiation does not exceed the factor of 7. The snap shot of the focused radiation is shown in Fig. 8.

Another experiment with holes was performed in APS at the 51D beamline6. The task was to demonstrate whether it ispossible to focus the undulator radiation at the long focusing distance, here 20 m. Such a long focusing distance requiresgenerally a very good quality of optical elements. The source-to-monochromator distance was 55 m. We decided to sagittalyfocus the 8.048 keY and 13 keY radiation. For these wavelengths and above described geometry the diameters D of theholes drilled into Si single crystals parallel to (1 1 1) crystallographic planes were 7.2 mm for 8.048 keV and 4.4 mm for 13keY. Fig. 9 shows one of these crystals; the hole at the middle has the diameter 6 mm but it was not used. The inner

Fig. 9. One of the crystals used in APS Fig. 10. General shape of a crystalexperiment. The channel on the left side for 3 wavelengthswas cut to compare the diffraction ina hole with that on a flat crystal

surface of the holes were polished and etched. This time the diameters of the holes were much larger than the size of thebeam (which was about 2 mm) so that the part of the surface hit by the beam was not too different from the parabola. Bothcrystals with holes were arranged into dispersive position and the plane of diffraction was vertical. The calculated FWHMhorizontal size of the demagnified image of the source at the place of the focus was about 308 .tm. The real (measured)FWHM horizontal size of the focus was 417 .tm. The FWHM horizontal size of unfocused radiation at the place of thefocus was 2. 1 3 mm. This result was the same for both energies. One can conclude that in the case of the point sourcethebeam would be squeezed to about 0.1 mm. Fig. 1 1 shows the plot of the sagital profile of the focused beam together with thebeam diffracted on the flat part of the crystals (the profile was measured at the place of the focus).The length of the crystals was 50mm for both wavelengths. Generally, various wavelengths may require various lengths ofholes. This may be solved very easily as shown in Fig. 10. It should be emphasised that such kind of focusingmonochromator creates a focus at certain place only for one ? and its close neighbourhood.

2.3 Hole or parabolic groove manufactured into an asymmetrically cut crystal

Korytar, Bohacek, and Ferrari78 suggested that the refraction effect leading to the sagittal deviation ö of the diffractedbeam from the plane of diffraction should be enhanced for the general asymmetric diffraction, i.e. for the case when thediffraction is partially asymmetric and partially inclined. By it we mean the case when the surface is not parallel to thediffracting planes but the diffraction is neither (purely) inclined nor (purely) asymmetric. Such a case may be also called a"rotated-inclined" diffraction (see the analogy with a rotated-inclined monochromator utilising this kind of diffraction).

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Korytar, Bohacek, and Ferrari7'8 formulated the corresponding equations and presented a numerical solution. We haveperformed an experiment on BM5 beamline in ESRF9 where the beam was diffracted on a wedge cut into a surface of anasymmetrically cut crystal. From the splitting of the diffracted beam we concluded that this effect really exists.

250

200

150

U)

100

C

50

0

Fig. 11. The plot of the sagital profile of the focused andunfocused radiation at the APS experiment

Korytar's numerical solution is complicated and thus not too convenient for fast design of sagittally focusingmonochromators based on a longitudinal groove in asymmetrically cut crystal. For this reason Hrdy'° derived a very simpleapproximate formula for for rotated-inclined diffraction. He showed that

where= K' tan 13,

K'=K[(2+b-t-1/b)/4cosa].

(5)

(6)

Here the meaning of 13 is following. First it is necessary to make an asymmetric cut such that the angle between thediffracting planes and the surface is a. Then let us imagine some plane P which is perpendicular to the intersection line ofthe diffracting planes and the surface. (The impinging beam lies in this plane.) The intersection line of the plane P with thesurface is o. The angle 1 is the angle by which we have now to rotate the surface about o to create the desired rotated-inclined case. This exactly describes the angle between the wall of a longitudinal groove machined into an asymmetricallycut crystal and the flat surface. The index of asymmetry b is here defined as

where a is positive at grazing incidence.

b=sin(O—a)/sin(O+a), (7)

To calculate a or the focusing distance f it is sufficient to replace in (3) K by K'. Fig. 12 shows the dependence of thefocusing distance f on Bragg angle for two grooves of the same parameter a but different a. It is clearly seen that thefocusing distance for a different from zero is smaller than that for a =0, particularly when 0 approaches a. From (6) and(7) it is also seen that the focusing distance is independent of the sign of a. It means that we have an option: either to use a

0 1000 2000 3000 4000

Sag ittal direction, Jim

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higher or lower resolution arrangement depending on the sign of a. On the other hand, to create the focus with anasymmetric crystal at the same focal distance as with symmetric crystal, it is sufficient to use smaller a; the parabola is thenmore "open" and the horizontal size of the beam which is to be focused may be increased (up to several cm in some cases).

Another advantage of using an asymmetric crystal is that the relative width of the sagittal fan of the diffracted beam (seeabove) with respect to ö is smaller and thus the

14000\ S=40000 mm, S:i(111), N=4

12000

10000aIph=O

Na=O 22800C

6000

4000 , ph=C

a=12.382mm/

2000 .

3 B-12 14 16 18 20 22 24 26 28 30

Bragg angle [deg]

Fig. 12. The dependence of a focusingdistance on 0 for symmetric and asymmetriccrystals

Fig 13. Longitudinal groove in asymmetriccrystal

focus should be sharper10.Like in the symmetrical case the best arrangement should be a dispersive (-,+,+,-) arrangement to reduce or cancel the

aberrations mentioned above. The design of the shape of the crystals for the focusing monochromator for two or morewavelengths will generally look like it is schematically shown in Fig. 14 for grooves and Fig. 15 for holes.

>• / / /// /// /// /7 / /

Fig. 14. Crystal with grooves for two A Fig. 15. Crystal with holes for two A

The sagittal focusing using the asymmetric crystal was first realised at BMS beamline at ESRF11. We have used thewavelength A = 0.15 nm, Si(1 11) crystals and a = 12.38°. The focusing distance was 2 m. For the simplicity we usedcrystals with holes instead of grooves. The diameter of the holes for this arrangement was D =4.4 mm. As in the previous

EEa(3CaU)V0)C

()0

4 10

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cases two such crystals were set into a dispersive position to create (-,+,+,-) geometry and the plane of diffraction washorizontal (in this case the vertical size of the source does not influence the size of the focus substantially). Fig. 16 showsthe beam, which is leaving the second crystal; its vertical size is 3 mm. Fig. 17 shows the image of the beam at the distanceof 215 cm from the crystals. The vertical size of the image is about 140 jtm. It is clearly demonstrated that the sagittalfocusing based on a groove or a hole produced into an asymmetrically cut crystal works as predicted.

Fig. 16. The beam leaving the focusing monochromatorwith holes drilled into asymmetrically cut Si (111) crystals

Fig. 17. The image of the beam 215 cm fromthe focusing monochromator

2.4 Tuning the wavelength and the focusing distanceThe above formulae give the relationship between the focusing distance f, focusing monochromator-to-source distance S,parameter a of parabola, wavelength A, and the angle of asymmetry a (a and A are hidden in K' ). Usual requirement is tokeep the focusing distance constant when changing A. It implies that the change of A requires the change of a or a. The first

Fig. 18. Longitudinal groove Fig. 19. Longitudinal groove withwith variable a. variable a

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approach was suggested by Hrdy and Siddons3 and is shown in Fig. 18. The parameter of the groove changes longitudinallyand the proper a is adjusted by the longitudinal translation of the crystal. The second approach was suggested

EE

C)C

U)

0)C')

00

Fig. 20. Four different grooves may ensure the focusing distance constantfor 0 from about 70 to about 26°.

by Artemiev5. When cutting the groove by a diamond wheel, then the end of groove represent the groove withlongitudinally variable a (Fig. 19). Then again like in the previous case a may be adjusted by the longitudinal translation ofthe crystal.

There exists, however, the third approach how to keep the focusing distance f approximately constant for various X. Asseen in Fig. 1 1 the dependence of f on 0 for the asymmetric crystal is in certain interval of X practically constant. Thissuggests the possibility to design several grooves or holes (like shown in Figs. 13 and 14) such, that by replacing one grooveby another it is possible to keep the focusing distance almost constant in broad wavelength interval, as is seen in Fig. 20.(From the geometrical reason the simple hole does not allow to utilise a broad interval of 0 without an additional surgery inthe crystals).

3. MERIDIONAL FOCUSING USING BRAGG DIFFRACTION ON TRANSVERSAL GROOVE

As was suggested by Hrdy and Hrda12 the SR x-ray beam diffracted on a crystal with the properly designed transversalgroove on its surface should be meridionally focused. The shape of the groove, however, is more complicated than in thecase of the meridional focusing and is determined as a numerical solution of a differential equation.Fig. 21 shows the shape of the transversal groove designed for an experiment on BM5 beamline in ESRF (5=40000 mm,f=2000 mm, X=0.15 nm, Si(1 1 1)). The main problem here is that the focusing is in the plane of diffraction and in SRbeamlines always two crystals in parallel position are needed. This means that if the groove is produced into one crystal andthe other crystal remains flat, then the diffraction condition is not completely fulfilled on both crystals for all rays at thesame time and a part of the radiation is not diffracted. If the groove is produced on both crystals, the situation may be evenworse. From the same reason the use of more crystals arrangement to compensate for the smearing is problematic. This doesnot happen in the case of sagittal focusing described above. Nevertheless, the first experiment performed in ESRF13 clearlyshowed that the focusing effect exists. Fig. 22 shows the snap-shot of the beam meridionally focussed at the focal distanceof 2 m on the transversal groove

Bragg angle [deg]

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0,2E 01

I • . I> -1,0 -0,5 0,0 0,5 1,0

Fig. 21. The shape of the transversal groove used for ESRF experiment. Theradiation is coming from the left side.

described above. The Si(1 11) double crystal monochromator is located upstream the grooved crystal. The intensity at thepeak is in this case about 3 times higher than the intensity of the radiation diffracted from the flat part of the crystal. Thedetails of the experiment will be discussed elsewhere13. As suggested by Hrdy and Hrda12 the tunability may be in principleassured by changing the parameters of the groove along the groove and by the transversal translation of the crystal.

Fig. 22. The snap shot of the 0.15 nm radiation diffracted on the crystalwith a transversal groove shown in Fig. 21. The radiation is coming fromabove. The sharp horizontal line is the focused radiation at 2m from thecrystal. Above and below this line are seen images of the radiation diffractedfrom the flat part of the crystal

4. POSSIBILITY OF TWO DIMENSIONAL FOCUSING

So far the focusing only in one direction was described. As suggested in Hrdy and Hrda12 it should be in principle possiblein (-,+,+,-) arrangement to produce longitudinal grooves into the second and third crystals and a transversal groove into theforth crystal. This should concentrate the beam in both directions. Another possibility may be to produce a properlydesigned depression into the surface of one crystal. Neither of the methods was tested experimentally.

x[mm]

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5. CONCLUSION

The theoretical and experimental work performed so far shows that the utilisation of the diffractive—refractivephenomena for the focusing of synchrotron radiation is feasible. The focusing monochromators based on these phenomenaare simple, compact and cheap, particularly in the case of sagittal focusing. In the case of meridional focusing the furtherresearch is still necessary. The angular acceptance grows with wavelength and with focusing distance.

The next step should be to determine what size of focus could be reached. The sagittal focusing with parabolic grooves in(-,+,+,-) arrangement seems to be practically without aberrations; nevertheless in our experiments we have not reached thetheoretical limit. This may be connected with the fact that it is very difficult to produce the precise shape of grooves withsingle crystal surface. The further experimental work in this direction is obviously needed. The influence of various possiblegeometrical factors are now being studied by the special ray tracing program developed for this purpose.

As the diffracted beam does not have to pass through any absorbing material (as compared with CRL), the above methodseems to be perspective for longer wavelengths.

ACKNOWLEDGEMENTS

All crystals for the above described experiments were grown and shaped in the Polovodice a.s. company in Praha. Thisresearch was partially supported by the Ministry of Industry and Trade of the Czech Republic (grant PZ-CH122) and theGrant Agency of the Academy of Sciences of the Czech Republic (grant A1010104/O1). The experiments performed inESRF were supported by ESRF and the experiment performed in APS was supported by the U.S. Department of Energy,Basic Energy Sciences, Office of Science, under Contract No. W-3 1-109-Eng-38 and DND-CAT.

REFERENCES

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W-shaped longitudinal groove", J. Synchrotron Rad. 7, 382-385, 2000.3. J. Hrdy, P. D. Siddons, "X-ray focusing using an inclined Bragg-reflection lens," J. Synchrotron Rad. 6, 973-978, 1999.4. N. Artemiev, J. Hrdy, S. Peredkov, A. Artemiev, A. Freund, and R. Tucoulu, "Sagital focusing of synchrotron radiation

diffracted on the walls of the longitudinal hole drilled into a single crystal monochromator," J. Synchrotron Rad.,submitted.

5. N. Artemiev, Thesis, Charles University Praha, to be published.6. J. Hrdy, J. Quintana, N. Artemiev, and F. Franc, to be published.7. D. Korytar, P. Bohacek, and C. Ferrari, "X-ray flat diffractor optics I", Czech J. Phys. 50, 841-849, 2000.8. D. Korytar, P. Bohacek, and C. Ferrari, "X-ray flat diffractor optics II",Czech J. Phys. 51, 35-47, 2001.9. D. Korytar, J. llrdy, N. Artemiev, C. Ferrari, and A. Freund, "Sagittal x-ray beam deviation at asymmetric inclined

diffractors", J. Synchrotron Rad., in print.10. J. Hrdy, J. Synchrotron Rad., submitted.11. N. Artemiev, J. Hoszowska, J. Hrdy, and A. Freund, to be published.12. J. Hrdy, J. Hrda, "Meridional focusing of x-rays diffracted onto a single crystal with a transversal groove (Bragg

diffraction asymmetric lens)," J. Synchrotron Rad. 7, 78-80, 2000.13. J. Hrdy, E. Ziegler, N. Artemiev, F. Franc, J. Hrda, T. Bigault, and A. Freund, to be published.

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