REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 70, NUMBER 2 FEBRUARY 1999

Monochromatic focusing of subpicosecond x-ray pulses in the keV rangeT. Missalla, I. Uschmann,a) and E. ForsterX-ray optics group, Institute of Optics and Quantum Electronics, Friedrich-Schiller-Universita¨t Jena,Max-Wien-Platz 1, 07743 Jena, Germany

G. Jenke and D. von der LindeInstitut fur Laser- and Plasmaphysik der Universita¨t Essen, 45117 Essen 1, Germany

~Received 1 September 1998; accepted for publication 18 November 1998!

An effective x-ray optical method to focus keV x-ray pulses shorter than one picosecond by usingspherically or toroidally bent crystals is presented. The spectral, spatial, and time-dependentproperties of focusing by two-dimensional bent crystals are calculated by considering geometricaleffects, physical limitation in high performance crystal optics, and reflectivities obtained by x-raydiffraction theory. These properties are compared with first experimental results of focusing x raysfrom a plasma created by a laser pulse with 4.5 mJ energy and 100 fs pulse length. The x-raysignals, simultaneously obtained from a von Ha´mos spectrometer and two-dimensional bent crystalsare compared and found in good agreement with theoretical data. The possibilities and aspects oflaser pump x-ray probe experiments using this type of x-ray optics system and currently availablelaser systems are discussed. ©1999 American Institute of Physics.@S0034-6748~99!05002-9#

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I. INTRODUCTION

Recent developments in short pulse laser physics sucthe advent of the chirped pulse amplification~CPA! haveopened new possibilities for using medium size or tablelaser systems to achieve ultrashort laser pulses with hpeak power.1 These pulses deliver focused intensities whare many orders higher than the threshold value for geneing plasmas.2 The interaction of ultrahigh intensity lasepulses with solids or gases can generate intense emissioVUV and x-ray radiation. There are different mechanismswhich x rays can be produced when high intensity femtosond laser light pulses are focused on solid targets.

At first, electrons are accelerated by the radiation fieldthe laser pulse up to energies of order of one keV to sevMeV.3 Some of the electrons are accelerated into the tarproducing Bremsstrahlung radiation and, e.g.,K-shell ion-ized atoms in the solid target.4 The resultingK-shell lineemission from the cold plasma as well as the Bremsstrahcan be used as an x-ray source. Furthermore, a veryplasma layer at the front of the solid target is heated uptemperature of several hundred eV and emits strong xline emission characteristic of highly ionized plasma. Fimeasurements of the temporal x-ray emission from a sucosecond laser produced plasma with a fast streak camdemonstrates that in selected cases the duration cashorter than the time resolution of the camera.5,6 Up to now,there is no standard method to determine the pulse lengtultrashort x-ray pulses because the temporal resolution lof high speed x-ray streak cameras is relatively long, slighless than 1 ps.7 In contrast, recent publications8 present simu-lation of x-ray emission spectra which show that in particuthe duration for selected satellites of resonance lines f

a!Electronic mail: ingo@roentgen.physik.uni-jena.de

1280034-6748/99/70(2)/1288/12/$15.00

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highly ionized ions should be much shorter than a picosond. The first experiment measuring an ultrafast changethe Bragg reflection from a Langmuir–Blodgett film in a fehundreds of femtoseconds proves that the duration ofSiKa radiation used in this experiment is lower than 600 f9

Intense ultrashort x-ray pulses can be used in differkinds of experiments such as x-ray absorption spectroscx-ray diffraction, or x-ray photoelectron spectroscopy.10 Insuch experiments, the time evolution of many physical pnomena induced by fast electron processes can be invgated, for instance, photodissociation, chemical reactionbreaking chemical bonds, changes of crystallographic sttures, atomic motion, or surface bondings. Several otherperiments may be envisaged, where such fast processeused to develop fast x-ray shutters based, for instancex-ray diffraction or x-ray absorption. On the other hanlarge aperture bent crystals, developed in recent years hbeen applied to diagnostics of hot temperature plasmasexample, Johann spectrometers for high resolution, highpersion spectra, and doubly bent crystals for monochromimaging of laser produced plasmas, e.g., fusion plasmasplasmas for x-ray lasers.11–14 Although such systems arhighly monochromatic due to their narrow reflection curvethey provide much higher luminosity than other hard x-roptics like total external reflection at grazing incidencezone plates because of their much larger solid aperture an

The aim of the present paper is to show that such crydevices can be applied, first to analyze the keV x-ray emsion from fs laser produced plasmas and to use this radiafor application in time-resolved x-ray experiments. Combing intense ultrashort x-ray pulses with Bragg reflecting otics can open a new wide field in pump-probe experimeapplied to spectroscopy or diffractometry.

8 © 1999 American Institute of Physics

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1289Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

II. X-RAY FOCUSING WITH BENT CRYSTALS

The fundamental focusing techniques with bent crysfor luminous high resolution spectroscopy were describedthe 1930’s15–18 and applied in many investigations, i.e.,solid state, atomic, and plasma physics. There was signifiprogress in the 1970’s and 1980’s in experimental investtions of highly perfect grown crystals by high resolutiox-ray topography and x-ray diffractometry combined wtheoretical investigations based on the dynamical theoryx-ray diffraction.19 The combination of such new qualitcrystals with high performance optics and replica techniqled to new standards in x-ray spectroscopy and x-imaging.20 Toroidally and spherically bent crystals were usto focus keV x rays from laser plasmas produced with hpower nanosecond and picosecond laser pulses. In thismonochromatic images were obtained from which sparesolved plasma parameters, i.e., electron temperaturesplasma densities could be derived.12 Narrow rocking curvesof the two-dimensional bent crystals, with widths of somarcseconds, may be measured by combining the bent crywith flat crystals in double-crystal x-ray diffractometry.21

This article analyzes the keV x-ray line emission fromplasma heated by a fs-laser pulse. The intensity of the xradiation focused by a bent crystal depends on the subtesolid angle, the reflection curve of the bent crystal, the sptral distribution of the x-ray emission, and the geometry, iposition of the source and magnification factor.

A. Geometry

Ideal point-to-point focusing of a source can be achievby a crystal with ellipsoidal lattice planes if both the sourand detector are placed at the foci.22,23 The incident angle ofthe x-ray radiation to the lattice planes varies over the crysurface which means that for crystal apertures typically uthe imaging is slightly polychromatic (Dl/l51022– 1023). If the imaging needs to be truly monochromatic the crystal should be ellipsoidally bent, but with tsurface in a toroidal shape, i.e., it cuts into the lattice planHere the horizontal radius on which both foci has toplaced is the radius of the Rowland circle. Because scrystals are not available in practice this article describesfocusing properties of crystals where the lattice planesparallel to the surface and bent either spherically or todally. In general, a toroidally bent crystal has horizontal a

FIG. 1. Focusing of x rays with a toroidally bent crystal, fulfilling thcondition Rv /Rh5sin2 Q0. The source and the focus are on the Rowlacircle.

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vertical radii of curvatureRh and Rv ~Figs. 1 and 2!. Thefocal lengths of the crystal in the horizontal and verticalrection f h and f v depend onRh , Rv and the Bragg angleQ0

with

f h5Rh

2sinQ0 , f v5

Rv

2 sinQ0.

To obtain a point-to-point focusingf h and f v have to beequal. This leads to

Rn

Rh5sin2 Q0 . ~1!

If the condition of Eq.~1! is not fulfilled ~for example,for a spherically bent crystal andQ,p/2, Fig. 3! the focalplanes are separated. This has to be taken into accounsmall x-ray point focus is desired. The reflected energy ocrystal is given by the following integral:24

Eref5Eamin

amaxEFmin

FmaxElmin

lmaxda dF dl G~a,F!J~l!

3CS s~a,F!2Dl

l0tanQ0D . ~2!

G(a,F) is the angular distribution of the energy emittedthe point source. The anglesa andF are the horizontal~dis-persion plane! and the vertical divergence angles.J(l) is the

FIG. 2. Focusing of x rays with a toroidally bent crystal. The position of tsource outside the Rowland circle leads to a focus position inside the Rland circle and to a magnification factork,1.

FIG. 3. Focusing of x rays with a spherically bent crystal. Due to differfocal lengthsf h and f v the horizontal and vertical foci are separated. BcauseRv5Rh5R and Rv.Rh sin2 Q0 the position of the vertical focus isbehind the position of the horizontal focus.

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1290 Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

energy distribution of the spectrum, C@s(a,F)2Dl/l0 tanQ0# is the reflection curve of the crystal ans(a,F)5Q2Q0 is the deviation of the Bragg angleQ fromthe central Bragg angleQ0 corresponding to different reflection points on the crystal. By assuming isotropic radiatiG(a,F) can be set to unity. The source is considered apoint source. The spectral range of the focusing is smenough to neglect the wavelength dependence of the retion curveC. A wavelength differencel2l0 simply shiftsthe Bragg peak though an angle (l2l0)tanQ0 /l0. In mostcases of Bragg reflection, the widthDQRC ~FWHM! of thereflection curveC of the crystal is small compared to thwidth DQL5DlL /l tanQ0 of a spectral line. Typical valueof DQRC andDQL are

DQRC592.3 arcsec54.4731024 rad,

DQL51220 arcsec55.9131023 rad

~corresponds toDlL510 mÅ!.

These values are for a quartz crystal in~10.0! reflection(2d10.058.5096 Å) and AlKa radiation (l58.339 Å, Q0

578.5°).The Bragg angleQ depends on the positions of th

source and the reflection point on the crystal. The devias from Q0 can be approximated21,25 for a spherically bentcrystal to

s'k21

2ka1

1

2 tanQ0a22

k21

4k

3tanQ0S 12k11

2ksin2 Q0DF2 ~3!

and for a toroidally bent crystal,

s'k21

2ka1

1

2 tanQ0a22

k21

4k

3tanQ0S 12k11

2k sin2 Q0DF2, ~4!

with

k5l b

l a5

1

2l a

Rh sinQ021

5Rh sinQ0

2l a2Rh sinQ0,

where l a and l b are the source-to-crystal and the crystal-focus distances.

Figures 4 and 5 show the comparison between the evalues ofs and those given by the approximations~3! and~4!. It should be mentioned that the crystal size in the vertidirection is ten times larger than in the horizontal one. Fthe crystals of diameters up to 20 mm used up to nowapproximation can be applied. For larger vertical sizes, thapproximations do not truly represent the Bragg angle. Nertheless, for Rowland circle geometry, the Bragg anvariation over most of the surface does not exceed the wof the reflection curve, about 30 arcsec in this case. Bcontributions are still smaller than the spectral linewidth. F

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magnification factorsk,0.7 andk.1.5, the Bragg anglevariation increases by a factor of 20 up to 40 compared wthe value at the Rowland circle position.

B. Source on the Rowland circle, k 51

The Bragg angle variations~a,F! is the smallest for thecase ofk51. This means the source is placed on the Ro

FIG. 4. Variation of the Bragg angles5Q2Q0 over the crystal surface fora quartz crystal in~10.0! reflection (2d10.058.5096 Å) assuming a poinsource on the Rowland circle~magnification factork51), Al Hea line (l57.7566 Å), ~a! toroidally bent crystal,~b! spherically bent crystal.

FIG. 5. Variation of the Bragg angles5Q2Q0 over the crystal surface ofa spherically bent quartz crystal in~10.0! reflection (2d10.058.5096 Å) as-suming a point source on the Rowland circle~magnification factork51), AlHea line (l57.7566 Å), ~a! k50.5, ~b! k50.711 ~minimized F depen-dence ofs!, ~c! k51.5.

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1291Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

land circle (l a5 l b5Rh sinQ0). In Eqs.~3! and ~4! only thea2 term remains. The variation ofQ in the vertical directionis negligible.

From Eq.~2! an average valueJaver.can be found for thelimits of lmin andlmax, for which the equation

Eref5Javer.Eamin

amaxEFmin

FmaxElmin

lmaxda dF dl

3CFs~a,F!2Dl

l0tanQ0G

is valid. Changing the integration variable froml to Q leadsto

Eref5Javer.Eamin

amaxdaE

Fmin

FmaxdF

dl

dQ EQmin

QmaxdQ

3C@s~a,F!2~Q2Q0!#,

Eref5Javer.Eamin

amaxdaE

Fmin

FmaxdF

l

tanQ0Rint ,

Eref'Javer.

Aeff sinQ0

l a2

l

tanQ0Rint ,

whereAeff is the effective reflecting crystal area and

Rint5EQmin

QmaxdQ C~Q2Q0!

is the integrated reflectivity. In the range of interest, it isthe order of the width of the reflection curve of the crysand much smaller than the angular widthDQL of the spectralline (s!DlL tanQ/l).

In this caseJaver. can be approximated by

Javer.'EL

DlL'Jmax,

whereJmax is the maximum of the spectral line profile anEL is the total energy of this emission line emitted in 1This leads to

Eref'EL

DlL

Aeff sinQ0

l a2

lRint

tanQ05

EL

DlL

Aeff

Rh2 sinQ0

lRint

tanQ0~5!

and to the number of reflected photonsNref :

Nref'NL

DlL

Aeff sinQ0

l a2

lRint

tanQ05

NL

DlL

Aeff

Rh2 sinQ0

lRint

tanQ0, ~6!

whereNL is the total number of photons of the spectral liemitted in 1 sr (EL5hcNL /l). The x-ray photons of anemission line are collected in an solid angleV5Aeff sinQ0 /la

2 with l a5Rh sinQ0. The spectral windowgiven by the widthDQRC of the reflection curveC is one totwo orders smaller than the widthDQL of the emission line.Then the number of reflected photons is reduced by a faDlRC/DlL with DlRC5lRint /tanQ05lcmaxDQRC/tanQ0.

C. Focusing in a spectral window Dl, kÞ1

If the source position is not on the Rowland circleEqs. ~3! and ~4!, the term linear ina becomes dominant

l

.

or

There is aF2 dependence ofs, which is small compared tothe a dependence ofs, but is limiting the crystal heightwhich would reflect radiation in a limited spectral windowdetermined by thea dependence according to the cryswidth Dscrystal and the magnification factork. TheF depen-dence ofs can still be minimized if theF2 term in Eqs.~3!and ~4! is zero. This is the case if

k5sin2 Q0

22sin2 Q0

in the case of a spherically bent crystals@Fig. 5~b!# and

k51

2 sin2 Q021

in the case of toroidally bent crystals. In the limits of thcrystal size mentioned above and at larger distances fromRowland circleu(k21)/ku.0.2, the quadratic term ina issmall compared to the linear one. Hence for a fixedk,

da

dQ'

da

ds'

2k

k215const. ~7!

The substitution of thea integration in Eq.~2! leads to

Eref'Elmin

lmaxdl J~l!E

Fmin

FmaxdF

da

dQ EQmin

QmaxdQ C~Q!,

Eref'2k

k21RintE

lmin

lmax

dl J~l!EFmin

FmaxdF

52k

k21RintE

lmin

lmax

dl J~l!Dhcrystal

l a, ~8!

whereDhcrystal is the crystal height in the vertical directioperpendicular to the dispersion plane.

With l a5(k11)/(2k)Rh sinQ0, one gets

Eref5EL

~2k!2

k221

DhcrystalRint

sinQ0Rh~9!

and

Nref5NL

~2k!2

k221

DhcrystalRint

sinQ0Rh. ~10!

Equations~9! and~10! are only valid when the spectral rangof the focusing setup

Dl'lUk21

k11U Dscrystal

Rh tanQ0~11!

is larger than the widthDlL of the emitted spectral banwidth of the source. The valueDscrystal is the width of thecrystal in the dispersion plane.

Figure 6 shows the dependence of the reflected x-energy in relation to the x-ray energy emitted from tsourceEref /EL for different magnification factorsk.

If the spectral window of the crystalDl is larger than thewidth DlL of the spectral line only a part of the horizontcrystal size~in the dispersion direction! contributes to thereflection of x rays from the spectral line. This effect growwith increasinguk21u, i.e., the source moves away from thRowland circle. Further, it can be seen that if the spec

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1292 Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

band width is rather large this function is mainly monotonIf the source is inside the Rowland circle, i.e.,k.1, the solidangle does not change strongly. On the other hand, formagnification, when the source is outside the Rowlacircle, the effective solid angle of the crystal continuousdecreases.

D. Bragg reflection, reflectivity of the crystal

The use of structurally perfect crystals provides the psibility of new x-ray optics devices, i.e., as monochromacrystals for synchrotron radiation or focusing crystal sptrometers @spectral resolution (Dl/l)cr51026– 1024#. Inparticular, silicon, germanium, or quartz crystals are suitacandidates because they are available as large perfect cryand they can be bent without plastical deformation unlother crystals like ammonium dihydrogen phosphate, ptaerythritol or others. With new techniques for elastic bening the lattice planes can be bent to an accuracy sufficienfocus x rays with precision comparable to visible reflectioptics. The maximum wavelength reflected by the crystalimited by the double lattice spacing 2dhkl which depends onthe crystallographic structure, the crystallographic oriention hkl, and the lattice constants. The Bragg law shows tin principle there is no lower limitation, since

l52dhkl sinQ, where Q is the Bragg angle.

The Bragg reflection differs fundamentally from visiblight reflection in that it takes place in a crystal volume, n

FIG. 6. Ratio of the reflected x ray radiation energy related to the eneemitted by the sourceEref /EL as a function of the magnification factork forfocusing of the Al Hea line (Q0565.71°, Rint50.8331024 rad) using atoroidally bent quartz crystal in~10.0! reflection. The values are calculatefor crystal sizesDscrystal5Dhcrystal50.1 Rh sinQ0 and assuming Gaussialineprofiles with different linewidthDlL . The rectangles indicate the valueof Eref /EL for the source position on the Rowland circle.

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just on the surface. The size of his volume is determinedthe dynamical x-ray diffraction theory for perfect flat anbent crystals and by the kinematical theory of x-ray diffration for mosaic crystals, i.e., they are strongly distortedcrystal imperfection.

The crystal reflectivity depends on a fundamental paraeter, the Fourier component of the dielectric susceptibixhkl :

xhkl5l2

p

r e

VFhkl .

In this expression,r e is the classical electron radius,Fhkl isthe structure amplitude of the crystal for thehkl reflection,andV is the volume of the unit cell. In the dynamical theorthe reflectivity of a plane perfect crystal is given in appromation of no absorption by

Rintper5

8

3

1

sin~2Q!

11cos~2Q!

2Axhkl•xhkl.

The dynamical value of reflectivity is the smallest for thhighest degree of crystal perfection. For crystals whichdistorted by lattice imperfection or elastically deformedcrystal bending26 the reflectivity generally increases. Thmaximum value of reflectivity occurs in the mosaic limit anis

Rintmos5

p2

2lm

1

sin~2Q!

11cos2~2Q!

2~xhkl•xhkl!,

wherem is the absorption coefficient for x rays of the wavlength l. In general, the two limits differ greatly for weaphotoelectric absorption and low structure amplitudes in ctrast converge for strong absorption. For bent crystals,reflectivity lies between these two limits depending on being radius, susceptibility, wavelength, and Bragg angle.27

E. Crystal selection

As mentioned above, quartz, silicon, and germaniumavailable as highly perfect crystals suitable for x-ray optiThe possible crystal orientation with corresponding doulattice plane spacing for the longest usable wavelengthlisted in Table I. The reflectivity for the perfect crystal anthe mosaic crystal are compared~for an assumed Braggangle of 70°!. The width of the reflection curve for a perfeccrystal sets the upper limit on the spectral resolution. Itclear that as the integrated reflectivity increases, the spe

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ivity,ity

TABLE I. Crystals for x-ray optics with double netplane spacing, dynamical, and kinematical reflectspectral resolution (l/Dl)cr due to the width of the reflection curve and maximal normalized reflectiv~calculated for a Bragg angle of 70°!.

Crystal Orientation2dhkl

~Å!Rint

dyn

~mrad!Rint

mos

~mrad!

Spectralresolution(l/Dl)cr. cmax5Imax/I0

Quartz 1010 8.5098 113 155 11800 0.37

Quartz 1011 6.6865 190 260 8720 0.47

Germanium 111 6.5327 438 558 3350 0.41Silicon 111 6.2710 159 200 7170 0.40

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1293Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

resolution decrease provided that there is no significchange in the maximum reflection, as can be seen in Tab

To achieve maximum intensity in point-to-point focuing @see Eq.~6!#, an important parameter is the aperturethe bent crystal. If a crystal disk is elastically bent to a twdimensional surface, the size is limited due to the elastress by the critical shear stress levelsc . In the case, wherea circular disk of radiusr is pressed onto a sphere of radiusRthe relative length change of the crystal edge shouldsmaller thansc /E1 (E1 is the Young’s modulus in isotropicapproximation!. This leads to the maximum crystal diamet2r :

2r 52A6sc

ELR .

For a spherically bent silicon crystal of bending radius150 mm, the diameter cannot exceed 18 mm. In casetoroidally bent crystal, initially in a rectangular shape, a crtal infinitesimally narrow in one direction, say the horizontcan be bent through a whole circle of 2p radians in thevertical direction. The dimension in the horizontal directiis limited by elastic stresses, since the length along theof the crystal and its central axis differ. This should be asmaller thansc /E1 . By assuming that the long crystal edgis bent to a radiusRv , which corresponds to a barrel likform, the crystal sizeDscrystal in the horizontal direction islimited by

Dscrystal5A8sc

ElRh sinQ. ~12!

For a Bragg angle of 65.7° and a horizontal bending radRh5150 mm, this gives a horizontal size ofDscrystal

59.5 mm for a silicon crystal. From Eqs.~6! and ~12! themaximum ratio of reflected to the total number of emittphotons is

S Nref

NLD

max

5S Eref

ELD

max

54pdhkl

DlLA8sc

Elsin2 Q cosQRint .

~13!

Silicon crystals can cover 0.01% of the whole solid an~4p steradians! for the case of a spherically bent crystal wicircular shape and 3.4% sinQ in case of the toroidally benone with a whole barrel shape.

The Bragg angle is determined by the choice of the crtal and spectral range~spectral line or continuum!.

Figure 7 shows the change with the Bragg angle ofspectral resolutionDl/l and the term (Rint cosQ sin2 Q),which is the angle dependent term for the reflected inten@see Eq.~13!#. A very important conclusion is that both threflected intensity and the spectral resolution increase wBragg angle.

For experiments where it is important to optimize tnumber of detected photons, for instance, an x-ray induphotoelectron spectroscopy with only a moderately narrspectral window, a useful crystal is one whose reflectcurve is as wide as possible, in order to reflect the whspectral line, e.g.,Ka line.

ntI.

f-ic

e

fa

-,

deo

s

e

-

e

ty

th

dwnle

For emission or absorption spectroscopy experimethe resolution must be equal or finer than the requiremwhich limits the maximum integrated reflectivity. In thicase, only the selection for maximum peak reflectivitycmax

is possible~see Table I!.

F. Temporal behavior of the focusing with bentcrystals

He and Wark28 and Chukhovskii and Fo¨rster29 studiedthe temporal behavior of the Bragg reflection of ultrashx-ray pulses for the limiting cases of perfect and moscrystals. This behavior becomes important if the time deDt1 due to different path lengths of the x rays reflectedthe crystal volume is comparable to the duration of theflected x-ray pulse.

For perfect crystals, where the absorption is weak copared to the extinction (te!ta ,te : extinction depth,ta : ab-sorption depth!, the time delay can be approximated by

Dt1'2te

c sinQ, ~14!

wherec is the speed of light. For weakly absorbing perfecrystals, the extinction depthte is given by30

te5l sinQ

2puxhruP,

which dominates the penetration depth inside the widththe total Bragg reflection of the crystal.P is a polarizationdependent factor with

P5H 1 for s polarization

cos~2Q! for p polarization.

Here uxhru is the absolute value of the complexhth Fouriercomponent of the real part of dielectric susceptibility. Thleads to

Dt2'l

2pc

2

uxhruP. ~15!

FIG. 7. Bragg angular, i.e., photon energy dependent contribution ofreflected x-ray energy by a silicon crystal in~111! reflectionRint sin2 Q cosQ and the corresponding spectral resolution. The integrareflectivitiesRint for a perfect Si crystal using the dynamical theory and fa ideal mosaic crystal using kinematical theory in~111! reflection are usedas lowest, i.e., highest reflectivity limit.

1294 Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

TABLE II. Comparison of temporal broadening of the x-ray pulse by x-ray diffractionDt1 and spectral bandwidth limitation Dt2 @FWHM of the Fourier transform of the crystal reflection curveC(n2n0); n,n0 : fre-quencies# for different crystal reflection and x-ray wavelength.

Crystal RadiationQ~°!

te

~mm!DQRC

~arcsec!Dt1

~fs!Dt2

~fs!

Quartz (1010) Al Ka 78.5 1.9 92 13 21.3

Quartz (1010) Al Hea 65.7 1.68 37 12 21.8

InSb~111! Si Ka 72.6 0.28 259 2.0 4.1Ge~111! continuum 42.25 0.32 59 3.2 3.4

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enin

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ally

which corresponds to the result of Chukhovskii and Fo¨rster29

for a point source and monochromatic radiation.The finite width DQRC of the reflection curves of the

crystals introduces a further limit on the duration of theflected x-ray pulse. This reason is that the finite band wicorresponds to a temporal broadening of the quasi-dpulses of the x rays, which can be determined by the Foutransformation of the reflection curve in the frequenspace.28 For a given reflection curve profile, the widt~FWHM! Dv in the frequency space and its Fourier tranform Dt2 in the time space are related by

DvDt25C1, ~16!

where the constantC1 depends on the reflection curve prfile in the frequency space. For a perfect symmetricallycrystal the dynamical theory of x-ray diffraction predictshalf width DQRC of the reflection curve in the weak absortion limit31

DQRC52uxhruPsin 2Q

.

This leads to

Dt25l

2pc

sin2 Q

uxhruPC1. ~17!

The temporal response timeDt2 of the Bragg reflection dueto the band width limitation on the focusing ultrashort x-rpulses is directly related to the extinction depthte .28 Typicalvalues forDQ, te , Dt1 , andDt2 @determined as the FWHMof the Fourier transform of the crystal reflection curveC(v2v0)# are shown in Table II.

In these examples, the values forDt1 and Dt2 aresmaller than 30 fs, and they do not introduce a significtemporal broadening of an x-ray pulse whose durationseveral hundreds of fs. The effect of the temporal broadendue to band limitation of the Bragg reflection and the refltion of the x-ray pulse in a certain volume in the crystal cincrease at smaller wavelength. Further, the elastic benof the crystal causes the penetration depth inside the cryto increase beyond the extinction depth. This effect widthe reflection curve and causes further temporal broaden

Another factor which determines the temporal resolutfor focusing ultrashort x-ray pulses is the time delayDt3 dueto different optical path length for the source-crystal-focdistances.Dt3 is zero only when a point source and an x-rfocus lie at the foci of a crystal bent to a rotational ellipsoFor spherically and toroidally bent crystals, the variation

-hltaer

-

t

tisg-

ngtalsg.

n

s

.f

the path length and thusDt3 depend on the Bragg angleQthe effective crystal aperture and the magnification factok.

For a toroidally bent crystal with an horizontal radiuscurvatureRh5164.0 mm, a Bragg angleQ565.7° ~Al Hea!and the position of the point source on the Rowland cir(k51) Dt3 has been determined by ray-tracing calculatiofor different diametersd of the crystal size:

d515 mm, Dt354 fs; d530 mm, Dt3520 fs;

d540 mm, Dt3560 fs; d550 mm, Dt35120 fs.

Furthermore, for a crystal diameterd515 mm, the temporalbroadening has been calculated for different magnificatfactorsk ~Fig. 8!. For k in the range 0.5,k,2, crystal aper-ture diameterd smaller than 15 mm and a point source, tvariation in the optical path lines is in a region of 50 fsmaller than the expected duration of an x-ray pulse.

III. EXPERIMENTS

The x-ray radiation of Al and Si targets was recordeddifferent focusing crystal optics to determine total numberthe emitted photons, to compare the photon density and cbrate the different optics, and to draw conclusions for psible application experiments. The experiments were pformed using an amplified CPM dye laser system operaat 620 nm. The maximum pulse energy was 4.5 mJ, andbackground energy due to amplified spontaneous emiswas less than 5% of the total energy. Second harmonictocorrelation measurements were performed covering anal range greater than eight orders of magnitude. Anample is shown in Fig. 9. These data show that the pu

FIG. 8. Temporal broadening of a x-ray delta-pulse focused by a toroidbent crystal for different magnification factorsk.

ok-

eau

as

udth

esalrgin

tfso

n-tedkelec-rted

2.

olidmea-

nd

ctra

as

e

data-ctrated

lifie

th

1295Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

duration is less than 100 fs and that the laser intensity drbelow 1027 of the maximum in less than 1 ps. The bacground signal in Fig. 9 at approximately 1028 is caused byscattering from the surfaces of the KDP crystal used for sond harmonic generation. The two satellite peaks in thetocorrelation are caused by reflections from the crystal sface and do not represent an actual structure of the lpulse.

The laser pulses were focused onto massive aluminand silicon targets at an angle of incidence of 45°. Theameters of the oval-shaped projected focal spots onsample surface were approximately 8 and 12mm for an ach-romatic lens withf 520 cm, and 3mm and 4.5mm for anachromat withf 56 cm. Examples of the measured profilof the intensity distribution in the focal plane for normincidence are depicted in Fig. 10. From the pulse enepulse duration and focal distribution the maximum peaktensity on target is estimated to be greater than 1017W/cm2.

To record Al and SiK-shell spectra a cylindrically benPentaerythriol~PET! crystal with a radius of curvature o100 mm was used in the von Ha´mos geometry. As detectorKodak DEF and Kodak SB film was used. The density

FIG. 9. Second harmonic autocorrelation measurement of the ampCPM dye laser pulse.

FIG. 10. Examples of measured profiles of the intensity distribution oflaser pulse focused by an achromatic lense (f 520 cm), normal incidence ofthe laser radiation.

ps

c-u-r-er

mi-e

y,-

n

films was digitized in two-dimensional scans on a microdesitometer. The values of the optical density were converto photon density on film by using the calibration of Henet al.32 By knowing the integrated reflectivity of the crystaand the transmission of the x-ray window used in the sptrometer, the photon density on the film has been conveto the spectral energy distributionJl emitted by the source in1 sr ~J/sr/Å!.

Typical Si and Al spectra are shown in Figs. 11 and 1The strongest lines of highly charged ions are the Si Hea andthe Al Hea lines. Intense SiKa and the AlKa radiationproduced mainly by hot electrons in colder plasma and starget regions have also been detected. There was nosurable Lya radiation. The Al Hea and the AlKa lines havebeen chosen to test the x-ray focusing with toroidally aspherically bent crystals. For the Al Hea line, a toroidallybent quartz crystal in 10.0 reflection was used. The spewere recorded simultaneously using a von Ha´mos spectrom-eter. A spherically bent quartz crystal in 10.0 reflection wused to focus the AlKa radiation. Taking into account thetransmission of the light protection foil used in front of thx-ray film the number of photonsNref reflected by the doublybent quartz crystals was determined. To compare thesewith those from the von Ha´mos spectra the number of photons emitted by the plasma was calculated from the spethrough the reflectivity of the PET crystal. These calcula

d

e

FIG. 11. Si spectra recorded with a von Ha´mos spectrometer. The SiKaand the Si Hea lines are the most intense lines.~a! ELaser52 mJ/pulse, laserfocus diameterdfoc'10mm on the target surface,~b!,~c! ELaser52 mJ/pulse,laser focus diameterdfoc'3 – 4mm on the target surface.

umDth

o

r

A

ahot

-tw

eo

the-oriftsbyeri-be-ri-

clee

gy.

y athe

1296 Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

photon emissivities of the source give estimates of the nber of photonsNref,theo which should be reflected by the 2bent crystals with respect to the focusing geometry andquartz crystal reflectivity. In this way comparingNref,theo,estimated from the von Ha´mos spectra withNref , actuallydetected with the 2D bent crystal, provide a comparisonthe calibration of both analyzers. The values forNref,theohavebeen related to the spectral line distributionJl determinedfrom the spectra recorded with the von Ha´mos spectromete~Fig. 13!.

A. Al He a focusing

X-ray monochromatic imaging was achieved for theHea line by using a toroidally bent~10.0! quartz crystal. Thedata of the setup are presented in Table III. The toroidoptimized for anastigmatic, i.e., two-dimensional focusingan angle of 65.7°, which is equal to the Bragg angle. Tlaser energy wasElaser54.5 mJ per pulse focused to a spdiameter of 10mm ~related to normal incidence! at 45° to thetarget surface delivering an intensity of 4.131016W/cm2.The laser light wasp polarized to obtain maximum absorption of the laser radiation at the target surface. Results ofexamples for Al Hea focusing are shown in Figs. 13~a! and13~b!. Table IV compares values ofNref,theoandNref obtainedin the experiments.

The spectral window of the toroidally bent crystal for thmagnification factor used is determined by the variation

FIG. 12. Al spectra recorded with a von Ha´mos spectrometer. The AlKaand the Al Hea lines are the most intense lines. Laser enerELaser54.5 mJ/pulse, laser focus diameterdfoc510mm on the target surfaceThe Al Hea line was fitted be two~a! or one~b! Lorentz functions and theAl Ka line, by one Lorentz function. For the AlKa line, the parameter ofthe fits are:~a! NL52.503107 photons/pulse/sr,DlL56.3 mÅ, ~b! NL

50.923107 photons/pulse/sr,DlL56.6 m Å.

-

e

f

l

iste

o

f

the incident angle along the horizontal crystal position@seeEquation~11!# and amounts toDl519.0 mÅ. Because thisis larger than the measured linewidth the crystal reflectsx-ray line only by half the horizontal aperture. A misalignment of the Bragg angle of 0.1°, which is the main error fthe calibration measurement with a 2D bent crystal, shthat crystal area, which reflects the x-ray line horizontally2.5 mm. For a circular aperture as was used in the expment, the effective aperture reflecting the line changescause the effective vertical crystal height varies with hozontal crystal positions. Allowing for this the values forNref

are in good agreement with the theoretical valuesNref,theo. Ifthe position of source would be placed on the Rowland cir(k51, l a5 l b5Rh sinQ0) and a rectangular crystal aperturwould be used withAeff5(0.1Rh sin2 Q0)

2, the number of re-flected photons for a Al Hea spectral line with a spectralwidth DlL of 8–10 mÅ could be increased up toNref,max

>2000 photons/pulse whenNL5107 photons/pulse.

FIG. 13. Al Hea line recorded with a von Ha´mos spectrometer and densitydistribution of reflected photons per shot in the detector plane obtained btoroidally bent crystal. Subtracting the backround and integration overphoton density distribution leads to total numberNref of reflected photonsper pulse~see Table IV!.

TABLE III. Geometry of the Al Hea focusing.

Crystal: toroidally bent quartz crystalOrientation: symmetrical~10.0! reflection

(2d10.058.5097 Å,Rint50.8331024 rad)Diameter of the crystal: d510 mmWavelength: l057.7566 ÅCentral Bragg angle: Q0565.7°Radii of curvature: Rh5167.870 mm

Rv5139.244 mmFocal lengths: f h576.51 mm

f v576.38 mmSource-to-crystal distance: l a5140.3 mmSource-to-crystal distance: l b'168 mmMagnification factor: k51.2

l,

1297Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

TABLE IV. Observed emission data of the Al Hea line: Nref—photons reflected by the toroidally bent crystaNL(EL)—photon number~energy! emitted by the source obtained by von Hamos signal,DlL—measured linewidth, Nref,theo photons should be focused by the crystal usingNL .

EL

S 1029 J

sr pulseDNL

S107 photons

sr pulse D DlL

~mÅ!

Nref,theo

Sphotons

pulse DNref

Sphotons

pulse D~a! Fig. 11~a! 1.77 0.69 7.6 328 247~b! Fig. 11~b! 2.70 1.05 10.4 422 364

ee

thdetioTh

e

-ds

iththaw

nsseaoue--

cios

foi

s

bDt t

ry

i-re-

ct?aspar-

ally

tohot

ion1

ct?a-ed

10lss-

eri-f thetal,

ces

ansion

ac-

B. Al K a focusing

In a further experiment, the AlKa radiation was focusedby a spherically bent quartz crystal~10.0! ~see Table V!.Because the different focal length in the horizontal and vtical direction are different at the Bragg angle of 78.5° ushere, two focal lines are formed in the horizontal and invertical focus position, respectively, and a spatially extenfocal area between them. The measured intensity distribucorresponding to these results are shown in Fig. 14.width of the focal lines are 30mm for the horizontal focusposition and 60mm for the vertical one. Table VI shows thresults for the reflected photonsNref . The calibration wasperformed to obtain the photon numberNL emitted by thesource or the emitted energyEL both related from the detected photon numberNref . Because the calibration depenon the spectral widthDlL of the AlKa line @see Eqs.~5! and~6!# and the Bragg angle adjustment, a typical linewidth w7 mÅ and ideal adjustment, where the Bragg angle oncrystal center corresponds to the line maximum weresumed. These photon numbers are in good agreementdata from previously recorded von Ha´mos spectra~Fig. 12!.

We have presented a qualitative study of the inteemission of keV line radiation produced by a 100 fs lafocused onto silicon or aluminum targets. The use of ansolutely calibrated spectrometer allows to determine a cversion ratio of laser energy into x-ray line energy of abo1.231025. These data are in good agreement with publishdata of Chenet al.33 From theoretical calculation, it is possible to derive of aboutTh>5.8– 15.8 keV for the hot electrons at intensity and the laser wavelength used.34 By usingabsolutely measured photon numbers of theKa line andconversion ratio of electrons intoKa photons of$1.1– 6.3%31023, the conversion ratio of laser energy into hot eletrons is of the order of 5%. By assuming that the emisstime of the x-ray radiation has the same value as the lapulse duration and the emission area is the same as thespot, the estimated x-ray intensity in the target position3.331011W/cm2, which is of a similar order as previoudata of Audebertet al.35

IV. DISCUSSION

Comparison of the recorded photons in x-ray lines otained with the von Ha´mos spectrometer and with the 2crystals shows satisfying agreement. This is evidence thameasured reflectivity’s of both crystals used@PET ~002! andquartz~10.0!# are in good agreement with diffraction theo

r-dedne

es-ith

erb-n-td

-nercals

-

he

at these wavelength.36 Furthermore, it confirms the theoretcal description of the calibration of the 2D bent crystals psented in this article.

What is the maximum photon number we can expeThe maximum photon number reflected by the crystal wmeasured to be about 3000 photons per pulse focusedtially in a relatively large area of 200mm diameter. Thefocusing was incomplete because the crystal was sphericbent (Rh5Rv) and hence astigmatic@Eq. ~1! was not ful-filled#. By using a toroidally bent crystal, it was possibleobtain higher average photon density of one photon per sper micron square. In this article, it was shown that a fract(3.5– 20)31025 of the photons emitted from the plasma insteradian were focused by a 2D bent crystal.

What is the smallest x-ray spot diameter we can expeFrom ray-tracing calculation, it is well known that aberrtions for toroidally bent crystals of the solid angle usspread out to the focus of a point source by not more thanmm.13 Measured spatial resolution of toroidally bent crystaare in good agreement with the ray tracing simulation. Asuming both the x-ray emission region shown by the expmental results are larger than the expected. Reasons odiscrepancy are first alignment errors of the bent cryswhich can produce image distortions.13 The requirements forthe alignment of the Bragg angle azimuth and focal distanare described by Dirksmo¨ller et al.13 If it is assumed theemission region of the x-rays is not considerably larger ththe focused laser beam, the diameter of the x-ray emisspot could be in the order of about 10mm.

V. FUTURE PERSPECTIVES

From the scaling laws for the temperatureTh and theflux of hot electrons delivered from the laser plasma inter

TABLE V. Geometry of the AlKa focusing.

Crystal: spherically bent quartz crystalOrientation: symmetrical~10.0! reflection

(2d10.058.5097 Å,Rint52.2131024 rad)Diameter of the crystal: d510 mmWavelength: l058.339 ÅCentral Bragg angle: Q0578.5°Radius of curvature: R5151.7 mmFocal lengths: f h574.33 mm

f v577.4 mmSource-to-crystal distance: l a5148.7 mmSource-to-crystal distance: ~a! l b5148.7 mm@Fig. 12~a!#

~b! l b5155.0 mm@Fig. 12~b!#~c! l b5161.5 mm@Fig. 12~c!#

Magnification factor: k51

n

reosp

o

trle

f

5ra

cutareiveto

he

ag-larg isare

c,to-

sp

ing

aac-entts,flat

yys-ityach

ingntonorizedhinri-thela-

ray

s-re,eir

nd

Y.

cto

1298 Rev. Sci. Instrum., Vol. 70, No. 2, February 1999 Missalla et al.

tion, it is possible to optimize the laser intensity for efficiefast electron production, to produce intenseKa lineradiation.33 From theK-shell electron cross section which aelectron energy dependent it is well known that the crsections reach a maximum at electron energies three ufour times of the binding energy of theK electrons. Accord-ingly the optimum laser intensity for a laser wavelength800 nm ~typical of a Ti-Sapphir system! should be 1016– 831017W/cm2 to generate hot electrons whose energy disbution has a maximum between 6 and 30 keV. Such etrons could produceKa-line radiation between 1.6 keV~sili-con! and 8.4 keV ~copper!. Recently, observed data oconversion of laser energy into energy of hot electrons37 andwell known conversion ratios of electrons intoKaradiation38 show that it should be possible to create3108– 109Ka photons per 100 mJ laser energy. With curently available laser systems where the pulse energiesbetween 30 and 200 mJ, it should be possible to fo104– 106 photons in a single shot with the described crysoptics here. In any case this signal is strong enough toister with sensitive detectors like cooled x-ray sensitCCDs which counts almost every x-ray photon in the phoenergy region mentioned.

The temporal broadening by the crystal optics of t

FIG. 14. Density distribution of reflected photons per pulse in the deteplane obtained by a spherically bent crystal, AlKa radiation. ~IntegratednumberNref of reflected photons per pulse see Table VI.!

t

sto

f

i-c-

-reslg-

n

x-ray pulse depends on the crystal rocking curve, the iming geometry and the wavelength used. For the particucases discussed in this article, the theoretical broadeninexpected to be shorter than 100 fs. Furthermore, therethree advantages of the 2D crystal optics:

~i! The reflected radiation is highly monochromatiwhich is essential for spectroscopy experiments or phoelectron spectroscopy.

~ii ! The point-to-point focusing of the 2D crystals allowthe size of the interaction region of the x rays with a pumlaser beam to be reduced, which is very important for gettgood time resolution.

~iii ! Generally, the combination of a bent crystal andflat one used in a x-ray double-crystal diffractometer iscompanied with a strong loss of intensity due to the differdispersion of both crystals. For diffraction experimenwhere an achromatic setup of a focusing 2D crystal and asample crystal is used,21 almost all Bragg-reflected intensitby the bent crystal will be also reflected by the sample crtal. There will be no significant lost of the reflected intensby the sample crystal because the reflectivity could re100% at the maximum of the reflection curve.

We have presented a new technique which is promisfor the application of ultrashort keV x-ray pulses to differekinds of x-ray experiments, like real time x-ray absorptispectroscopy and x-ray diffractometry. A typical setup fsuch experiments was proposed in Ref. 39 and was realin Ref. 9. In case the modification of the sample, i.e., a tcrystalline layer is very fast, this kind of correlation expements could be used to measure the x-ray pulse width insubpicosecond regime. The rapid progress in short pulseser development should provide data from such short x-pulse experiments in the near future.

ACKNOWLEDGMENTS

The authors acknowledge P. Gibbon for fruitful discusions in the field of laser-plasma interaction. Furthermothey also acknowledge O. Wehrhan and A. Saupe for thhelp in crystal preparation and characterization.

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TABLE VI. Observed emission data of the AlKa line: Nref—photons re-flected by the spherically bent crystal,NL(EL)—photon number~energy!emitted by the source obtained byNref , l b—distance crystal detector.

l b

~mm!

Nref

Sphotons

pulse DNL

S107 photons

sr pulse DEL

S 1029 J

sr pulseD~a! Fig. 12~a! 148.7 2030 1.1 2.6~b! Fig. 12~b! 155.0 3090 1.7 3.9~c! Fig. 12~c! 161.5 1490 0.8 1.9

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Au

FK.en

i,

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M

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ich-

le-

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ri-mi-

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