Modern Developments in X-Ray and Neutron Optics Episode 11 docx

More recently, both lat-erally and depth-graded multilayers have been designed and fabricated; they alloweither reflection of divergent beams or over a broad angular or wavelength range,

(a) (b)

Fig 24.2 Cross-sectional HR TEM images of the as-deposited (a) and collapsed (b) Ni/C ML whose XRR and GIXDS simulation parameters are given in Table 24.2.

Black bars at the bottom correspond to 10 nm (a) and 20 nm (b)

energy-dispersive X-ray and electron diffraction analyses while the matrix

is formed by C and a small fraction of fine granular fcc Ni Obviously, theannealing stimulates the growth and coalescence of the original Ni grainsfound in the as-deposited state, which governs also interface morphology asreflected in an increase of the lateral correlation length and a decrease of thevertical correlation of the interface roughness (decay of interface conformity)

As soon as Ni grains are well developed and the ML becomes discontinuous,diffusion of C along Ni grain boundaries may also contribute to the ML break-down A complete diffusion of C into Ni layers was reported in the past [21].The observed thermal stability is comparable to sputtered and PLD Ni/Cmirrors [18, 19, 25]

It is worth noting that the ML period steadily increases and C layer sities decrease when increasing the temperature above 100◦C Similar effects

den-were reported in sputtered Ni/C MLs with larger periods [21, 42] and den-wereattributed to a transformation of the amorphous into the graphitic-like struc-ture Although we were not able to trace this effect directly by HR TEM orX-ray diffraction due to very thin C layers, the observed growth and coales-cence of Ni grains connected with a long-distance collective diffusion of Niatoms across C regions may induce graphitization It was shown that such

a metal-driven graphitization is preferred to carbide formation when C is inexcess [43] which explains also the absence of carbide formation in the tem-perature range applied Once initiated, the graphitization due to Ni diffusionproceeds even at RT as evidenced by ≈8% increase of the ML period which

was observed on the sample annealed at 300◦C after a 16-month RT storage.

24 Multilayers with Ultra-Short Periods 399

Table 24.3 The XRR simulation parameters of the as-deposited Ni/B4C ML (400periods)

tNi tB 4 C Λ σeff Ni-on-B 4 C σeff B 4 C-on-Ni

Ni/B4C MLs prepared by DECR do not suffer from agglomeration effects

in Ni as a ML with a period below 2 nm and an extremely small interfacewidth below 0.3 nm could be deposited (Table 24.3) Obviously, much higheradatom mobilities with DECR sputtering than with UHV deposition have

a healing effect on the geometrical interface roughness and layer continuity

at small thicknesses which goes along with the amorphous character of thelayers as confirmed by X-ray and electron diffraction On the other hand,the presence of compound layers does not favour mixing effects Significantly,substitution of C by B4C has no detrimental effect on the theoretical optical

contrast Due to very smooth interfaces, a large number of periods (N = 400)

was necessary to visualize GIXDS effects in RSM Only the GIXDS aroundthe first ML Bragg maximum is visible (Fig 24.3a) as the second maximumemerges directly from the instrumental background and the higher orderscannot be seen at all This fact is a consequence of the ultra-short ML periodwhen the measurements of RSM are especially instructive A concentration

of GIXDS in the form of a sheet around the Bragg peak (resonant diffusescattering) is a clear sign of a partial vertical correlation of the interfaceroughness while its distinct asymmetry is exceptional HR TEM revealed thatthe interfaces are wavy and partly copy each other

The as-deposited Ni/B4C ML was first exposed to several isothermalannealings at 300◦C up to 8 h of total time with no significant changes of

XRR but a slight improvement of the peak reflectivity on the first Braggmaximum Contrarily, an additional annealing at 400◦C for 2 h destroyed the

ML completely and the Bragg peak disappeared Therefore, an intermediate

350◦ C/2 h annealing, which brought about a decrease of the peak

reflectiv-ity by≈1 order of magnitude, was done independently Such a pre-annealed

sample was then processed by a series of 5- or 10-min RTAs with a step-likeincrease of temperature up to 520◦C which resulted in a further severe reduc-

tion of the peak reflectivity and a reduction of the ML period from 1.59 to1.54 nm, the first Bragg maximum being still well resolved This reductionmay be attributed to the annealing-out of the excess free volume typical foramorphous structure HR TEM inspection showed that the layered structurewithout mixed regions but with many topological defects was still preserved.The RSM (Fig 24.3b) exhibits substantial changes in comparison with the as-deposited state Though it was not possible to simulate RSM with commoncorrelation functions presented in the previous section, a qualitative changedue to the annealing could be simulated when doubling the lateral correlation

(b)

Fig 24.3 Reciprocal space maps of the (a) as-deposited and (b) annealed up

to 520◦ C Ni/B4C ML around the first Bragg peak In the first map, the shadow

of a beam stopper protecting a position-sensitive detector hides the XRR whosesimulation parameters are shown in Table 24.3 The second map was measured with

a point detector The labels denote the logarithms of the extreme values of the

intensity

length from 10 to 20 nm This increase cannot be attributed to the graingrowth as in the case of Ni/C couple as no crystallization inside the MLwas observed However, large re-crystallized regions in the substrate at theinterface with the ML, presumably Ni silicide grains, could be seen locally

by TEM After the ML breakdown, the original layered structure was formed into an inhomogeneous amorphous-like structure with only one diffusering in electron diffraction pattern This collapsed ML still keeps a sharpinterface with the substrate HR TEM revealed a rare occurrence of crys-tallographically ordered regions, probably (111) planes of fcc Ni This factsuggests that Ni diffusion controls the ML breakdown as in Ni/C couple butthe mechanism of decay of the compound layers is unclear in the absence of a

trans-24 Multilayers with Ultra-Short Periods 401

Table 24.4 The XRR and GIXDS simulation parameters of the as-deposited Sc/Cr

stability of Ni/B4C mirror adversely

Ultra-short periods could be achieved and many periods (N = 250) were

deposited also for the Sc/Cr couple using ECR ion source Similarly as for

Ni/B4C, only the region around the first Bragg maximum could be used forGIXDS analysis due to the ultra-short ML periods (Table 24.4) The presence

of the resonant diffuse scattering sheet in RSM (Fig 24.4a) suggests here aswell a partial vertical correlation of the interface roughness In contrast toprevious cases, where the lateral correlation function according to (24.1) wasused, PSD given by (24.2) proved to be convenient to simulate GIXDS forSc/Cr MLs (Table 24.4; Fig 24.4b) Different cuts throughout the calculatedRSM gave a good quantitative agreement with particular scans measured sep-arately The vertical correlation length represents only≈8% of the total ML

thickness The σeff and σ values are practically identical with a slight

asym-metry between Sc/Cr and Cr/Sc interfaces, showing no mixing as expectedbecause of the negligible miscibility of the elements No mixing and a high reg-ularity of the ML stack were confirmed by HR TEM and TEM, respectively.Electron diffraction showed only one ring typical for an amorphous struc-

ture but narrower than that for Ni/B4C which implies a higher short-rangeordering

Thermal stability tests were performed on a ML with a period of 1.25 nmrepeated 150 times The sample was first exposed to several isothermal anneal-ings at 280◦C up to 33 h of total time which brought about a decrease of the

peak reflectivity on the first Bragg maximum by ≈25% This decrease was

connected with an increase of the initial interface roughness of 0.28 nm to thefinal value of 0.30 nm for Cr-on-Sc interfaces while a similar increase from0.26 to 0.29 nm was observed for Sc-on-Cr interfaces Simultaneously, the MLperiod increased to 1.28 nm which may be ascribed to a structural orderinginside the layers A further 450◦ C/4 h annealing decreased the peak reflec-

tivity by one order of magnitude and a subsequent 650◦ C/4 h one destroyed

the layered structure completely, as reflected in the disappearance of the MLBragg maxima No crystalline phase could be detected by X-ray diffraction.Presumably, a fine granular structure typical for immiscible components wasformed

(b)

Fig 24.4 Measured (a) and simulated (b) reciprocal space maps of the Sc/Cr ML

around the first Bragg peak The simulation parameters are shown in Table 24.4

The labels denote the logarithms of the extreme values of the intensity

24.5 Conclusions and Outlook

The ability to form thermally stable ultra-short period MLs was tested formaterial couples of different compositions and miscibilities Such periods areinherently required for high-quality imaging in the water window or for veryhard X-rays and put strong requirements on the interface quality

The Cu/Si couple is attractive for very hard X-rays because it is lowabsorbing and forms very regular ML stacks easily down to small periods Onthe other hand, it is a miscible pair of materials We found a limit of≈2 nm for

the ML period which still yields a well-resolved ML Bragg peak in the XRRcurve for UHV-deposited MLs and which compares well with dc-sputtered

24 Multilayers with Ultra-Short Periods 403ones prepared previously Thermal stability restricts their use to below 100◦C

and therefore does not qualify this material pair to be used for ML mirrorsworking in high heat load conditions

Ni/C mirrors were studied mostly for applications below the C–K edge(284 eV) or for hard X-rays of several keV (e.g G¨obel mirrors) A shift to ultra-short periods would render them useful for applications close to 100 keV with alow-absorption coefficient A better thermal stability than for the Cu/Si couplemay also be expected due to a low mutual solubility of Ni and C We found

a minimum limit of ≈2.5 nm for the ML period imposed by agglomeration

effects in polycrystalline Ni layers Such a ML is stable up to 300◦C, the

breakdown being controlled by the growth and coalescence of Ni grains andNi-induced graphitization of C The thermal stability is comparable to that

of Ni/C MLs with larger periods studied previously, which suggests that it

is independent of the ML period when continuous layers are formed Verticalcorrelation of the interface roughness is much weaker than for Cu/Si MLs withpositive implications for their use when a good specular imaging contrast isrequired

From the technological point of view, UHV deposition with an in situsubstrate heating proved to be a cost-effective promising alternative to insitu ion-beam etching for Cu/Si and Ni/C couples Nevertheless, ultra-shortperiods down to 2 nm or less are not accessible with them

An ultra-short ML period far below 2 nm could be achieved by a stitution of C by B4C in Ni/C MLs and application of DECR sputtering

sub-for deposition of Ni/B4C MLs A high adatom energy allows formation ofextremely smooth amorphous layers which are continuous at very small thick-nesses The presence of compound layers enhances thermal stability to 350◦C

on a long-term annealing and above 500◦C on RTA The amorphous character

of the layers plays also some role as it excludes fast grain boundary diffusion.The mirror collapses by Ni diffusion and decomposition of B4C layers withoutformation of a well-developed crystalline phase

The Sc/Cr couple is also able to provide MLs with an ultra-short periodfar below 2 nm using ECR sputtering The MLs have excellent quality andgood thermal stability up to 500◦C as expected for the elements with neg-

ligible miscibility The vertical correlation length of the interface roughnessconstitutes less than 10% of the total ML thickness, so that its detrimen-tal effect on the specular imaging contrast is minimized The ML breakdown

is presumably controlled by the formation of fine granular phase typical forimmiscible elements

For the future, new material combinations suitable for ultra-short period

ML mirrors aimed at specific applications have to be tested in detail in away similar to that demonstrated in this work In particular, material pairsyielding structures with minimum interface roughness have to be searchedfor It has to be stressed that even traditional material pairs represent newchallenges for technology when MLs with layer thicknesses below 1 nm andseveral hundred periods have to be deposited At present, an optimization of

the reflectivity in UHV electron-beam evaporated ML stacks is done in situ byion milling and partial material removal after deposition of each layer In thisway the layers, which have been deposited thicker than nominal values, arethinned to optimum thicknesses with in situ control For the mirrors deposited

by sputtering, a high quality of the ML stack is achieved by employing stablesputtering parameters and a precise control of the motion of the sample withrespect to the targets ECR sputtering with the inert gas pressure far belowthe thermalization threshold proves to be superior to other techniques in terms

of yielding extremely smooth interfaces To fabricate regular ML stacks withultra-short periods, further progress is expected by making a still tighter con-trol of the deposition process including an in situ monitoring with a resolutionbelow 10−1nm to provide feedback for subsequent correction procedures.

Thermal stability requirements are also more critical with ultra-short ods below 2 nm, particularly in view of the advent of free electron laser (FEL)sources whose brilliance is several orders of magnitude higher than that atpresent synchrotron beam lines To set and manipulate thermal stability lim-its of ultra-short period mirrors working with femtosecond intense pulses, MLresponse to short-time power loads far below a second and with the rampedge of several 100 K s−1 must be studied Here, advanced time-resolved in

peri-situ diagnostics of the processes at the interfaces, like in peri-situ ellipsometry withseveral ms read-out, must be applied

Though much knowledge has already been gained, the research and opment of ultra-short period ML mirrors with the individual layer thicknesses

devel-on the sub-nanometre scale is still open to new creative ideas The design andthe production of mirrors with atomic level control will shift frontiers not only

in the ML optics but also in nanotechnology as a whole

Acknowledgements

The authors are grateful to M Yamamoto from IMRAM Tohoku University,Sendai, for providing Sc/Cr ML samples and to V Hol´y from Charles Uni-versity, Prague, for providing DWBA software Z Bochn´ıˇcek from MasarykUniversity, Brno, is acknowledged for X-ray reciprocal space measurements.Supports by COST P7 Action, Scientific Grant Agency VEGA (contracts

no 2/4101/26 and 2/6030/26) and Center of Excellence SAS project CE-PII/2/2006 are acknowledged

References

1 M Yamamoto, T Namioka, Appl Optics 31, 1622 (1992)

2 E Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham,

1994)

3 D.G Stearns, M.B Stearns, Y Cheng, J.H Smith, N.M Ceglio, J Appl Phys

67, 2415 (1990)

24 Multilayers with Ultra-Short Periods 405

4 T Kingetsu, M Yamamoto, Surf Sci Rep 45, 79 (2002)

5 V.V Kondratenko, Yu.P Pershin, O.V Poltseva, A.I Fedorenko, E.N Zubarev,S.A Yuchin, I.V Kozhevnikov, S.I Sagitov, V.A Chirkov, V.E Levatsov, A

Vinogradov, Appl Optics 32, 1811 (1993)

6 M Hansen, K Anderko, Constitution of Binary Alloys (McGraw-Hill, New

York, 1958)

7 H.J Stock, U Kleineberg, A Kloidt, B Schmiedeskamp, U Heinzmann, M.Krumrey, P M¨uller, F Scholze, Appl Phys Lett 63, 2207 (1993)

8 R Sender´ak, M Jergel, ˇS Luby, E Majkov´a, V Hol´y, G Haindl, F Hamelmann,

U Kleineberg, U Heinzmann, J Appl Phys 81, 2229 (1997)

9 A Patelli, J Ravagnan, V Rigato, G Salmaso, D Silvestrini, E Bontempi,

L.E Depero, Appl Surf Sci 238, 262 (2004)

10 D Morris, C.J Buckley, G.R Morrison, A.G Michette, P.A.F Anastasi, M.T.Browne, R.E Burge, P.S Charalambous, G.F Foster, J.R Palmer, P.J Duke,

Scanning 13, 7 (1991)

11 A Nefedov, H Zabel, F Sch¨afers, Nucl Instrum Meth A 467–468, 345 (2001)

12 H Takenaka, H Ito, K Nagai, Y Marumatsu, E Gullikson, R.C Perera, Nucl

16 M.S Bibishkin, A.A Fraerman, A.E Pestov, K.A Prokhorov, N.N

Salashchenko, Yu.A Vainer, Nucl Instrum Meth A 543, 333 (2005)

17 D.L Windt, Appl Phys Lett 74, 2890 (1999)

18 G.S Lodha, S Pandita, A Gupta, N.V Nandedkar, K Yamashita, J Electron

Spectrosc Rel Phenom 80, 453 (1996)

19 K Nakajima, S Aoki, S Sudo, S Fujiwara, Jpn J Appl Phys 31, 2864 (1992)

20 J Friedrich, I Diel, C Kunz, S Di Fonzo, B.R M¨uller, W Jark, Appl Optics

36, 6329 (1997)

21 V Dupuis, M.F Ravet, C Tete, M Piecuch, Y Lepetre, R Rivoira, E Ziegler,

J Appl Phys 58, 5146 (1990)

22 C Borchers, C Michaelsen, Philos Mag A 82, 1195 (2002)

23 M Ulmeanu, A Serghei, I.N Mihailescu, P Budau, M Enachescu, Appl Surf

Sci 165, 109 (2000)

24 N.V Kovalenko, S.V Mytnichenko, V.A Chernov, JETP Lett 77, 80 (2003)

25 V.A Chernov, E.D Chkalo, N.V Kovalenko, S.V Mytnichenko, Nucl Instrum

Meth A 448, 276 (2000)

26 R Dietsch, T Holz, T Weissbach, R Scholz, Appl Surf Sci 197, 169 (2002)

27 Y.P Pershin, Y.N Zubarev, V.V Kondratenko, O.V Poltseva, A.G

Pono-marenko, V.A Servryukova, J Verhoeven, Metallofiz Nov Tekhnol 24,

795 (2002)

28 E.I Puik, M.J Van der Wiel, H Zeulenmaker, J Verhoeven, Appl Surf Sci

47, 251 (1991)

29 A.F Jankowski, L.R Schrawyer, M.A Wall, J Appl Phys 68, 5162 (1990)

30 T Bottger, D.C Meyer, P Paufler, S Braun, M Moss, H Mai, E Beyer, Thin

Solid Films 444, 165 (2003)

31 A.F Jankowski, C.K Saw, C.C Walton, J.P Hayes, J Nilsen, Thin Solid Films

469, 372 (2004)

32 Ch Morawe, J.-C Peffen, O Hignette, E Ziegler, Proc SPIE 3773, 90 (1999)

33 N.N Salashchenko, E.A Shamov, Opt Commun 134, 7 (1997)

34 F Sch¨afers, H.C Mertins, F Schmolla, I Packe, N.N Salashchenko, E.A

Shamov, Appl Optics 37, 719 (1998)

35 J Birch, F Eriksson, G.A Johansson, H.M Hertz, Vacuum 68, 275 (2002)

36 T Kuhlmann, S Yulin, T Feigl, N Kaiser, T Gorelik, U Kaiser, W Richter,

Appl Optics 41, 2048 (2002)

37 K Sakano, M Yamamoto, Proc SPIE 3767, 238 (1999)

38 J.H Underwood, T.W Barbee, AIP Conf Proc 75, 170 (1981)

39 V Hol´y, T Baumbach, Phys Rev B 49, 10668 (1994)

40 S.K Sinha, E.B Sirota, S Garoff, H.B Stanley, Phys Rev B 38, 2297 (1998)

41 C Borchers, P Ricardo, C Michaelsen, Philos Mag A 80, 1669 (2000)

42 T Djavanbakht, V Carrier, J.M Andre, R Barchewitz, P Troussel, J Phys

IV 10, 281 (2000)

43 R Sinclair, T Itoh, R Chin, Microsc Microanal 8, 288 (2002)

Specially Designed Multilayers

J.I Larruquert, A.G Michette, Ch Morawe, Ch Borel, and B Vidal

Abstract Periodic multilayers, utilising Bragg reflection at single angle or

wave-length, are established as efficient reflectors from the hard X-ray down to the extremeultraviolet (XUV) region of the electromagnetic spectrum More recently, both lat-erally and depth-graded multilayers have been designed and fabricated; they alloweither reflection of divergent beams or over a broad angular or wavelength range,

or a combination of both Recent developments in aperiodic structures, along withadvances in ultra-short period and transmission mutilayers, are discussed in thischapter Modelling methods to provide designs for specific purposes are described,

as are advances in manufacturing techniques and quality control In addition to peakreflectivity at a specific wavelength or angle, high integrated reflectivity over a givenwavelength or angular range is considered, along with flat reflectivity profiles Impor-tant potential applications of flat response mirrors are X-ray micro-spectroscopy,X-ray diffraction, XUV polarimetry, and any other technique requiring both highreflectivity and broad bandwidth

25.1 Introduction

Periodic multilayers, utilizing Bragg reflection at single angles or wavelengths,are established as efficient reflectors from the hard X-ray to the extremeultraviolet (XUV) region of the electromagnetic spectrum Recently, laterallygraded multilayers allowing reflection of divergent beams and depth-gradedmultilayers providing broad angular or wavelength ranges have been designedand made; combinations having both types grading have also been considered.Recent developments in such aperiodic structures, along with advances inultra-short period and transmission multilayers, are discussed in this chapter.Modeling methods to provide designs for specific applications are described,but manufacturing techniques and quality control are not discussed explicitly,

as these are similar to those used for conventional multilayers In addition topeak reflectivity at a specific wavelength or angle, high integrated reflectiv-ity over a given wavelength or angular range is considered, along with flatreflectivity profiles

Important potential applications of flat response mirrors include X-raymicrospectroscopy, X-ray diffraction, XUV polarimetry, and any other tech-nique requiring both high reflectivity and broad bandwidth Importantly, inthe recent years, the use of multilayer-based X-ray optics at third-generationsynchrotron sources has seen a considerable growth [1,2] The principal appli-cations are focusing or collimating optics and broadband monochromators.

25.1.1 Periodic Multilayers

For many years, the only effective way of manipulating X-ray beams wasthrough the phenomenon of “total” external reflection For this, the graz-ing angle of incidence must be less than the critical angle which, for mostmaterials, is less than about 1◦ for X-rays Such small grazing angles result

in several major disadvantages including aberrations and small effectiveapertures However, at low energies, single surface reflectors could be used,while at high energies crystal optics were (and still are) employed

The problem was in the intermediate energy range; it was this gap thatperiodic multilayer mirrors were initially designed to fill Such coatings can, inprinciple, be deposited on substrates of any form and can be used as reflectorsfor energies of∼0.01–10 keV at incidence angles ranging up to, in some cases,

normal incidence Periodic multilayers have a typical energy resolution of1–10% which is about two orders of magnitude larger than possible with per-fect crystals, leading to a significant gain in flux At the same time, they allowfor larger effective apertures than grazing incidence mirrors, making them con-siderably shorter and easier to handle When properly prepared, they conservethe properties, such as coherence, of the X-ray beam, and they can be tailored

to specific applications using the refraction-corrected Bragg equation

mλ = 2d

Here, m is the diffraction order, λ is the (vacuum) wavelength, d is the tilayer period, n is the average refractive index in the multilayer stack, and

mul-θ is the glancing angle at the multilayer surface However, periodic

multilay-ers work only within specific wavelength or angular ranges, where (25.1) isapproximately satisfied, which limits their applicability Such restrictions can

be alleviated by using optimized designs

25.2 Optimized Multilayers

In any geometry where the X-ray beam is incident over a range of glancingangles, the multilayer structure has to be deposited with a lateral thick-ness gradient to fulfill the Bragg condition (25.1) along the full length ofthe mirror [3] Similarly, whenever broader reflectivity profiles are required,

25 Specially Designed Multilayers 409depth-graded or nonperiodic layered structures have to be deposited [4] Non-periodic and laterally graded multilayers can be combined into single devicesproviding fixed focus optics over broad energy intervals [5]; such optics havewide applicability in, e.g., microspectroscopy and diffraction experiments.

25.2.1 Laterally Graded Multilayers

Most multilayer-based synchrotron optics, including focusing and collimatingdevices using curved multilayers and flat monochromators [6], require lateralthickness gradients to fulfill the Bragg condition for fixed photon energy alongthe full beam footprint Depending on the geometry, different lateral gradientshave to be applied; three commonly used geometries are elliptical, parabolic,and flat For an elliptical surface,

sin2θ = b

2

where θ is the glancing angle, b is the semiminor axis of the ellipse, and p and

q are the object and image distances For a parabolic profile,

where s is the perpendicular distance from the surface to the source point.

To obtain the corresponding bilayer lateral thickness gradients, the anglesfor the appropriate surface must be used in (25.1) Strictly speaking, the localangle of incidence varies also with depth, but this effect is typically negligiblecompared to the widths of the multilayer Bragg peaks

In practice, lateral thickness gradients are obtained by differential coatingtechniques In a static setup, masks modify the particle fluxes while dynamicapproaches use moving masks or the relative motion between source andsubstrate In the sputter deposition system at the European SynchrotronRadiation Facility (ESRF), for example, the substrate remains fixed whilethe sources are scanned with variable speeds [1]

Figure 25.1 illustrates the case of a laterally graded parabolic W/B4Cmultilayer collimator [1] Here, both W and B4C layer thicknesses were variedindividually to optimize the peak reflectivity along the total length of theoptic The agreement between theory and experiment is remarkable For themultilayer to be efficient, the required accuracy of the thickness profile must

be considerably better than the width of the multilayer Bragg peak With the

present ESRF equipment, relative errors of <0.5% RMS can be achieved.

0 2 4 6 8

Distance along mirror [cm]

d

B4C

W

Fig 25.1 Thickness profile of a laterally graded parabolic W/B4C multilayer

Broken lines indicate the ideal case deduced from the given geometry Solid lines show a simulation based on realistic parameters of the coating system The symbols

are experimental data obtained from X-ray reflectivity measurements

25.2.2 Depth-Graded Multilayers

One way of increasing the bandwidth of a multilayer is simply to reduce thenumber of reflecting bilayers For hard X-rays, this approach is penalized

by the rapidly decreasing reflectivity A more efficient option is to vary the

d-spacing in such a way that X-rays of different energies, or incident at

dif-ferent angles, are reflected from different depths within the total stack Thisconcept of depth grading was first introduced in the design of supermirrorsfor neutron optics [7], but for X-rays, the stronger absorption complicates thedesign However, different mathematical models have been developed, lead-ing to nonperiodic layered structures [8–10] These typically involve startingwith a periodic structure, or some other analytically derived layer thicknessdistribution, and then randomly changing the position of a randomly selectedboundary between two layers in at attempt to improve some “merit function”(MF) If, for example, the requirement is for a flat reflectivity response over agiven angular range at a fixed wavelength, then an appropriate MF would be

MF =

N



i=1 [R(θ i)− R0]2, (25.5)

25 Specially Designed Multilayers 411

(b)

Fig 25.2 (a) Specular reflectivity of an aperiodic Ni/B4C multilayer as a

func-tion of the incidence angle at an X-ray energy of 8.048 keV Circles indicate the experimental data points; the line is a simulation based on the bilayer structure

of (b)

where R(θ i ) is the reflectivity at a selected angle from N sample points and R0

is the target reflectivity The reflectivities are calculated using the standardParrat recursive formalism [11]

Figure 25.2a shows the reflectivity spectrum of such a multilayer, designedand made at the ESRF [4] It was designed to provide a flat reflectivity profile

over an angular bandwidth of about 20% at 8 keV It consists of 43 Ni/B4Cbilayers of variable thickness as indicated in Fig 25.2b Numerical fits to thedata indicate that the residual undulations on the flat plateau are mainlycaused by oxide formation on top of the sample It is mandatory that all keyparameters of the sample structure (thickness, density, interdiffusion, oxida-tion) have to be known in advance, since they enter directly into the designalgorithm

Such mirrors can also be designed for lower energies at larger grazingincidence angles An example of the performance of a W/Si mirror designed toact as a spectral filter on a titanium Kα(4.5 keV) X-ray source, for applications

in radiobiology, is shown in Fig 25.3 The mirror was designed and made incollaboration with Tongji University, Shanghai, and the measurements weredone at the Gray Cancer Institute, UK

Similar depth-graded multilayers can be designed to work over a range

of energies at fixed incidence angle A simple example of this would be toarrange a series of periodic mirrors, one on top of the other, such that eachreflects a specific energy with the upper mirrors being relatively transparent

to the radiation reflected from the lower mirrors An example of the stackstructure of such a mirror [12–16], consisting of 20 W/Si bilayers, and thecorresponding reflectivities are shown in Fig 25.4 The calculated reflectivity

at a grazing incidence angle of 0.6 ◦ is over 40% in the energy range 6–9 keV.

The first six bilayers (zone 1) define the shape of the calculated reflectivity

Fig 25.3 Reflectivity of a broad angular range multilayer designed for an energy

of 4.5 keV The thin curve is the design reflectivity (with the aim of a flat response of 20%) and the thick curve is a fit to the measured points with an interlayer roughness

of 0.4 nm

Fig 25.4 Layer thickness distribution of a five-zone depth-graded multilayer (left)

and the corresponding reflectivities (right); the measured reflectivities are indicated

by the line with error bars The overall reflectivity takes into account absorption in

the upper layers, and the sudden drop at an energy of just over 10 keV is due to thetungsten Lβabsorption edge

curve for energies up to 9.5 keV The layers of zones 2–5 are primarily sible for the reflectivity at 9–10 keV In the 6–9 keV range, the reflectivitycurve is flat with some small oscillations which can be attributed to the mix-ing of second-order reflectivity between zone 1 and the deeper layers of themirror The decrease in the total reflectivity at energies higher than 9 keV ismainly due to absorption in the tungsten layers and to the effects of rough-ness, which increase exponentially with the square of the photon energy Theexperimental reflectivity is higher than the calculated values, especially for the

respon-25 Specially Designed Multilayers 413

Fig 25.5 A depth-graded Mo/Si multilayer designed for a flat reflectivity profile

in the wavelength range 15–17 nm at the Brewster angle The left-hand plot shows the layer thickness distribution and the right-hand graph shows the calculated (line)

and measured reflectivities

“total” reflectivity part of the spectrum (below 5 keV) and at energies greaterthan 8 keV These differences are partially due to a±0.01 ◦systematic shift in

the experimental angle corresponding to the goniometer angular precision

A more flexible approach allows each layer thickness to be varied in theoptimization An appropriate MF is then

where R(E i ) and R(λ i) are the reflectivities at selected energies and

wave-lengths, respectively, from N sample points, and R0is the target reflectivity.Here, examples of mirrors designed for extreme ultraviolet (EUV) radiation,where wavelength is the more usual variable, are considered [17–22]

Figure 25.5 shows the parameters and performance of a depth-gradedMo/Si multilayer designed for a reflectivity of 50%, in the wavelength range15–17 nm, at the Brewster angle which, since the refractive indices are veryclose to unity, is very close to 45◦ The lower measured reflectivities may be

attributed to interlayer roughness or diffusion with a length scale of about

1 nm The performance of a similar mirror, designed for a wider wavelengthrange, is shown for comparison in Fig 25.6; because of the wider wavelengthrange, the measured performance is compromised compared to that of themirror of Fig 25.5 The mirrors were designed and made in collaborationwith Tongji University, Shanghai, and the measurements were carried out atthe BESSY synchrotron

Since these mirrors were designed to work at the Brewster angle, they alsoact as polarizing elements The plotted reflectivities are for the s-component

of polarization, Rs Figure 25.6 also shows the design and measured polarizing

capabilities, P , of the two mirrors, where

P = Rs− Rp

Fig 25.6 Calculated and measured reflectivities (left) of a Mo/Si multilayer

designed for a reflectivity of 35% in the wavelength range 14–18 nm The hand plot shows the calculated (light curve) and measured (open circles) polarizing

right-capabilities of this mirror as well as those of the mirror of Fig 25.5

Similar multilayers can also be made on transmissive substrates Whenused in conjunction with reflective multilayers, in a polarizer/analyzer arrange-ment, these allow polarization analysis to be carried out over an extendedwavelength range without having to change the optics [22]

25.2.3 Doubly Graded Multilayers

The concepts of laterally and depth-graded multilayers may be combined intosingle structures; such a device would be capable of broadband focusing withfixed focal distance [23] Rewriting the Bragg equation (25.1) leads to theenergy dispersion relation for a multilayer

by the appropriate expression from (25.2) to (25.4)

The concept is similar to that of DuMond diagrams [24] and can be

illus-trated by plotting the dispersion of the energy E of the reflected photons against the focal distance q along the optics The example in Fig 25.7 shows the case of two elliptically shaped Ru/B4C multilayer mirrors Periodic mul-tilayers without lateral gradient are characterized by the curved thick solidlines Such structures reflect X-rays only in a narrow “intersection volume”

in (E, q) space, defined by the intrinsic multilayer line width, which is

typ-ically a few percent With a lateral period thickness gradient, the energyresponse becomes constant (thick horizontal solid line in Figure 25.7) In prac-tice, this has the advantage that the multilayer reflects at the same energyover the whole mirror length An aperiodic multilayer without lateral gradientwidens the energy bandpass and therefore opens the dispersion area, thereby

increasing the intersection zone in (E, q) space The most attractive solution,

25 Specially Designed Multilayers 415

Fig 25.7 Energy dispersion of different Ru/B4C multilayers, with periods of

6.7 nm, showing the reflected energy bands against the distance q from the focal spot along the multilayer surface Thick solid lines represent the periodic multilayers

either with or without lateral thickness gradient A purely depth-graded multilayercovers the area limited by the thin curves A combination of both is indicated by

the zone between the straight lines The lighter shaded areas show the responses of two Ru/B4C Kirkpatrick–Baez mirrors

however, is a combination of lateral and depth gradient It provides a broadand constant energy bandwidth over its whole length

As described in Sect 25.2.2, the spectral response of a nonperiodic

mul-tilayer can be designed locally The overall intensity spectrum I q of a curvedfocusing device, however, depends on the additional lateral gradient and onthe geometrical influence of the curved mirror The local reflectivity profilealong the mirror can be calculated based on the knowledge of both lateral and

depth gradients In addition, the local intensity dI q /dα q as a function of the

angular acceptance dα q has to be taken into account (see inset in Fig 25.8)

In the given configuration, the X-ray source can be considered to be isotropicover the angular range used In the case of an elliptical profile, and at a given

incidence angle θ, the local intensity contribution dI q in the focus coming

from a given angular range, dα q , is proportional to the angle, dα p, seen fromthe source Since, at the reflection point of the beam,

integrates over the whole focal spot The results for the two Ru/B4C mirrors

of Fig 25.7 are shown in Fig 25.8 Only the relevant branch, with p > q, is

shown The minimum possible grazing angle occurs at the vertex of the ellipse

where p = q and where the local intensity is normalized to unity Very similar

results can be obtained for parabolic mirrors [5] The total spectral responsecan be obtained by integrating the local reflectivity spectra multiplied by(25.10) over the full length of the optics

Fig 25.8 Relative intensity after reflection from two elliptically shaped mirrors as

a function of the incidence angle θ The curves were calculated using (25.10) and correspond to the branch p > q of the ellipse showing both multilayer mirrors The illuminated (active) angular zone and the smallest angles of incidence (for p = q) are indicated by dotted lines

Fig 25.9 Gray scale images of the experimental (left) and simulated (right)

X-ray reflectivity data of multilayer 170, plotted as functions of focal distance q and photon energy E

25 Specially Designed Multilayers 417

Figure 25.9 (left image) shows a two-dimensional plot in (E, q) space

based on the experimental data set of the multilayer 170 (see Fig 25.8) and

the corresponding simulation (right image) The area in (E, q) space selected

here is indicated by the dotted rectangle in Fig 25.7 The plots show broadhorizontal bands of high reflectivity and confirm the more general theoreticalapproach described above The intensity fluctuations of the experimental data

of Fig 25.7 are caused by subtle deviations from the ideal layer structure Inthe main features, however, there is a good agreement

25.3 Multilayers with Strongly Absorbing Materials

In the discussions of the previous sections, although absorption had to betaken into account in determining the multilayer properties, at the wave-lengths considered, it is sufficiently low that reasonable performances canstill be obtained However, reflective multilayer coatings for the wavelengthrange 50–105 nm, in the EUV region of the electromagnetic spectrum, havenot previously been developed because all conventional materials are stronglyabsorbing Due to this strong absorption, radiation would be mostly absorbed

in the few outermost layers of a multilayer; only single layer coatings areuseable A new concept, sub-quarter-wave multilayers (SQWMs), has beenrecently developed; with this, a reflectivity enhancement is obtained over that

of a single layer even when the materials are strongly absorbing For lengths shorter than ∼50 nm, the intrinsic absorption decreases but remains

wave-moderately high Thus, reflective coatings based on two-material multilayersare feasible, but even so SQWMs may provide valuable reflectivity increases.According to this new multilayer concept, enhancements in reflectivity can

be obtained by the superposition of thin films of two or more materials whencertain material selection rules are satisfied The main results on SQWMsare summarized in the following sections Section 25.3.1 highlights the mainsteps in the derivation to obtain the new multilayers Section 25.3.2 exhibits

a few examples of multilayers involving materials with a strong absorption.Section 25.3.3 demonstrates that the new multilayers can also be applicable

in spectral regions of moderate absorption where standard two-material tilayers are already possible Finally, Sect 25.4 summarizes new algorithmsfor multilayer optimization

mul-25.3.1 Sub-Quarter-Wave Multilayers

A general multilayer coating can be considered to consist of m thin films of

different absorbing materials on an opaque substrate Figure 25.10 shows adiagram of such a general multilayer The complex refractive indices of thefilms and substrate are referred to as ˜n i , i = 1, 2, , m + 1, starting with

the outermost film; ˜n0 is the refractive index of the incidence medium Sincethe materials absorb radiation, all the ˜n are complex numbers, and in the