62 (2007) 615–621www.elsevier.com/locate/sab

Spectrochimica Acta Part B

Computer simulations and experimental results onair-gap X-ray waveguides☆

D. Pelliccia a,b, I. Bukreeva b,d, M. Ilie b, W. Jark c, A. Cedola b, F. Scarinci b, S. Lagomarsino b,⁎

a Universita' di Roma “La Sapienza”, Phys. Dept. and I.N.F.N. Roma1, P.le A. Moro 2, 00185 Roma, Italyb Istituto di Fotonica e Nanotecnologie — CNR, Via Cineto Romano 42, 00156 Roma, Italy

c Elettra — Sincrotrone Trieste in Area Science Park, S.S. 14, Km 163.5, 34012 Basovizza Trieste, Italyd RAS — P.N. Lebedev Physics Institute, Leninsky pr. 53, 119991 Moscow, Russia

Received 20 December 2005; accepted 22 February 2007Available online 2 March 2007

Abstract

X-ray Waveguides (WG) are optical elements able to provide nanometer size coherent X-ray beams. Both Resonant beam Coupling (RBC) andFront Coupling (FC) can be implemented to couple the incident radiation with the WG. In this paper we implemented a computer code based onfinite difference scheme solutions of parabolic wave equations to simulate the coupling and the propagation of electromagnetic waves into air-gapwaveguides in Front Coupling mode. Both a synchrotron source and a table-top laboratory sources have been considered. In this last case thedegree of coherence of the incoming beam has been explicitly taken into consideration. Air-gap WGs have been fabricated following simpleprocedures, and tested at the Elettra synchrotron source. The results are in good agreement with theoretical expectations.© 2007 Elsevier B.V. All rights reserved.

Keywords: X-ray optics; X-ray waveguide; Synchrotron radiation; Computer simulation

1. Introduction

Planar and 2D X-ray waveguides (WG) are optical elementsof considerable interest both from a fundamental point of view,and for their use in microprobe (i.e. microimaging, microdif-fraction, microfluorescence, etc.) experiments. After the firstdemonstration of the feasibility to provide a submicrometerbeam with planar WG [1,2] and with 2D WG [3], manydifferent schemes and many applications have been demon-strated. For a recent short review see [4]. Further investigationsof X-ray waveguides are particularly important as they have aprominent position in the recently started discussion of thesmallest possible X-ray spots. In fact first Bergemann et al. [5]argued that single mode X-ray waveguides will provide the

☆ This paper was presented at the “18th International Congress on X-rayOptics and Microanalysis” (ICXOM-18) held in Frascati, Rome (Italy), 25–30September 2005, and is published in the Special Issue of Spectrochimica ActaPart B, dedicated to that conference.⁎ Corresponding author.E-mail address: stefano.lagomarsino@ifn.cnr.it (S. Lagomarsino).

0584-8547/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.sab.2007.02.018

smallest X-ray spots with dimensions of around 10 nm.Interestingly Pfeiffer et al. [6] produced already a one-dimensional waveguide that was operated almost at this limit.Later Schroer and Lengeler [7] show that refractive lenses withparabolic concave surfaces and especially in the lightenedFresnel version can ultimately provide smaller spots. Thefeasibility of smaller spots had also been shown already byMichette et al. [8] for reflective Bragg Fresnel optics. As far asproduction technology is concerned the single mode X-raywaveguides are obviously feasible by use of state-of-the-artinstrumentation as was shown already by Pfeiffer et al. [6]. Theother objects instead are more demanding and in fact similarspot sizes are not yet reported for them.

In the following we will treat essentially planar WGs. In thecase of 2D waveguides with rectangular section, the twoorthogonal directions can be considered as independent, andthus the extension of the present discussion to two dimensions isquite straightforward. The incident radiation can be coupled intothe guiding layer of X-ray WGs in two different ways: ResonantBeam Coupling (RBC) and front coupling (FC). RBC [1,2]

Fig. 2. Intensity distribution in front of and along a front coupling WG inmonomodal regime for: a) first mode; b) second mode.

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takes place in a three layer WG, with the incident beam atgrazing angle transmitted by the very thin upper layer andtrapped by the intermediate guiding layer (see Fig. 1a); with thisscheme the incoming beam of several tens of microns can becompressed down to nanometer dimensions and significanteffective gain (N100) can be achieved [9]. In the FC scheme[10] the incoming radiation is directly side-coupled with theWG aperture, and the spatial acceptance is therefore equal to theWG gap (see Fig. 1b). Moreover, several modes can be excitedsimultaneously in the WG. A variant of the front couplingscheme can be adopted with a reflection of the incoming beamjust in front of the WG entrance, at one of the resonance angles(see Fig. 1c). In this way only one particular mode will beselected. In this geometry the spatial acceptance of the WG istwice the WG gap. However, not only the spatial acceptanceΔSmust be taken into account, but also the angular acceptanceΔϕ:the WG can essentially accept and transmit spatially coherentradiation, which covers a phase space volume given by: ΔSΔϕ=Cλ, where C is a numerical constant depending onwhether the spatial ΔS and the angular extension Δϕ are takenas FWHM (C=0.44) or standard deviation (C=1 /4π) of theirrespective distributions [11]. Therefore for maximum outputflux a pre-focusing optics can be used which optimizes thecoupling [12]. Now the application of waveguides based on thinfilm guiding layers is limited to the hard X-ray range due to theunavoidable absorption losses in the guiding layer. On the otherhand, front coupling WGs with vacuum/air gap can be usedvirtually at any photon energy [13], which makes themattractive also for present and future Free Electron Lasers(FEL) installations. In this perspective the behavior of WGswhen illuminated by very short pulses has been very recentlyinvestigated [14]. Scope of this paper is to verify the predictionsof computer simulations for the mode transmission with

Fig. 1. Scheme of the different coupling modes: a) Resonant beam couplingthrough the cover layer. b) Front coupling directly into the guiding layer.c) Front coupling with pre-reflection and generation of a standing wave pattern.

experimental results of air-gap WGs. The computer simulationswere carried out (Section 2) using a code based on finitedifference scheme solutions of parabolic wave equations [15].For the incident radiation both synchrotron radiation source andtable-top laboratory source have been simulated. The WG werefabricated following a quite simple procedure, described below(Section 3.1). The experimental results, obtained at themicrofluorescence beam-line at Elettra (Trieste) are describedin Section 3.2. A general discussion and the conclusions arereported in Section 4.

2. Computer simulations

The code we used to simulate the mode structure inside andoutside of the planarwaveguide is based on the numerical solutionof the parabolic wave equation

2ik∂U∂x

þ ∂2U∂y2

þ k2ðe−1ÞUðx; yÞ ¼ 0 ð1Þ

Eðx; yÞ ¼ Uðx; yÞexpðikxÞ; ð2Þ

where U(x,y) is the complex amplitude of the electric field E(x,y)which changes slowly along the optical axis 0X. The coordinatesystem is chosen so that x is parallel to interfaces in the plane ofincidence, y is perpendicular to guiding layer and z is parallel to it(normal to the paper). It is assumed that the dimensions along z

Fig. 3. Vertical intensity distribution in the guiding layer of an X-rayWG for first(a) and second (b) modes calculated with computer code (full line) and withEq. (4) (open circles).

Fig. 4. Variation of the phase of the electric field complex amplitude (from(Eq. (2))) in front of and along a front coupling WG in monomodal regime forfirst (a) and second (b) mode.

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aremuch higher than along y, therefore the field can be consideredconstant along z.

In Eq. (1) ε=1−δ+ iγ is the dielectric constant of thematerial (δ and γ are optical constants), θ is the grazingincidence angle of the beam, k=2π /λ is the wave vector, and λis the wavelength of radiation. The code has been used tosimulate the response of the WG both to plane wave and toradiation from an incoherent source.

2.1. Plane wave incoming radiation

Due to the high brilliance and to the large distance betweensource and WG, synchrotron radiation is well approximated, ingeneral, by a plane wave. In the pre-reflection geometry theplane wave with grazing incidence angle θm is totally reflectedjust in front of the WG entrance, giving rise to a standing waveabove the reflecting surface with the periodicity correspondingto a resonance mode. θm corresponds to the resonance modecondition

tanksinhmd−ðm−1Þp

2

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2hm−cos2hc

psinhm

; ð3Þ

where d is the width of the vacuum gap between the twocladding layers, m is the mode number, θc≈δ1/2 is the criticalangle for total reflection, and ε=1−δ+ iγ is the dielectricconstant of the cladding material. Simulations of Fig. 2a) and b)

refer, respectively to first and second resonance from aWGwithPt cladding layers and gap of 100 nm at λ≈0.154 nm. Thespatial intensity distribution in front, inside and after the WG isshown. Fig. 3a) and b) report in more detail the comparison forthe same WG, respectively for first and second mode, betweenthe simulation result of the intensity distribution perpendicularto the WG length (bold lines) and the analytical formulation(empty circles) given by [16]

UmðyÞ ¼expðdmyÞ yb0

cosðjmyÞ þ ðdm=jmÞsinðjmyÞ 0VyVd½cosðjmdÞ þ ðdm=jmÞsinðjmdÞ�exp½−dmðy−dÞ� yNd

8<:

ð4Þwhere κm=ksinθm and δm=[(1−ε)k 2−κm2 ]1 / 2.

Fig. 4(a,b) show instead the variation of the phase of thecomplex amplitude for first (a) and second (b) modes. Changeof color from black to white corresponds to change of phasefrom 0 to π.

Fig. 5 shows in more detail the distribution of intensity forthe first mode in free space close to the WG exit. It coincideswith the diffraction pattern of a confined wave with amplitudegiven by Eq. (4) by a slit with size d corresponding to the WGgap. In the near field zone, i.e. Δxbxdif∼d2 /4λ=1.6 ·104 nm,where Δx is the distance from the WG end, the beam crosssection is practically constant and equal to the WG gap d. AtΔx=xdif it is possible to note a focusing effect, with a clear

Fig. 5. a) Intensity distribution of the first mode in the vicinity of the exit from theWG. b) Vertical intensity distribution directly at the exit of the waveguide (thickline) and at a distance from the waveguide end of x=xdif=d

2 /4λ (thin line).

Fig. 6. Waveguide in front-coupling scheme illuminated by an extended sourceat short distance from the entrance.

Fig. 7. Intensity distribution in front of and inside a WG illuminated by anextended source. Case a) is for a distance Lmin=2 cm, which corresponds to thelower limit for spatially coherent illumination for a gap d=100 nm, a source sizeS=15 μm and λ=0.154 nm (Pt cladding). In case b) at L=1 cm the illuminationis spatially incoherent under the same conditions.

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narrowing of the cross section (Fig. 5b). In the far field zonexNxdif the beam diverges with the divergence given by λ /d.

2.2. Radiation from incoherent source at short distance

As mentioned before, the WG accepts and transmits thecoherent part of radiation. WGs are used mostly withsynchrotron radiation sources, but it has been demonstratedthat they can provide useful intensity also with table-toplaboratory sources [17,18]. For the latter condition we have toinvestigate the effect of the illumination of the waveguideentrance with spatially incoherent radiation on the properties ofthe exiting beam. This may allow us to derive the geometricalcondition for optimized coupling. For excitation of independentmodes in the front coupling waveguide the next condition,based on the definition of the transverse coherence length Lc,should be fulfilled:

2dVLc ¼ k L=S ð5Þ

where L is the distance between source and waveguide, d is thethickness of the guiding layer, 2d is the spatial acceptance of thewaveguide, S is the size of the source, and λ is the wavelengthof the incoming radiation. Fig. 6 shows a schematic view of thesource and waveguide. If the equality sign holds in Eq. (5), the

distance L can be considered as Lmin, the minimum distance forcoherent irradiation. If d=100 nm, S=15 μm and λ=0.154 nmLmin=20 mm. It is evident that optics with a small entranceaperture can collect only a small part of the photons radiated bythe source. The number of photons F that is coupled by thewaveguide located at a distance L from the source can beapproximately estimated as:

Fc2U0

pdzdyL2

ð6Þ

where Φ0 is the number of photons that the source radiates insolid angle π, 2dzdy is the entrance aperture in the z and ydirections, respectively. As in the previous paragraph we willconsider planar waveguides with dz≫dy.

Fig. 8. a) and b): Intensity distribution in the vicinity of the exit of the waveguide illuminated by an incoherent source located at the distance Lmin=2 cm (a)and L=1 cm (b). c) and d): Vertical intensity distribution at a distance from the waveguide end corresponding to xdif (see text) for respectively L=2 cm andL=1 cm.

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In order to simulate the behavior of the WG, we should startwith the simulation of the incoherent source. X-ray radiation froma typical X-ray tube is emitted with a coherence time of the orderof 10−14 s. On the other hand, typical count rates do not exceed ingeneral 108 ph/s. Thus the average interval between consecutivepulses will be 10–8 s, several orders of magnitude larger than thecoherence time; under these conditions the process of detection ofphotons can be considered as a completely random process.

The average intensity of radiation is therefore the normalizedsum of the intensity of independent photons:

hIðYr Þi ¼ 1T

Z T

0EðYr; tÞE⁎ðYr; tÞdt ¼

Xn

In: ð7Þ

Computer simulation gives the same result if the source ofradiation is considered as a set of N radiators with random initialphases:

hIðYr Þi ¼ 1N

XNm¼1

EmðYr ÞE⁎mðYr Þ ¼Xn

In ð8Þ

The elementary radiator gives rise to a spherical wave withorigin positioned randomly within the source area S. The initialphase is also randomly distributed within the interval [−π, π].Fig. 7a) and b) report the intensity distribution, calculatedfollowing the above described code, in front and inside a planarWG with Pt cladding layer, for respectively a distance L=Lmin

and Lmin /2. The gap was d=100 nm, the source size S=15 μmand λ=0.154 nm. Fig. 8a) and b) show in more detail thedistribution at the exit of the WG, and Fig. 8c) and d) the crosssection profile at a distance x=xdif for respectively L=Lmin andL=Lmin /2. As can be seen, for L=Lmin the WG can provide asingle mode coherent beam, whilst for L=Lmin /2 a clear modemixing takes place.

2.3. Material and absorption considerations

In front coupling mode with air (or vacuum) gap, absorptionlosses are due only to the penetration of the tails of the intensitydistribution into the cladding material. Rigorous calculationsshould involve physical optics considerations, but simple ray-tracing can give very good approximate values for estimation ofabsorption losses.

In case of independent mode propagation, the variation ofpower along the WG length X is given by (θm≪θc) [19]

W ðX Þ ¼ W0dRNFrðhmÞcW0exp −2

g

d3=2Xdh2m

� �ð9Þ

W0 is the energy of radiation coupled by the waveguide, RFr

is the Fresnel coefficient of reflection for the vacuum-claddinglayer boundary, N=Xθm /d is the number of reflections whichthe ray undergoes, θm is the grazing incidence anglecorresponding to the resonance condition (Eq. (3)), m is the

Fig. 9. a) Variation of the intensity for the second mode at λ=0.1 nm along aWG with Pt cladding layer and vacuum gap 100 nm calculated with computercode (black line) and with Eq. (9) (closed circles). b) Intensity distribution insidethe WG for the case described by curve a).

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mode number, d is the WG gap and ε=1−δ+ iγ is the dielectricconstant of the cladding material. The same function W(X) canbe calculated using the computer code. Fig. 9 shows a very goodcorrespondence between the calculated Xabs following Eq. (9)for the second mode of the above described WG (Pt claddinglayer), and the result of computer simulation based on solutionof parabolic wave equation.

Taking into account Eq. (9) and the approximate resonancecondition 2d sinθm≈mλ, the absorption length along the opticalaxis of the WG can be written as:

Xabsc2m2

d3=2

g

d3

k2; ð10Þ

A quadratic dependence of Xabs on the mode number becomesevident. In Table 1 the absorption length is given for differentcladdingmaterials and modes 1, 2 and 3. Calculations correspondto λ=0.154 nm and thickness of guiding layer d=100 nm.

Table 1

Material Mode

1 2 3

Pt 9.6 ·106 nm 2.4 ·106 nm 1.1 ·106 nmC 2.5 ·108 nm 6.4 ·107 nm 2.8 ·107 nmSi 1.7 ·107 nm 4.3 ·106 nm 1.9 ·106 nm

3. Experimental

3.1. Fabrication of air-gap waveguide

The fabrication principle for vacuum gap planar waveguidesis quite simple; the main step processes are the followings:

1) Coat a Si wafer with a Cr layer of the desired resonatorthickness.

2) Spin an optical resist on top of it.3) Define the channel (width ≫1 μm) by optical lithography

with a very simple mask.4) Etch away the exposed resist and the Cr layer underneath it.5) Cut or cleave the wafer to the desired waveguide length.6) Etch away the remaining resist and clean the pieces

thoroughly (dust free).7) Cover the channel with another Si wafer.8) Assure constant channel thickness by pressing both wafers

against each other.

With this procedure we prepared several air-gap waveguidesthat we tested with synchrotron radiation.

Fig. 10. a) Experimental result on an air-gap waveguide measured at themicrofluorescence beam line at Elettra (Trieste). b) Angular positions ofresonant modes; closed points: experiment; open triangles: calculationconsidering a gap of 430 nm.

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3.2. Test of waveguides

The tests were carried out at the microfluorescence beamlineof the Sincrotrone Elettra in Trieste. The beam from a bendingmagnet was monochromatized by a double-crystal Si (111)monochromator, dispersing in the vertical plane, and limitedby slits at about 1.8 mm (H)×0.2 mm (V) before the WG.The WG was put with its surface in the horizontal plane on amotorized support with 6 degrees of freedom, allowing a carefulorientation of the waveguide with respect to the incident beam.A CCD camera with nominal pixel size of 6.7 μm and measuredPoint Spread Function of about 12 μm was placed downstreamof the waveguide. Fig. 10a) shows the result of the measurementson one waveguide. In the figure the abscissa is the incidenceangle and the ordinate is the exit angle with respect to the WGtangential direction. Up to 10modes are visible in the figure, withan angular distance between the modes corresponding to a gap ofabout 430 nm. The distribution of the main maxima (brighterspots) with secondary maxima (less bright spots) in between, is atypical result of the waveguide effect [10]. In Fig. 10b) we reportthe experimental angular separation (closed points) and thetheoretically expected one (open triangles) for a gap of 430 nm.The agreement is very good for all the resonances. Themeasurements were taken at energy of 7665 eV. It is to benoted that a WG with a gap of 430 nm should sustain up to abouttwenty resonances. The fact that only ten resonances are visiblecan be explained by a slight tapering, i.e. a non-perfect parallelismof the two wafers, with the larger aperture in the entrance part.Simple calculations, taking account of the WG length which isnearly 20 mm, show that a tapering angle of about 0.0044° willavoid that resonances higher than the tenth order can propagate upto the end of the WG.

Data analysis for other WGs with smaller gaps down to about40 nm is still underway.

4. Discussion and conclusions

The theoretical and experimental properties for X-ray beamsexiting from air gap waveguides have been presented for thecase of front coupling in combination with preselection of theresonant modes in a pre-reflection scheme. The simulationcode, based on solution of parabolic wave equation, is able topredict the transmission through the WG both in the case of anideal plane wave, and in the case of an incoherent source placedat short distance from the waveguide, as could be the case for alaboratory source. The simulation of the beam at the exit of thewaveguide indicates an interesting focusing effect due todiffraction at the WG boundary. Thus the minimum beam crosssection does not occur, as generally stated, at the WG exit, but ata distance given by xdif=d

2 /4λ. This can be a useful indicationfor optimization of microprobe experiments. Another usefulindication of theoretical considerations regards the attenuationalong the WG length: from Eq. (10) and Table 1 results aquadratic dependence of the attenuation length vs. the modenumber. In case of direct front coupling where different modescan simultaneously be excited, a careful choice of thewaveguide length can discriminate between different mode

propagation, allowing only the lowest order resonances to arriveat the end of the waveguide with enough intensity.

The simple procedure for fabrication of air (or vacuum) gapwaveguides presented here, and successfully tested withsynchrotron radiation measurements, gives good perspectivesfor inexpensive and efficient waveguides. The procedure cancertainly be improved, especially to guarantee a uniform gapthrough all the waveguide length. Extension to 2D waveguidesis also possible with this scheme, but particular care must bedevoted, in this case, to the quality of the walls.

Acknowledgements

Partial financial support from Italian project SPARX isgratefully acknowledged.

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