Prediction of the transmission through thin-®lm waveguides for X-ray microscopy Werner Jark* and Silvia Di Fonzo Sincrotrone Trieste, SS 14 km 163.5 in Area Science Park, I-34012 Basovizza (TS), Italy. E-mail: werner.jark@elettra.trieste.it Thin-®lm slab waveguides can con®ne incident X-ray beams in one direction in guiding layers as thin as 10 nm. Consequently they can provide attractive beam dimensions for microscopy purposes. This report presents a simple model and analytical equations for the transmission calculation, which provide results consistent with the rigorous calculations based on recursion techniques. By using these results the waveguide transmission can be compared directly with other microscopy objectives. Ideally X-ray waveguides can ®lter the spatially coherent content out of an incident radiation beam with an ef®ciency of 1. The transmissions measured for state-of-the-art one- and two-dimensional waveguides are found to correspond to experimental ef®ciencies of 0.5 for each con®nement direction. Waveguides with thinner guiding layers cannot be used ef®ciently in highly collimated beams; instead the beam divergence in unfocused beamlines at state-of-the-art synchrotron radiation sources may eventually have to be increased to the larger angular acceptance of these waveguides by use of other focusing optics. Keywords: X-ray optics; X-ray microscopy; X-ray microbeams; spatial coherence; waveguides. 1. Introduction Long undulators in third-generation synchrotron radiation sources and in future free-electron lasers (Andruszkow et al. , 2000) will concentrate the emitted radiation into very narrow emission cones. Consequently one can consider the creation of a still intense and signi®cantly demagni®ed image of the source with optical compo- nents of given focal lengths at increasingly large source distances. Ultimately this image size will be limited to the diffraction-limited spot size for a particular focusing objective. With this future perspective, discussions on the smallest possible spot size behind an optical component began. Bergemann et al. (2003) showed that an X-ray beam cannot be further focused in a tapered double-plate waveguide when the plate distance W falls below W  =2' c : 1 Here  is the wavelength and ' c is the critical angle for the material of the plates. In the hard X-ray range the refractive index n of a material is usually written as n =1    i, where  and  are small compared with unity and are related to the number of atoms per unit volume N e by (Henke et al. , 1993)   r e  2 N e f 1 2 ;   r e  2 N e f 2 2 : 2 Here f 1 and f 2 are the atomic scattering factors, which are tabulated by Henke et al. (1993) and Chantler et al. (2003), and r e = 2.818  10 15 m is the classical electron radius. The critical angle can then be obtained from ' c 2 1=2 ; 3 and, as f 1 is constant for a given material in the hard X-ray range, W is a constant in this range. For gold-coated plates, W = 8 nm. Bergemann et al. (2003) then argue that this number is the natural lower limit for the smallest spot size to which X-rays can be focused by diffraction, for example also in a Fresnel zone plate. Obviously the same limit also applies for slab waveguides with a low-Z guiding layer. Indeed a beam originating from the termination of a slab waveguide with a guiding layer thickness as small as 10.4 nm was detected by Pfeiffer (1999). Spot sizes of this order have not been detected as yet by any other means. The next-closest spot size of 20 nm is reported by Chao et al. (2003) for Fresnel zone plates operated in the soft X-ray range with a photon energy of E = 600 eV ( = 2.07 nm). In this case the observed spot size is actually the diffraction limit of Fresnel zone plates, which is given by the width of the outermost opaque zone (Schmahl & Rudolph, 1969; Attwood, 1999). In this case the reported number is not the ultimate natural limit, but is imposed by the present manufacturing technology. State- of-the-art Fresnel zone plates for hard X-rays still have larger zone widths (Di Fabrizio et al. , 1999) and thus provide larger spot sizes of the order of 80±90 nm (Yun et al., 1999). Spot sizes in the same 80± 90 nm range have also been obtained by Hignette (2003) by re¯ecting crossed KB mirror pairs (Kirkpatrick & Baez, 1948), while compound refractive lenses (CRLs) (Snigirev et al. , 1996) recently provided 210 nm in one direction (Schroer et al., 2003). Smaller numbers are reported by Pfeiffer et al. (2002) for the ®rst two-dimensional waveguides of 33 nm  66 nm, and for multibounce capillaries providing a beam diameter of 50 nm (Bilderback et al., 1994). Note that the beam compression in waveguides in one direction (Spiller & Segmu È ller, 1974; Feng et al, 1995; Lagomarsino et al. , 1996) can now routinely provide beam sizes of the order of 35 nm (Di Fonzo et al. , 2000; Jark et al., 2001), while a value of 26 nm is reported by Zwanenburg et al. (2000) for a tapered air-gap waveguide. Fresnel zone plates, mirrors and CRLs fall into the categories of diffracting, re¯ecting and refracting objectives, which are also used in the visible range for imaging and microscopy purposes. Waveguides and capillaries are beam-compressing devices, which have no equivalent for microbeam production in the visible. Now Fresnel zone plates, mirrors and CRLs have predictable properties for the component aperture and transmission (Attwood, 1999; Born & Wolf, 1980; Lengeler et al., 1999), and the experimental performance of state-of-the-art objects is always rather close to the prediction. Consequently the performance of these objectives can readily be compared. For X-ray waveguides the derivation of the parameters that can be used for this comparison has not yet been made. Consequently experimentally determined transmission ef®ciencies for state-of-the-art real objects are not yet available. This report will address both problems, and will thus create a ®rst basis that will allow comparison of the X-ray waveguide performance with that of the other objectives for both aspects, the predicted performance and the experimentally determined one. 2. Theoretical considerations This study assumes a monochromatic incident X-ray beam with a bandwidth E/E < 10 3 as is mostly used in experimental set-ups at synchrotron radiation sources. This was found to be narrower than could be accepted in X-ray waveguides (Cedola et al., 1998). Note that a similar spectral resolution is needed in order to obtain the diffraction-limited spot size behind chromatic Fresnel zone plates. On the other hand, achromatic mirrors do not need a monochromatic beam, while chromatic CRLs can focus the incident radiation with a larger bandwidth (Jark, 2004). research papers 386 # 2004 International Union of Crystallography DOI: 10.1107/S0909049504016826 J. Synchrotron Rad. (2004). 11, 386±392