Compound refractive lenses (CRL) for x rays¹ are unique devices for hard x-ray microscopy. By creating a microscopically small x-ray beam, a sample can be analyzed with high spatial resolution in a scanning microscopy manner. By this, they facilitate microdiffraction, microfluorescence, and microspectroscopy, and reveal two- or three-dimensional chemical and structural material properties on a microscopic scale.^2,3,4,5,6,7 As well, full field imaging is possible, when x-ray light which is diffracted from an object is imaged onto an x-ray detector by means of CRLs.⁸

In analogy to visible light lenses, x-ray refractive lenses introduce a phase shift to the electromagnetic waves traveling through the lensmaterial. Knowing the refractive index and the shape of a CRL, the optical characteristics can be derived in equivalence to the known laws of visible light optics.⁹

In our letter we describe the use of CRLs as optical elements for high resolution x-ray diffraction. We use the fact that a lens acts as an angular analyzer that may create a diffraction pattern of an object near its back focal plane¹⁰ that can easily be recorded with a two-dimensional detector.

We verified this concept with a diffraction experiment at the undulator beamline ID18F of the European Synchrotron Radiation Facility (ESRF). The setup is schematically shown in Fig. 1. The essential quantity for the diffraction scheme is the small source located at a large distance r from the sample. From the undulator device we obtain an effective vertical source size of 60  µm¹¹ and a horizontal source size of 800  µm, which we further reduce to 50  µm at a distance of 32  m by means of horizontal slits. From the location in the experimental hutch at r = 58  m, the effective source appears inside a solid angle = /r (the angular source size) of 1.9  µrad horizontally and 1  µrad vertically.

Figure 1.

A wavelength of 0.443  Å was selected by a double crystal monochromator, with a relative bandwidth of 1.4×10^–4. We place a parabolic CRL¹² into the beam path, centered on the optical axis, and behind, at a distance s = 1314  mm a high-resolution x-ray camera. The CRL images the source in demagnification onto the image plane, and r and s are the corresponding object and image distances. The x-ray camera is a CCD-based two-dimensional detector with a spatial resolution of 2  µm (line spread function).¹³ The CRL is fabricated of N individual lenses (N = 112) stacked behind each other on a common axis. An individual lens is made of a polycrystalline Al sheet into which two paraboloids are pressed from both sides. The radius of curvature at the apex of each parabola is R = 200  µm. The increasing absorption of x rays towards the outer torus of the lens limits its aperture to an effective size of ~200  µm in diameter (FWHM). As a test sample, we use a two-dimensional photonic crystal.¹⁴ It consists of a hexagonal whole pattern, which is fabricated into a 150  µm thick Si wafer [Fig. 2(a)]. The two vectors describing the hexagonal grid, a and b, have a length of 4.2  µm, and an included angle of 120°. The holes are almost quadratic in shape and have a size of around 2  µm. The photonic crystal forms a two-dimensional lattice, and the resulting lattice lines d₀₁ and d₁₁, expressed with Miller indices, have a distance of 3.6 and 2.1  µm, respectively. This photonic crystal is placed immediately after the lens (Fig. 1, plane A). At the image plane, we record the two-dimensional x-ray diffraction pattern of the crystal [Fig. 2(b)]. The successive orders of the {01}-family and {11}-family reflections are observable with the orientation according to the lattice spacing and the symmetry of the photonic crystal. The diffraction angles are multiples of 12.3 and 21.0  µrad for the {01}-reflections and the {11}-reflection, respectively. These values agree with the calculated diffraction angles 2= /d.

Figure 2.

In reciprocal space, the corresponding lattice vectors g₀₁ and g₁₁ we are able to resolve have the lengths 1.75×10^–3 and 2.99×10^–3  nm^–1, respectively. We may have a closer look at the resolution in reciprocal space that can be achieved with this method. Given the wave vectors of the incident and the scattered radiation, k₀ and k, the momentum transfer q is defined by q = k–k₀. The perpendicular momentum transfer resolution q of this kind of diffraction instrument can be derived in terms of resolvable detail at the detector plane. It is given by the fact that only diffraction angles 2 larger then the angular size of the demagnified source can be resolved. This leads to

where |k₀| = 2/. At a given x-ray energy, the momentum resolution depends on the angular source size and is independent of the divergence angles (or ). In our experiment, the beam illuminating the sample has an angular spread of around 150  µrad, which is large compared to the diffraction angles.

The momentum transfer resolution also reflects the largest correlation length between scattering centers to give rise to interference. This correlation length is ultimately set by the transverse coherence length of the incoming x-ray beam, which after free propagation from the source to point at distance r has the value¹⁵ = ·r/= /. From here we derive the same result for the momentum resolution as in Eq. 1. The transverse coherence lengths for our particular experiment are 23  µm horizontally and 44  µm vertically [Fig. 2(a)].

Nearly similar momentum resolution in forward scattering may be achieved at small angle x-ray scattering (SAXS) instruments at high brilliance synchrotron radiation sources.^16,17,18 Usually, the incoming beam divergence is kept small, in order to minimize the instrumental resolution function R(q), as the scattered intensity I(q) is obtained by convoluting the cross section of the sample with R(q): I(q)S(q)R(q–q)dq. Often additional collimation and angular analysis is applied by means of single crystal optics.¹⁹ These methods, however, do not account for the transverse coherence attained in the incoming beam, as was already pointed out by Sinha ²⁰

Instead, we follow the standard Fresnel–Kirchhoff diffraction theory for paraxial optics,¹⁰ for the particular case of our experimental instrument (Fig. 1). The amplitude at the detector plane P may be described by:

with

_lens and _object are complex transmission functions of the lens and the sample, respectively. The phase shift induced by the CRL and the corresponding transmission function are known:

expressed with the Cartesian coordinates x, and y, centered on the optical axis in plane A (a constant phase shift due to finite axial thickness is omitted). The quantity is the decrement of the refractive index (for Al at 28  keV, = 6.89×10^–7). The focal length f of the CRL is given as 1/f = 1/r + 1/s = 2N/R. From Eqs. 2,3 follows, that the phase-modulation induced by the lens reduces the Fresnel–Kirchhoff integral to a Fourier transformation between the angular amplitude distribution U(_x,_y) and the object transmission function _object(x,y). The intensity distribution at the detector plane is therefore the diffraction pattern of the sample.

We introduced an x-ray optical system that allows to access momentum transfer resolutions in the order of 10^–4  nm^–1. Up until now, even at the current generation synchrotron sources, such a resolution has rarely been obtained, and only with additional angular collimation by means of Bonse–Hart-type crystal collimators.¹⁹ In the future, with the straightforward fast acquisition of two-dimensional high-resolution diffraction patterns with fast cameras one would be able to study static and dynamic properties within distinct volumes of the reciprocal space, with correlation lengths of up to several tens of micrometers. At present, this is possible only with visible light scattering. Moreover, the constraint to conserve the angular divergence is actually not necessary, and one may design high-resolution diffraction instruments in a compact way (present high resolution small-angle x-ray scattering beamlines contain drift spaces of more than 10  m in length). Recent progress in the making of x-ray lenses allows access into the photon energy regime above 100  keV.²¹ Then the attenuation length for most media exceeds several hundred micrometers. This opens up applications in hard condensed matter and material science problems, which otherwise can be carried out with neutron radiation only. For example, a high-energy x-ray reflection scheme based on CRLs has recently been used for the study of buried interfaces.^22,23

The authors would like to thank R. Rueffer, A. Chumakov, O. Leupold, and J.-P. Celse from beamline ID18 for their excellent support. As well, they appreciate Viktor Kohn's critical remarks during the preparation of the manuscript.