Fresnel zone plates consisting of alternating transmissive and opaque circular rings can be used to focus X-rays. The spatial resolution that can be achieved with these devices is of the order of the width of the outermost zone and is therefore limited by the smallest structure (20-40 nm) that can be fabricated by lithography today. Here we show that a large number of pinholes distributed appropriately over the Fresnel zones make it possible to focus soft X-rays to spot sizes smaller than the diameter of the smallest pinhole. In addition, higher orders of diffraction and secondary maxima can be suppressed by several orders of magnitude. In combination with the next generation of synchrotron light sources (free-electron lasers) these 'photon sieves' offer new opportunities for high-resolution X-ray microscopy and spectroscopy in physical and life sciences.

An advanced type of diffractive optical element is presented that combines the concept of the photon sieve with an offaxis, off-normal incidence reflection geometry. Compared to transmission optical elements, the signal-to-background ratio is significantly increased by separating the first from other diffraction orders without drastically reducing the size of the smallest diffractive element. The reflection photon sieve produces sharp foci at maximum contrast and offers the advantages of effective heat dissipation and a large working space above the focal plane. Experimental results for a device working at a photon energy of 100 eV are presented and compared to theory. © 2011 Optical Society of America OCIS codes: 230.1950, 340.7480. The parallel progress in nanofabrication and in the generation of highly brilliant synchrotron radiation has significantly advanced many spectroscopy, scattering, and imaging techniques by pushing them to nanometer-scale spatial resolution over a large part of the electromagnetic spectrum [1,2]. In the extreme UV to soft x-ray regimes, where strong absorption rules out the use of refractive lenses [3], transmission Fresnel zone plates [3-6] currently provide the best spatial resolution. Yet these diffractive optical elements commonly suffer from high background signals due to overlapping diffraction orders, strong secondary intensity maxima resulting from finite size effects, and limited working space between apertures and samples. Here we introduce a novel type of diffraction optics-a reflection photon sieve-that overcomes these limitations by using properly placed nanomirrors or nanoabsorbers [7] to obtain nanometer-scale foci of electromagnetic radiation in an off-axis, off-normal incidence reflection geometry. In addition to excellent focusing properties and good sample access, the design of the device allows for effective heat dissipation. The diffractive imaging of a pointlike source by a reflection photon sieve is schematically illustrated in Fig. 1. When monochromatic light illuminates the optical element under an angle of incidence α, the wave is both directly reflected (zero-order diffraction) and focused into different spots under different diffraction angles β (firstand higher-order diffraction). The curved dark areas on the device indicate the wave paths that contribute constructively to the focal spots. In the actual device these areas would be highly reflective, while the surrounding area would be made of a highly absorbing material. In coordinates of r and s, the condition for constructive interference is given bywhere p and q are the object and image distance, m is the diffraction order, and nλ the optical retardation between different diffraction areas (λ, wavelength; n, integer). Their boundaries can be calculated by setting the optical retardation to nλ AE λ 4 . The optical element described so far is essentially a reflection zone plate [8][9][10]. It becomes a reflection photon sieve by centering appropriately shaped nanomirrors, e.g., cylinders, on the dark...