Incoherent x rays in the wavelength interval from approximately 0.5–2 Å have been focused with refractive lenses. A single lens would have a long focal length because the refractive index of any material is close to unity; but with a stack of N lens elements the focal length is reduced by the factor N, and such a lens is termed a compound refractive lens (CRL). Misalignment of the parabolic lens elements does not alter the focusing properties and results in only a small reduction in transmission. Based on the principle of spontaneous emission amplification in a FEL wiggler, coherent x-ray sources are being developed with wavelengths of 1–1.5 Å and source diameters of 50–80 µm; and the CRL can be used to provide a small, intense image. Chromatic aberration increases the image size by an amount comparable with the diffraction-limited size, and so chromatic correction is important. Pulse broadening through the lens that is due to material dispersion is negligible. The performance of a CRL used in conjunction with a coherent source is analyzed by means of the Kirchhoff integral. For typical parameters, intensity gain is 105–106, where gain is defined as the intensity ratio in an image plane with and without the lens in place. (There may be some confusion concerning the usage of the word intensity. As employed in this manuscript, intensity, also called irradiance, refers to power per unit area. This is a commonly accepted usage for intensity, although there are places in the literature where the term radiant incidence is reserved for this definition and intensity refers to power per unit solid angle.) The image intensity is maximized when the CRL is placed 100–200 m from the source, and the diameter of the diffraction-limited spot is approximately 0.12 µm.
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