Abstract
Achromatic focusing systems for hard X-rays are examined which consist of a refractive lens paired with a diffractive lens. Compared with previous analyses, we take into account the behaviour of thick refractive lenses, such as compound refractive lenses and waveguide gradient index refractive lenses, in which both the focal length and the position of the principal planes vary with wavelength. Achromatic systems formed by the combination of such a thick refractive lens with a multilayer Laue lens are found that can operate at a focusing resolution of about 3 nm, over a relative bandwidth of about 1%. With the appropriate distance between the refractive and diffractive lenses, apochromatic systems can also be found, which operate over relative bandwidth greater than 10%. These systems can be used to focus short pulses without distorting them in time by more than several attoseconds. Such systems are suitable for high-flux scanning microscopy and for creating high intensities from attosecond X-ray pulses.
Highlights
Diffractive optics such as multilayer Laue lenses (MLLs) and sputter-sliced zone plates are currently under development as a means to achieve imaging at 1 nm resolution for X-ray wavelengths of 0.1 nm or less [1,2]
Given that the focal length of a refractive lens is proportional to 1/(n − 1) and inversely proportional to the square of the X-ray wavelength, the dispersion of a refractive lens in the X-ray regime is given by V = −2, which is twice the dispersion experienced by diffractive lenses
While it is generally well appreciated that diffractive lenses such as zone plates and MLLs exhibit a strong dependence of focal length on wavelength and, relatedly, an increase in the duration of short pulses due to the differences in path lengths of rays propagating from the lens to the focus, it is perhaps not as well known that such effects are even greater in refractive lenses
Summary
Diffractive optics such as multilayer Laue lenses (MLLs) and sputter-sliced zone plates are currently under development as a means to achieve imaging at 1 nm resolution for X-ray wavelengths of 0.1 nm or less [1,2]. An achromat design nulls the linear term in a series expansion of the focal length of the optical system as a function of the relative wavelength deviation λ/λ, leaving a quadratic dependence so that an equal focal length can be obtained for two distinct wavelengths This typically provides a bandwidth of 1% or more [11,12,13], even with diffractive lenses with tens of thousands of layers. While the focusing achromat design requires the refractive lens to be diverging (that is, have a negative focal length), both lenses must have comparable power.. As with the doublet lens (for which d = 0), achromatic focusing conditions can only be found for separated lenses when the refractive lens has a negative focal length and the diffractive lens is positive. Calculation of propagation times through the lens systems can serve as an independent check of the derivations of the achromatic conditions and are given in appendix A
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