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In the present chapter, the detailed structures of the field distributions in arbitrary X-ray femtosecond pulses passing through straight and tapered capillary waveguides are determined using scalar diffraction theory and the method of images. At the guide entrance the femtosecond pulses are represented in the form of Fourier integrals. The numerical analysis showed that the ultrashort X-ray wave propagates down the guide as the superposition of many pulses (transient modes) that diffract in the off-axis direction and interfere with each other. In opposition to the eigenmode theory of waveguides (solving a boundary condition problem), the field at the guide entrance can satisfy neither the guide wave equation nor the boundary conditions. That gives the possibility to consider not only the mode-matched waves, but also the pulses having more complicated properties at the guide entrance. For the straight multimode guide the usual result is recovered, namely that the spatial profile of the wave is undistorted along its propagation path so long as the input pulse is mode matched. For non mode-matched waves or tapered guides, the transverse intensity distribution of the pulse depends on the propagation path and on the parameters of the guide and the input wave. In addition, the computations show that changes of the temporal profile and duration of the pulse during propagation are negligibly small for a wide range of guide parameters, at least down to pulse widths of few femtoseconds. Finally, for tapered guides having very small guide exits (a few tens of nanometres), a very interesting behaviour of the femtosecond pulses was found; the numerical analysis showed that the output pulses have uniform transverse intensity distribution with very little dependence on the parameters of the guide and the input wave.

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