Dispersion properties of nano- and micropores in track membranes

A theoretical explanation of the constancy of the phase and group velocities of waves in through pores of polymer track membranes in the hard X-ray range, which was discovered in earlier works and not satisfactory explained by the theory of waveguide modes, is proposed. The developed X-ray propagation theory is based on the analytical solution of the parabolic equation in a waveguide by an integral transformation method. Using the $3D$ parabolic equation and a finite-difference method a numeric simulation of the X-ray propagation at two soft X-ray wavelengths was conducted, which demonstrated that the constancy of the phase and group velocities in through pores of track membranes holds in this case as well but with a lower precision. However, the product of the phase and group velocities is not equal to the square of the vacuum light speed anymore. It was also shown that constancy of the wave velocities in a pore breaks down when several propagating waveguides modes appear in it, which leads to oscillations of the wave velocities due to modes' interference.