Abstract: An X-ray and Gamma ray waveguide, cavity, and method for directing electromagnetic radiation of the X-ray, Gamma ray, and extreme ultraviolet wavelengths. A hollow fiber is used as the waveguide and is manufactured from a material having an index of refraction less than unity for these wavelengths. The internal diameter of the hollow fiber waveguide and the radius of curvature for the waveguide are selectively predetermined in light of the wavelength of the transmitted radiation to minimize losses. The electromagnetic radiation is obtained from any suitable source and upon introduction into the waveguide is transmitted along a curvilinear path. The waveguide may be formed as a closed loop to create a cavity or may be used to direct the electromagnetic radiation to a utilization site.