From simple waveguide theory it is shown that the characteristic wave impedance of air-filled (or evacuated) and dielectric-filled waveguides of equal cross-section propagating E-modes can be matched for any free-space wavelength and relative permittivity of the dielectric. For low-permittivity dielectrics, reflectionless transmission may be obtained for a bandwidth of 20%. It is possible to design a waveguide window on these lines, the thickness of which—unlike that of the resonant-iris window—is not critical. The simplicity of the structure appears suitable for high-power microwave valves. A universal design-chart is derived from which the window diameter for the matched condition may be read off directly for any free-space wavelength and relative permittivity of the dielectric. A window design with broadband mode-transformers is described.