We developed an extreme ultraviolet (EUV) polarimeter, which employs laser-generated high-order harmonics as the light source. This relatively high-flux, directional EUV source has available wavelengths between 10 nm and 47 nm with easily rotatable linear polarization. The polarimeter has allowed us to characterize the optical constants of materials that may be useful for EUV optics. The instrument has a versatile positioning system and a spectral resolution of about 180, and we have demonstrated that reflectance as low as 0.1% can be measured repeatably at EUV wavelengths. We investigate the high harmonic source used for polarimetry measurements by documenting the spatial evolution of the generating laser in a semi-infinite helium-filled gas cell under conditions suitable for harmonic generation. The laser is observed to focus, diverge, and refocus, accompanied by a flattop beam profile and extended harmonic phase matching. We numerically simulate the propagation to investigate these experimental results. We find that harmonic energy comes from the forward portion of the laser pulse, whereas the latter portion gives rise to the incidental double laser focusing. Good phase matching for the harmonics arises in large measure from a balance between the linear phase delay of the neutral atoms and the Gouy shift, which is elongated and nearly linearized when an aperture is partially closed on the beam. We compare reflectance data taken with the polarimeter instrument with expected results from well-characterized samples and find that they agree. To improve repeatability and reduce the effects of systematic measurement errors we have measured the ratio of p- to s-polarized reflectance and shown that optical constants can be extracted from this data as efficiently as from absolute reflectance measurements. These ratio measurements allow more accurate recovery of optical constants than our absolute reflectance measurements for our well-characterized samples. We use the polarimeter instrument and the ratio reflectance technique to determine the optical constants of copper, uranium, and their natural oxides from 10-47 nm. For copper, this measurement resolves previously conflicting data sets, where disagreement on optical-constant values arises from failure to keep samples from oxidizing before measurement. Uranium has been proposed as a high-reflectance material in the EUV for several years, however difficulties with oxidation have prevented its careful characterization previous to this work. We find that measured optical constants for uranium do not agree well with previously accepted theoretical calculations.