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It has been possible to obtain mirrors of very high reflectivity by following the simple four-step procedure described herein. The key to success is the ability to measure the scattering and other losses of the substrates and dielectric coatings to ensure that the specifications are being met. These measurements are especially critical in the important cleaning process. The cavity-attenuated phase-shift (CAPS) method is ideally suited for performing these important measurements, permitting us to obtain mirrors with reflectivities of R = 0.99975 ± 0.00005.