We demonstrate, both theoretically and experimentally, that special spectral-purity-enhancing multilayer mirror systems can be designed and fabricated to substantially reduce the level of out-of-band radiation expected in an extreme ultraviolet lithographic tool. A first proof of principle of applying such spectral-purity-enhancement layers showed reduced out-of-band reflectance by a factor of five, while the in-band reflectance is only 4.5% (absolute) less than for a standard capped multilayer.