A comprehensive study has been conducted on ultra-broadband optically pumped quantum dot (QD) reflective semiconductor optical amplifiers (QD-RSOAs). Furthermore, little work has been done on broadband QD-RSOAs with an optical pump. About 1 $\mu$m optical bandwidth, spanning 800 nm up to 1800 nm, is supported for the suggested device by superimposing nine groups of QDs. It has been shown that the device can be engineered to amplify a selected window or a group of desired windows. Moreover, the operation of the device has been thoroughly investigated by solving the coupled differential rate and signal propagation equations. A numerical algorithm has been suggested to solve these equations. As far as we are concerned, a broadband optically pumped QD-RSOA that can operate as a filter has been introduced.