The invention relates to an illumination system(100), a high-pressure discharge lamp (90) and an image projection system. The illumination system comprises a high-pressure discharge lamp and aback reflector (30) reflecting the light emitted by the high-pressure discharge lamp towards a light exit window (50). The back reflector comprises an optical axis (55). The high-pressure discharge lamp comprises a discharge vessel (90) comprising two electrodes (98, 99) between which, during operation, a discharge arc is produced. The discharge vessel comprises a first part (10) arranged at least partially between the discharge arc and the back reflector, and a second part (20) arranged at least partially between the discharge arc and the light exit window. The second part has a different shape compared to the first part, thereby forming a refractive element in the second part for reducing an angular distribution at the light exit window of the light emitted from the discharge arc and refracted By the second part. The measures according to the invention have the effect that,due to the reduction of the angular distribution at the light exit window,the specific refractive property of the second part improves the efficiency of the illumination system.