In　this　article,　we　consider　a　new　robust　estimation　procedure　for　functional　linear　models　with　both　slope　function　and　functional　predictor　approximated　by　functional　principal　component　basis　functions.　A　modified　Huber＇s　function　with　tail　function　substituted　by　the　exponential　squared　loss　(ESL)　is　applied　to　the　estimation　procedure　for　achieving　robustness　against　outliers.　The　tuning　parameters　of　the　new　estimation　method　are　data-driven,　which　enables　us　to　reach　better　robustness　and　efficiency　than　other　robust　methods　in　the　presence　of　outliers　or　heavy-tailed　error　distribution.　We　will　show　that　the　resulting　estimator　for　the　slope　function　achieves　the　optimal　convergence　rate　as　the　least-squares　estimator　does　in　the　classical　functional　linear　regression.　The　convergence　rate　of　the　prediction　in　terms　of　conditional　mean　squared　prediction　error　is　also　established.　The　proposed　method　is　illustrated　with　simulation　studies　and　a　real　data　example.