I stared at this in awe and admiration at the Natural Art History Museum in LA last week and I cannot find any information about who drew it, when it was drawn, or even whether it's Harun Al Rashid. Also, does anyone have sources to a higher resolution picture?
It looks Safavid to me, compare it to the portraits of Reza Abbasi: https://www.google.co.uk/search?q=reza+abbasi&rlz=1C5CHFA_enJO569GB570&espv=210&es_sm=91&source=lnms&tbm=isch&sa=X&ei=Go44U7L6OYWBhAeDioCIDQ&ved=0CAgQ_AUoAQ&biw=1276&bih=584
So 16th or 17th century. Unfortunately I can't find any references or better scans of it either. The only reference to it being Harun al-Rashid seems to be from wikipedia, and it's citationless.
Hopefully this might point someone with more time/knowledge in the right direction?
Do you mean you saw it at LACMA? You can email their curators for Islamic Art directly to ask (contact info at the bottom of this page).