Old English beam originally "living tree," but by late 10c. also "rafter, post, ship's timber," from Proto-Germanic *baumaz "tree" (source also of Old Frisian bam "tree, gallows, beam," Middle Dutch boom, Old High German boum, German Baum "tree," and perhaps also (with unexplained sound changes) Old Norse baðmr, Gothic bagms), which is of uncertain etymology (according to Boutkan probably a substrate word). The shift from *-au- to -ea- is regular in Old English.
Meaning "ray of light" developed in Old English, probably because beam was used by Bede to render Latin columna (lucis), the Biblical "pillar of fire." Meaning "directed flow of radiation" is from 1906. To be on the beam "going in the right direction" (1941) originally was an aviator's term for "to follow the course indicated by a radio beam." Nautical sense of "one of the horizontal transverse timbers holding a ship together" is from early 13c., hence "greatest breadth of a ship," and slang broad in the beam, by 1894 of ships, of persons, "wide-hipped," by 1938.
"emit rays of light," c. 1400, from beam (n.) in the "ray of light" sense. Sense of "shine radiantly" is from 1630s; that of "smile radiantly" is from 1804; that of "to direct radio transmissions" is from 1927. Related: Beamed; beaming.