Jewish Israelis emigrated in distinct waves throughout post war period. First mast influx occurred in 1948-1951 when most of Middle East Jews (Iran, Iraq, Turkey, Yemen, etc) and Eastern European Jews came (mainly Poland, Romania, and Bulgaria).
Between 1952 - 1971 most migration was from North African (Morocco and Algeria) and Eastern Europe. The only large movement of Jews from Middle East into Israel was from Egypt (I'm assuming 1956 Suez War was the cause). Itzik informed me his family move from Iraq in 1952 which was after most Jewish Iraqi's left.
The 1980's were a lull of immigration, with Ethiopian Jews beginning to come.
Finally, at the end of Cold War Former USSR (mainly Russia and Ukraine) countries allowed Jew's to emigrate to Israel. This ongoing migration contained over 1 million Jews and the dramatic increase in Israeli Population is shown below.
To get a better understanding of growth rates in Israel Log population values of the same data are shown below.
Also note that in 1981 number of Druze went up suddenly because Israel (all but in name) annexed the Golan Heights. I'm not sure why the Christian population increased in mid 1990's and decreased...