What was the secret to the growth of the Persian empire?

Answer 1

It harnessed the resources of the countries it progressively conquered, and used them for further conquests.

Examples:

Army - the Persians first conquered the Medes, and then absorbed its army, and the combined force then took over the Babylonian empire, and then these combined armies moved further on. Against the southern Greeks at the battle of Plataia, a third of the Persian army was from Greek cities in central Greece.

Navy - Persia did not have a navy, it gained one by levying contingents from Egypt, Phoenicia and the Greek cities of Asia Minor.