The Persian Empire begins with the Achaemenid dynasty who ruled an area from the Aegean to the Indus. The founder was Cyrus the Great, who called himself king of Babylon, king of Elam, and “Cyrus, king of the world, king of Anshan, son of Cambyses, king of Anshan…” [Waters, quoting a brick from Ur, from C.J. Gadd et al., Ur Excavation Texts (London, 1927), Vol. I – Plates, pl. 194 for the text and Vol. I – Texts, p. 58 for the transliteration; Schaudig, Inschriften, 549.]. He was the second Cyrus ruler in the tribal line, but the first Cyrus to rule the newly formed empire. Cyrus cites lineage back to his great-grandfather Teispes, who founded the royal line in which Cyrus was born, but he doesn’t claim lineage from an eponymous Achaemenes.
Cyrus is connected with the Achaemenids through marriage to the daughter of Pharnaspe, a member of the Median royal house, although Achaemenes isn’t mentioned until Darius’ Behistun Inscription. “Darius the king proclaims: My father is Hystaspes, the father of Hystaspes is Arsames, the father of Arsames was Ariaramnes, the father of Ariaramnes was Teispes, the father of Teispes was Achaemenes. Darius the king proclaims: For this reason we are called ‘Achaemenids’.” The genealogy Darius provides presents difficulties, and Achaemenes may or may not be a convenient fiction, meaning the whole dynasty is named for what could be Darius’ invented founder. Whether or not Cyrus was an Achaemenes by birth or only through marriage, claiming Cyrus’ Achaemenid blood gave legitimacy to Darius’ bid for the Persian throne. See: Waters, M. (2004). Cyrus and the Achaemenids. Iran, 42, 91-102. .
Because of this, some, like Jennifer Finn [FINN, J. (2011). GODS, KINGS, MEN: Trilingual Inscriptions and Symbolic Visualizations in the Achaemenid Empire. Ars Orientalis, 41, 219-275.], refer to Darius the Great as the first Achaemenid.