The time-dependent characteristic function for the vibratlonal quanta of two ions separately trapped in two caacaded optical eavities is obtained analytieally beyond the bad cavity linit, in which tfie motional mode of the ion in the first cavfiy is coupled to lhe cavity, field via an effective parametrc interacion and that of the ion in the second cavity" via a liaear-mixing interaction as proposed by Peng and Parkias Pfiys. Rev. A 65, 062323 (2002l]. It is fond, that the two ions develop in a two-mode mixed Gaussian entangle state if there exists a unidirecfional coupling from the first cavitv to the seecond one.