  In this paper we explore the concept of depth of a ring extension when the
overall algebra factorises as a product of two subalgebras, in particular the
case of finite dimensional Hopf algebras. As a result we generalise the results
by Kadison and Young \cite{HKY} on depth of a Hopf algebra $H$ in its smash
product with a finite dimensional left $H$-module algebra $A$, $A#H$ to the
context of generalised smash products $Q^{*op}#_\psi H$ \cite{Bz1} where $Q$ is
the quotient module coalgebra associated to the extension $R\subseteq H$ of
finite dimensional Hopf algebras \cite{Ka2}\cite{HKY}\cite{H}. Moreover,
following the construction of double crossed products in \cite{Ma} and
\cite{Ma1} we use our result on factorisation algebras to get a general result
on the depth of the extension of a Hopf algebra $H$ in its Drinfel\vtick d
double $D(H)$. $\mathbf{Keywords : }$ Depth, Factorisation Algebra, Smash
Product, Drinfel\vtick d Double, Double Crossed Product, Normal Extension.
$\mathbf{Subject}$ $\mathbf{classification: }$ $20C05$, $20G05$, $16W30$,
$17B37$, $13E10$.
