= Solution
For a real determinant-one map, bounded generic repeated transport requires eigenvalues $e^{\pm i\mu}$, equivalently $|\operatorname{tr}M|<2$. Here both planes have $\operatorname{tr}M=2-L_1L_2/f^2$, so
$$
\boxed{0<\frac{L_1L_2}{f^2}<4,\qquad \cos\mu=1-\frac{L_1L_2}{2f^2}.}
$$
For positive drift lengths and $f>0$, this means $f>\sqrt{L_1L_2}/2$. The equalities are marginal and generally produce a Jordan-block growth rather than stable motion. Moving the reference point changes the map by a similarity transformation, $M\to PMP^{-1}$, so the eigenvalues and full-cell phase advance are unchanged.
Write the stable map using the <Courant-Snyder parameters>, also called Twiss parameters:
$$
M=\begin{pmatrix}\cos\mu+\alpha\sin\mu&\beta\sin\mu\\-\gamma_T\sin\mu&\cos\mu-\alpha\sin\mu\end{pmatrix},\qquad \beta\gamma_T-\alpha^2=1.
$$
At the chosen reference point $\beta=M_{12}/\sin\mu$, $\alpha=(M_{11}-M_{22})/(2\sin\mu)$, and $\gamma_T=-M_{21}/\sin\mu$. Repeated traversal gives
$$
\boxed{M^n=\begin{pmatrix}\cos n\mu+\alpha\sin n\mu&\beta\sin n\mu\\-\gamma_T\sin n\mu&\cos n\mu-\alpha\sin n\mu\end{pmatrix}.}
$$
If the full ring contains $N_c$ identical cells, its one-turn phase is $N_c\mu$ modulo $2\pi$; $n$ ring turns use $nN_c\mu$. The displayed $M^n$ is for $n$ repetitions of the specified cell.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012-s19-cell-stability.png]
{title=Stability and phase advance of the equal-strength quadrupole cell}
{description=Both transverse planes have trace 2 minus L_1 L_2/f squared. The strict stable interval excludes the two marginal boundaries.}
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