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Use transverse coordinates $(x,x^{\prime })$ and a signed focusing strength $K$ in Hill's equation, $x^{\prime \prime }+Kx=0$. A drift of length $\ell$ has beam transfer matrixFor a uniform focusing quadrupole magnet with $K={\kappa }^2>0$, and a defocusing one with $K=-{\kappa }^2$,The focusing plane has oscillatory trajectories, while the defocusing plane has hyperbolic growth and decay. A magnetic quadrupole focuses one transverse plane and defocuses the other; reversing its gradient interchanges their roles.

If $|K|{\ell }^2\ll 1$ and the element's internal drift can be neglected or assigned separately, its kick may be concentrated at one position. With $1/f=|K|\ell$,Both kicks preserve position and change the slope linearly with displacement. All these ideal uncoupled maps have determinant one.

Use transverse coordinates $(x,x^{\prime })$ and define a positive signed focal length as focusing in that plane. A thin quadrupole magnet of length $\ell$ and signed gradient $G$ has integrated strength $1/f=|e|G\ell /p_0$; the electron charge and chosen field orientation are absorbed into the sign of $G$. Its beam transfer matrix isFor total electron energy $E_0$, $p_0=\sqrt{E_0^2-m_e^2{c}^4}/c$, so $f\simeq E_0/(|e|cG\ell )$ in the ultrarelativistic limit. The other transverse plane has the opposite signed focal length. A quadrupole magnet therefore focuses one plane while defocusing the other; the longitudinal magnetic field component is negligible in this paraxial idealization.

Let $D(d)=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)$ be the drift beam transfer matrix. Start immediately before the first lens and end immediately before its counterpart in the next FODO cell. Matrices act on column vectors from right to left. With $A=1/f_1$ and $B=1/f_2$, where both focal lengths are signed in the chosen plane,The two drifts account for both gaps. Direct multiplication gives $\det M=1$ andFor the equal-strength focusing-defocusing case, $f_1=f>0$ and $f_2=-f$:Choosing a different start within the cell gives a conjugate matrix, with the same trace and eigenvalues. If $f_2$ is specified as a positive magnitude of a defocusing lens, its signed value in these formulas is $-f_2$.