  We consider a Josephson junction with ${\rm F_1 F_2 F_3}$ ferromagnetic
trilayers in the ballistic regime, where the magnetization in each ferromagnet
${\rm F}_i (i=1,2,3)$, can have arbitrary orientations and magnetization
strengths. The trilayers are sandwiched between two $s$-wave superconductors
with a macroscopic phase difference $\Delta \varphi$. A broad range of
magnetization strengths of the central $\rm F_2$ layer are considered, from an
unpolarized normal metal (N) to a half-metallic phase, supporting only one spin
species. Our results reveal that when the magnetization configuration in ${\rm
F_1 F_2 F_3}$ has three orthogonal components, a supercurrent can flow at
$\Delta \varphi=0$, and a strong second harmonic in the current-phase relation
appears. Upon increasing the magnetization strength in the central ferromagnet
layer up to the half-metallic limit, the self-biased current and second
harmonic component become dramatically enhanced, and the critical supercurrent
reaches its maximum value. The higher harmonics in the current-phase relations
can be controlled by the relative magnetization orientations, with negligible
current damping compared to the corresponding ${\rm F_1 N F_3}$ counterparts.
For a broad range of exchange field strengths in the central ferromagnet ${\rm
F}_2$, the ground state of the system can be tuned to an arbitrary phase
difference $\varphi_0$ by rotating the magnetization in the outer ferromagnet
$\rm F_3$. For intermediate exchange field strengths in ${\rm F}_2$, a
$\varphi_0$ state can arise that creates a superconducting diode effect,
whereby $\Delta\varphi$ can be tuned to create a one-way dissipationless
current flow. The density of states demonstrates the emergence of zero energy
peaks for the mutually orthogonal magnetization configurations, which is
strongest in the half-metallic phase.
