We study long Josephson junctions with the critical current density alternating along the junction. New equilibrium states, which we call the field synchronized or FS states, are shown to exist if the applied field is from narrow intervals centered a
round equidistant series of resonant fields, $H_m$. The values of $H_m$ are much higher than the flux penetration field, $H_s$. The flux per period of the alternating critical current density, $phi_i$, is fixed for each of the FS states. In the $m$-th FS state the value of $phi_i$ is equal to an integer amount of flux quanta, $phi_i =mphi_0$. Two types of single Josephson vortices carrying fluxes $phi_0$ or/and $phi_0/2$ can exist in the FS states. Specific stepwise resonances in the current-voltage characteristics are caused by periodic motion of these vortices between the edges of the junction.
We treat theoretically Shapiro steps in tunnel Josephson junctions with spatially alternating critical current density. Explicit analytical formulas for the width of the first integer (normal) and half-integer (anomalous) Shapiro steps are derived fo
r short junctions. We develop coarse-graining approach, which describes Shapiro steps in the voltage-current curves of the asymmetric grain boundaries in YBCO thin films and different superconductor-ferromagnet-superconductor Josephson-type heterostructures.
