Spatial photonic Ising machines (SPIMs) are promising computation devices that can be used to find the ground states of different spin Hamiltonians and solve large-scale optimization problems. The photonic architecture leverages the matrix multiplexing ability of light to accelerate the computing of spin Hamiltonian via free space light transform. However, the intrinsic long-range nature of spatial light only allows for uncontrolled all-to-all spin interaction. Here, we explore the ability to establish arbitrary spin Hamiltonian by modulating the momentum of light, and arbitrary displacement-dependent spin interactions can be computed from different momenta of light. We formulate this principle as a generalized Plancherel theorem, which allows us to implement a SPIM with a minimal optical operation to obtain the Hamiltonian of customized spin interaction. Experimentally, we unveil the exotic magnetic phase diagram of the generalized J1-J2-J3 model, shedding light on the ab initio magnetic states of iron chalcogenides. Moreover, we observe Berezinskii-Kosterlitz-Thouless dynamics by implementing an XY model. Our study opens an avenue to controlling arbitrary spin interaction from the momentum space of light, offering a promising method for on-demand spin model simulation with a simple spatial light platform.