An iterative method with bandwidth constraint strategy is proposed to design phase holograms for high-quality speckle-free optical reconstruction. The bandwidth properties of the reconstructed field are analyzed theoretically based on the sampling theory, which helps in properly allocating the sampling resources for efficiently describing the speckles and artifacts in the reconstructed field. Iterative calculation with bandwidth constraint strategy of the reconstructed field and quadratic initial phase can optimize the phase hologram without stagnation problem, which provides effective controls of the reconstructed intensity fluctuations and helps to suppress the speckles and artifacts. Numerical and optical experiments have been performed to validate the proposed method can achieve excellent image fidelity.