Dose distributions distorted by a periodic structure, such as a ridge filter, are analytically investigated. Based on the beam optics, the fluence distributions of scanned beams passing through the ridge filter are traced. It is shown that the periodic lateral dose distribution blurred by multiple Coulomb scattering can be expressed by a sum of cosine functions through Fourier transform. The result shows that the dose homogeneity decreases exponentially as the period of the structure becomes longer. This analysis is applied to the example case of a mini-ridge filter. The mini-ridge filter is designed to broaden sharp Bragg peaks for an energy-stacking irradiation method. The dose distributions depend on the period of the ridge filter structure and the angular straggling at the ridge filter position. Several cases are prepared where the period and angular straggling are supposed to be probable values. In these cases, the lateral distributions obtained by the analytical method are compared to Monte Carlo simulation results. Both distributions show good agreement with each other within 1%, which means that this analysis allows estimation of the dose distribution downstream of the ridge filter quantitatively. The appropriate period of grooves and scatterer width can be determined which ensures sufficient homogeneity.