This paper considers random balls in a D-dimensional Euclidean space whose centers are prescribed by a homogeneous Poisson point process and whose radii are prescribed by a specific power law. A random field is constructed by counting the number of covering balls at each point. Even though it is not Gaussian, this field shares the same covariance function as the fractional Brownian field (fBf). By analogy it is called fractional Poisson field (fPf). In this paper, we are mainly interested in the simulation of fPfs with index H in (0,1/2) and in the estimation of the H index. Our method is based on the analysis of structure functions. The fPf exhibits a multifractal behavior, contrary to that of the fBf which is monofractal.