Fundamental properties for the coefficients of a second-order finite difference approximation of the fractional Laplacian in d≥2 dimensions are derived in this paper. The obtained decay rate of the coefficients implies that the coefficients (with no closed-form expression) can be approximated via d-dimensional inverse fast Fourier transform of size K per dimension with accuracy O(K), where α∈(0,2) is the order of the fractional Laplacian. For solving fractional partial differential equations on regular grids, the coefficient matrix is a d-level Toeplitz matrix that can be preconditioned by the d-level {ω}-circulant matrix. Here, a spectral analysis of the difference matrix is derived. The purpose of this work is also to justify some observations presented by Hao et al. (2021). Numerical experiments in two-dimension and three-dimension illustrate that {ω}-circulant preconditioner has better performance over T. Chan's circulant preconditioner.