The arcuate fasciculus (AF) is a fiber pathway in the human brain relevant for language processes and has recently been characterized by means of diffusion tensor tractography. The observations made concerning the left and right hemisphere AF include a characterization of the trajectories and quantification of physical properties such as fractional anisotropy, DTI-based fiber density and volume. However, these observations were based on the diffusion of water, which is not particular to either the intra- or extra-axonal compartments, and thus its usefulness for tissue characterization is limited. If the diffusion properties and in turn the geometric properties of only one tissue compartment can be isolated and characterized, a better microstructural characterization of AF is possible. In this study, water-based diffusion tensor probabilistic mapping was first implemented to segment the AF. Subsequently, diffusion tensor spectroscopic measurements of N-acetyl aspartate (NAA) were performed to measure the intra-axonal specific diffusion in left and right AF. Diffusion properties of NAA, which solely reflect the intra-axonal space, indicated possible leftward asymmetry in axonal diameter, where those of water, which are not compartment-specific, showed laterality to a lesser extent.