In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state {. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T=UH where U is a Haar unitary and U and H are V-free. (When Ker T=[0] this is equivalent to that (T, T*) is an R-diagonal pair in the sense of Nica and Speicher.) The measure + T is expressed explicitly in terms of the S-transform of the distribution + T *T of the positive operator T *T. In case T is a circular element, i.e., T=(X 1 +iX 2 )Â-2 where (X 1 , X 2 ) is a free semicircular system, then sp T=D , the closed unit disk, and + T has constant density 1Â? on D . Academic Press