Many laser interaction models assume that incident focused laser fields are Gaussian and use either the approximate ${\mathrm{TEM}}_{00}$ series model or the exact integral Gaussian angular-spectrum solution. Many practical laser systems, however, produce flat-top transverse intensity profiles, and indeed, such profiles are often desired. Here, an exact, integral solution is derived for all of the vector components having a general flattened Gaussian profile using the angular-spectrum method. This solution includes the pure and annular Gaussian modes as special cases. The resulting integrals are solved for tight focusing conditions exactly by making use of a Fourier–Gegenbauer expansion. This technique follows closely that of Sepke and Umstadter [ Opt. Lett. 31, 1447 (2006)] but, by redefining the expansion coefficients, the simplicity of the model is greatly enhanced and the computation time reduced by roughly a factor of 2 beyond the 2 orders of magnitude improvement obtained previously. This series solution is stable at all points and converges after $S\sim 20w_0$ terms, where $w_0$ is the $1∕e$ waist normalized to the laser wavelength.