This paper analyzes the impact of directional antennas in improving the transmission capacity, defined as the maximum allowable spatial node density of successful transmissions multiplied by their data rate with a given outage constraint, in wireless networks. We consider the case where the gain G_m for the mainlobe of beamwidth can scale at an arbitrarily large rate. Under the beamwidth scaling model, the transmission capacity is analyzed for all path-loss attenuation regimes for the following two network configurations. In dense networks, in which the spatial node density increases with the antenna gain G_m, the transmission capacity scales as G_m^4/α, where α denotes the path-loss exponent. On the other hand, in extended networks of fixed node density, the transmission capacity scales logarithmically in G_m. For comparison, we also show an ideal antenna model where there is no sidelobe beam. In addition, computer simulations are performed, which show trends consistent with our analytical behaviors. Our analysis sheds light on a new understanding of the fundamental limit of outage-constrained ad hoc networks operating in the directional mode.