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We have used a Monte Carlo simulation of a polar cap (PC) model of γ-ray pulsars to estimate light curves and phase-resolved spectra for sources whose rotational and magnetic axes are oriented so that only one of the magnetic poles produces emission directed at Earth. In this single polar cap (SPC) scenario, even sources whose light curves have two distinct peaks (Crab, Vela, Geminga, and PSR B1951+32) are due to emission concentrated near the rim of a single PC. If the inclination α is comparable to the half-width of the PC γ-beam, α ∼θ<SUB>b</SUB>, the peak-to-peak phase separation can have the large values (∼0.4-0.5) observed from these sources. In the model presented here we attribute the observed interpeak emission to pair cascades above the PC interior. Our simulation assumes the physics of conventional PC models, in which the γ-rays are due to photon-pair cascades initiated by curvature radiation from the acceleration of electrons above the PCs. In this work we assume that the acceleration occurs over a finite region that may extend up to several radii above the neutron star surface. We find that the combined effects of moderately enlarged PC dimensions and extended acceleration zones resolve a major difficulty with earlier PC models, namely their small beamwidths (and hence small detection probabilities). Our best fits to the observed light curves are obtained from models in which the accelerated electrons have a uniform surface density over the PC interior and a sharp density increase of ∼3-5 near the rim. Based on these assumptions, our model results for Vela reproduce key features of the light curves and phase-resolved spectra. We also consider the compatibility of the model with observations at X-ray, radio, and optical wavelengths.