Binarity has been hypothesised to play an important, if not ubiquitous, role in the formation of planetary nebulae (PNe). Yet there remains a severe paucity of known binary central stars required to test the binary hypothesis and to place strong constraints on the physics of the common-envelope (CE) phase of binary stellar evolution. Large photometric surveys offer an unrivalled opportunity to efficiently discover many binary central stars. We have combined photometry from the OGLE microlensing survey with the largest sample of PNe towards the Galactic bulge to systematically search for new binaries. A total of 21 periodic binaries were found thereby more than doubling the known sample. The orbital period distribution was found to be best described by CE population synthesis models when no correlation between primary and secondary masses is assumed for the initial mass ratio distribution. A comparison with post-CE white dwarf binaries indicates both distributions are representative of the true post-CE period distribution with most binaries exhibiting periods less than one day. An estimated close binary fraction of 12-21% is derived and is the first robust and independent validation of the prevailing 10-15% fraction estimated by Bond (2000, in Asymmetrical Planetary Nebulae II: From Origins to Microstructures, 199, 115). This suggests that binarity is not a precondition for the formation of PNe and that close binaries do not play a dominant role in the shaping of nebular morphologies. Systematic effects and biases of the survey are discussed with implications for future photometric surveys.