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The structure of the field transmitted through a metal plate perforated by subwavelength holes and illuminated by a beam is analyzed. Starting from a spectral representation of the transmitted field involving the plate's plane wave transmission coefficient and the incident beam's angular Fourier transform, the transmitted field is shown to asymptotically comprise saddle point, (transmission coefficient) pole, and (transmission coefficient) branch cut contributions. For incident beams with narrow angular support only the saddle point contribution is significant. The transmitted field is an attenuated, laterally, and angularly shifted replica of the incident beam. For beams with wide angular support all three saddle point, pole, and branch cut contributions can be significant. The saddle point contribution dominates the transmitted field far from the plate as it gives rise to a geometrical ray. The pole and branch point contributions are observed nearby the structure as they give rise to leaky and a new type of lateral waves, respectively. These contributions relate to so-called resonant and Rayleigh Wood anomalies; the former play a key role in the buildup of enhanced transmission phenomena. The influence of the plate thickness and the proximity of the plane wave transmission coefficient poles and branch points on the structure of the transmitted field is elucidated both for beams with narrow and wide angular support. This study is relevant not only for characterizing transmission properties of hole perforated plates but also those of many other gratings. The investigated structure and phenomena can be applied and exploited in the construction of novel probes, antennas, and microwave and optical filters.